nLab
geometry of physics

This page is growing incrementally as a series of lecture series proceeds.

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A set of lecture notes on differential geometry and theoretical fundamental physics, combining an introduction to traditional notions with an exposition of their formulation and refinement by higher geometry and extended prequantum field theory. With an eye towards Hilbert's sixth problem.

Divided into two parts:

• Part I) Geometry

• Part II) Physics

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Table

of chapters

• About this text

• References

Part I) Geometry

1. Coordinate systems

2. Smooth spaces

3. Differential forms

4. Differentiation

5. Smooth homotopy types

6. Groups

7. Principal bundles

8. Structure sheaves

9. Manifolds and Orbifolds

10. Reduction of structure groups

11. Representations and Associated bundles

12. Modules

13. Flat connections

14. de Rham coefficients

15. Maurer-Cartan forms

16. Principal connections

17. Characteristic classes

18. Integration

19. Super-geometry

Part II) Physics

1. Physics in Higher Geometry: Motivation and Survey

2. Fields

3. Lagrangians and Action functionals

4. Equations of motion

5. Local (topological) prequantum field theory

6. Prequantum Gauge theory and Gravity

7. Quantum mechanics

8. Geometric quantization

About this text

This page is going to contain an introduction to aspects of differential geometry and their application in fundamental physics: the gauge theory appearing in the standard model of particle physics and the Riemannian geometry appearing in the standard model of cosmology, as well as the symplectic geometry appearing in the quantization of both.

Scope and perspective

The intended topic scope and readership of the first layer of this page – the Model Layer – is much like that of the book (Frankel), only that here we make use of a more modern and more transparent conceptual toolbox. We also discuss in two other layers, the Semantic Layer and the Syntactic Layer deeper mechanisms at work in the background.

Notably, where traditional expositions of differential geometry proceed by generalizing the geometry of abstract coordinate systems ${ℝ}^n$ to smooth manifolds, here we instead begin by generalizing, in Smooth spaces – Model Layer, coordinate systems right away to smooth spaces, which happens to be both more expressive as well as actually much easier. In parallel (and to be read independently or not at all) we discuss in Smooth spaces – Semantic Layer how this means that we are working in the sheaf topos over abstract coordinate systems. Smooth manifolds are then introduced later as an intermediate notion, together with that of diffeological spaces. (Many of the constructions in differential geometry applied in physics do not actually need the notion of a smooth manifold, and, more importantly, for many notions in modern theoretical physics smooth manifolds are not actually sufficiently general.)

In fact we introduce smooth manifolds only after we introduce smooth groupoids (below in Smooth homotopy type - Model Layer - Smooth groupoids), which are differential geometric structures that are still simpler than smooth manifolds, and of course even more expressive than smooth spaces. Moreover, smooth groupoids are at the very heart of the geometry of physics: modern fundamental physics is all based on the “gauge principle” and in Model Layer – Gauge transformations in electromagnetism we explain how, mathematically, this is essentially nothing but the theory of smooth groupoids. As further background information we discuss in Smooth homotopy types - Semantic Layer how this means that we are working in a higher topos over abstract coordinate systems, and in Smooth homotopy type - Syntactic Layer how this means that we are reasoning about physics using the natural deduction rules of homotopy type theory.

From this setup then naturally flow all the many structures and phenomena seen in the geometry of physics:

Layers of exposition

We discuss each topic below in three stages, in three layers.

1. The first layer, called the Model Layer, deals with concrete explicit constructions as familiar from traditional textbooks on differential geometry and physics. This layer is supposed to be readable and useful all in itself and the reader who feels that this is all he or she wants to see can stick to this and ignore the other layers. In particular, while the Model Layer does invoke the basic notion of a category and of a functor – which are as simple as the notions of group or algebra –, it does not use any actual category theory.

1. The second layer, called the Semantic Layer, makes explicit the (higher) category theory and (higher) topos theory at work in the background. This puts the concrete constructions of the Model Layer into a more general context and helps to see certain organizational patterns that underlie the seemingly different phenomena. It provides some powerful theorems which the Model Layer is secretly benefitting from. For instance this layer gives a systematic rule for generalizing everything at the beginning Model Layers from ordinary differential geometry to what is called supergeometry, which is the context in which fermionic particles are formalized: the matter constituents of the observable universe.

1. The third layer, called the Syntactic Layer, makes explicit the expression of these phenomena in the formal internal language of the topos of smooth spaces – which is dependent type theory – and of the higher topos of smooth higher groupoids – which is homotopy type theory. This makes more transparent various constructions in (higher) topos theory used in Semantic Layer, and in fact it provides a natural construction principle for objects in a (higher) topos that model some intended meaning – which is precisely what mathematical physics is all about. This is meant for readers who enjoy seeing fundamental physics naturally rooted in genuinely fundamental mathematics, in natural deduction from practical foundations, as it were. Everybody else should ignore this.

The three layers

• Model Layer – concrete particular: models

• Semantic Layer – concrete general: categorical semantics in higher topos theory

• Syntactic Layer – abstract general: syntax in homotopy-type theory

This topos-theoretic perspective on fundamental physics which is discussed here is mostly original in the identifications it makes (Schreiber), but it draws insights and inspiration from (and maybe realizes) a vision already expressed since the 1960s by William Lawvere, one of the central figures in the development of topos theory and categorical logic. Lawvere links the very inception of topos theory to the motivation to axiomatize physics:

My own motivation $[$ for developing topos theory $]$ came from my earlier study of physics. The foundation of the continuum physics of general materials, $[...]$ involves powerful and clear physical ideas which unfortunately have been submerged under a mathematical apparatus including not only Cauchy sequences and countably additive measures, but also ad hoc choices of charts for manifolds and of inverse limits of Sobolev Hilbert spaces, to get at the simple nuclear spaces of intensively and extensively variable quantities. But, as Fichera lamented, all this apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? (Lawvere, 2000)

More historical pointers along these lines and further related material can also be found at higher category theory and physics.

To give a survey of how the exposition below proceeds in the fashion of these three layers, the following section The full story in a few formal words provides what may be read as commented index to the central themes of the following text. Whereas the exposition below is organized to start each topic with the discussion of its concrete models in a Model layer, then pass to a general abstract semantics in a Semantic Layer and then finally to the abstract formal syntax in a Syntactic Layer, these tables indicates how this passage to abstract syntax usefully reflects back onto the concrete theory:

The leftmost columns of the following tables formulate concepts in terms of ordinary language. The second columns translate that ordinary language fairly directly to the formal language of (homotopy) type theory. The third columns then interprets these formal syntactical expressions as universal constructions in a (higher, cohesive) topos by the rules of categorical semantics. Finally, the fourth columns indicate what this universal construction amounts to when concretely realized in the model given by smooth spaces and smooth ∞-groupoids. Finally the rightmost columns point to the chapters in the text below that deal with the given construction.

These tables show that fairly evident and naïve sounding statements in ordinary language turn under this translation into what is generally regarded as fairly sophisticated constructions. In fact some of these constructions have only been found by translating along the categorical semantics dictionary this way. So the following tables also serves to show how the general abstract discussion here is a means to facilitate reasoning about seemingly complicated concepts underlying fundamental physics:

The full story in a few formal words

The fundamental physics of the observed world is governed by what is called quantum theory. (This is explicitly so for the standard model of particle physics and induced from this all fundamental physics ever tested in laboratories; but by all that is known also the remaining ingredient of gravity is fundamentally a quantum theory, see at quantum gravity for comments).

Two major axiomatizations of quantum theory are known, namely

1. FQFT where one axiomatizes the assignment of spaces of states to pieces of worldvolume (the “Schrödinger picture” of quantum theory)

   fragments of which involve:

   • finite quantum mechanics in terms of dagger-compact categories

   • spectral triples and graph field theory quantum mechanics

   • 2-spectral triples and string sigma-models

   • Reshetikhin-Turaev construction for 3d TQFT

   • FRS-construction of 2d CFT from this via holography

   • extended topological quantum field theories

2. AQFT where one axiomatizes the assignment of algebras of observables to pieces of worldvolume (the “Heisenberg picture” of quantum theory)

   fragments of which involve:

   • Wightman axioms, Haag-Kastler axioms

   • Bohr toposes, Bohrification of local nets of observables

   • factorization algebras, factorization homology, topological chiral homology,

(For an attempt at a survey of the state of the art as of 2011 see the collection (Sati-Schreiber)).

But all fundamental quantum field theories observed in (or conjectured to underlie) nature arise by a process called quantization from structures in differential geometry (or are induced via a mechanism called the holographic principle from such that do).

This differential geometric data involves

• smooth functionals – called action functionals

• on smooth "spaces" – called moduli stacks

• of differential geometric structures such as fiber bundles and connections – called gauge force fields

• as well as sections of associated bundles – called matter fields.

Similar differential geometric structures are involved in the geometric quantization of such an action functional to an actual quantum field theory.

Hence there is a sequence:

differential geometry	$\to$	geometric quantization	$\to$	quantum field theory

We discuss a formalization of central aspects of this entire sequence. Our development proceeds – as befits a theory of physics and hence of nature – via natural deduction from practical foundations.

$$

Fundamentally, a language for physics is to be a language about existence; a language in which we can express judgements of the form:

There is a thing $x$ of type $X$.

For instance:

There is a gauge field $\nabla$ in the standard model $[X,B{(U\left(1\right)\times \mathrm{SU}\left(2\right)\times \mathrm{SU}\left(3\right))}_{\mathrm{conn}}]$ of gauge fields on spacetime $X$.

(Here the square bracket expression for a moduli stack of gauge fields will be incrementally explained in the following.)

To be predictive, a language for physics is moreover to be a language in which we can make natural deductions to deduce further such judgements from given ones. For instance:

Given a gauge field $\nabla$ as above, there is an underlying instanton sector, $\mathrm{UnderlyingBundle}(\nabla )$, in the collection $[X,B(U\left(1\right)\times \mathrm{SU}\left(2\right)\times \mathrm{SU}\left(3\right))]$ of instanton configurations in the standard model.

Quantum superpositions of such Yang-Mills instantons are the very substrate out of which the vacuum of the observed world is build: the instanton liquid in quantum chromodynamics. (For more see at Yang-Mills theory below.) We consider here a language to reason about such phenomena formally.

The formal language for such natural deduction of judgements about there being terms of some type is called type theory.

Expressions in (dependent) type theory:

(read columns 1+2 first, then 3+4)

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
There is…	$⊢\dots$	We speak in the context of a (higher) topos $H$, a place where things may be. (For the time being a (higher) locally cartesian closed category is sufficient.)	A topos for synthetic differential geometry, such as $H=$ Sh$($SmthMfd$)$. Eventually a higher such topos: $H=$ Smooth∞Grpd or SynthDiff∞Grpd or SmoothSuper∞Grpd or …	Smooth spaces and Smooth homotopy types
There is a thing $x$ of type $X$.	$⊢x:X$	An element $(*\stackrel{x}{\to }X)\in \mathrm{Mor}(H)$ of an object $X$ of $H$.	A point $x$ in a smooth moduli stack $X$.	Judgements about types and terms
There is a type $X$ of things $x$.	$⊢X:\mathrm{Type}$	An element $(*\stackrel{⊢X}{\to }\mathrm{Obj})\in \mathrm{Mor}(H)$ of the small-object classifier $\mathrm{Obj}$ of $H$.	A point in the moduli stack of all small moduli stacks.	Judgements about types and terms
Given a thing $x$ of type $X$ there is a thing $a(x)$ of type $A(x).$	$x:X⊢a(x):A(x)$	An element of a morphism $(A\to X)$ $\left(\begin{array}{ccccc}X& & \stackrel{a}{\to }& & A\\ & {}_{\mathrm{id}}\searrow & ⇙& {\swarrow }_{}\\ & & X\end{array}\right)$ in the slice topos $H_{/X}$.	An $X$-family in a moduli stack bundle $A$ over $X$.	Slice categories and Slice toposes and Slice ∞-Toposes
There is the collection of all things $a(x)$ for all $x$.	$⊢\left({\sum }_{x:X}A\left(x\right)\right):\mathrm{Type}$	The dependent sum/left adjoint to the product: $\begin{array}{ccc}{H}_{/X}& \stackrel{X_{!}}{\to }& H\\ (A\to X)& \mapsto & A\in H\end{array}$	The total space of a bundle.	Natural deduction rules for dependent sum types
There is a thing $t$ in the collection of all things $a(x)$ for all $x$.	$⊢t:{\sum }_{x:X}A(x)$	An element $*\stackrel{t}{\to }A$ of the total space object.	A point in the moduli stack $A$ over $X$.
There is an assignment $f$ of an $a(x)$ to each $x$.	$⊢f:{\prod }_{x:X}A(x)$.	An element in the internal object of sections $*\stackrel{f}{\to }[X,A{]}_X$	A point in the smooth relative mapping space of smooth sections.	Natural deduction rules for dependent product types
There is the collection of assignments of an $a(x)$ to each $x$.	$⊢\left({\prod }_{x:X}A\left(x\right)\right):\mathrm{Type}$	internal space of sections $[X,A{]}_X\in H$	A smooth relative mapping space of smooth sections.
In particular, there is the collection of such assignments when $A$ does not depend on $x$, the collection of functions from $X$ to $A$.	$⊢(X\to A)≔\left({\prod }_{x:X}A\right):\mathrm{Type}$	The internal hom object $[X,A]\in H$.	A smooth mapping space.	Smooth mapping spaces and smooth moduli spaces
There is a proof $p$ that it is true that there is $x$ of type $X$.	$⊢p:[X]$	An element $*\stackrel{p}{\to }{\tau }_{-1}(X)$ of the (-1)-truncation of the object $X$.	A point in the smooth space of equivalence classes of points in $X$.	Subobjects
There is a proof $p$ that it is true that there is an $a(x)$ for some $x$.	$⊢p:\left({\exists }_{x:X}A\left(x\right)\right)≔\left[{\sum }_{x:X}A\left(x\right)\right]$

In order to describe a structured reality, our language needs to be able to speak about comparison of things.

Fundamental physics rests on the gauge principle: it is meaningless to say that two things – such as two gauge fields $\nabla$ as above – are equal; instead they are gauge equivalent if there is a gauge transformation between them.

So our language needs to express judgements of the form:

There is a gauge equivalence between gauge fields ${\nabla }_1$ and ${\nabla }_2$.

And the language needs to be able to make natural deductions from such judgements to arrive at:

Given an equivalence $\lambda :{\nabla }_1\simeq {\nabla }_2$ there is an equivalence $\mathrm{UnderlyingBundle}(\lambda ):\mathrm{UnderlyingBundle}({\nabla }_1)\simeq \mathrm{UnderlyingBundle}({\nabla }_2)$ between the underlying instanton sectors.

The formal language based of the dependent type theory which we have so far that contains these statements is type theory with propositional equality. In this language we have judgements such as the following.

Expressions in dependent type theory with propositional equality:

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
Given $x,x\prime$, there is the collection of equivalences between $x$ and $x\prime$ equivalent.	$x,x\prime :X⊢(x\simeq x\prime ):\mathrm{Type}$.	The mapping cocone object $\begin{array}{ccc}{P}_{x,x\prime }X& \to & *\\ \downarrow & {⇙}_e& {\downarrow }^x\\ *& \stackrel{x\prime }{\to }& X\end{array}$	The moduli stack of gauge transformations between $x$ and $x\prime$.	Identity types
There is an equivalence $e$ between $x$ and $x\prime$.	$\begin{array}{c}⊢e:(x\simeq x\prime )\\ \mathrm{or}\\ ⊢e:(x⇝x\prime )\end{array}$	An element of the mapping cocone object.	A gauge transformation between $x$ and $x\prime$.
Given $x,x\prime$, there is the collection of proofs that it is true that $x$ and $x\prime$ are equivalent.	$x,x\prime :X⊢[x\simeq x\prime ]:\mathrm{Type}$.	The (-1)-truncation fo the mapping cocone.	The smooth space of equivalence classes of gauge transformations from $x$ to $x\prime$.

But the gauge principle reaches deeper: gauge transformations themselves are subject to the gauge principle.

In general it is meaningless to ask if two gauge transformations are equal, but we may ask if there is a higher gauge transformation that transforms one gauge transformation into the other. In the physics literature such gauge-of-gauge transformations are best known in their incarnation as ghost-of-ghost fields in what is called the BRST complex of the given gauge theory.

Careful analysis for instance of the Dirac charge quantization of magnetic charge shows that already quite mundane physical phenomena exhibit such higher gauge transformations. But more famously they are known to arise in various guises in string theory, which is a hypothetical refinement of the standard model of particle physics and gravity.

In either case, our formal language should not allow the deduction that gauge equivalences are themselves either equal or not, but only allow judgements of the following form:

There is a gauge-of-gauge equivalence $\rho :({\lambda }_1\simeq {\lambda }_2)$ between two given gauge equivalences ${\lambda }_1,{\lambda }_2:({\nabla }_1\simeq {\nabla }_2)$ between two given gauge fields ${\nabla }_1,{\nabla }_2$.

The flavor of type theory with propositional equality for which this is the case is called intensional type theory.

Since therefore a type $X$ in intensional type theory may contain homotopies between its terms of arbitrary order, we call it a homotopy type.

The homotopy-type nature of the type of gauge connections $[X,B{G}_{\mathrm{conn}}]$ is most familiar in the physics literature in its infinitesimal approximation, which is the (off-shell) BRST complex of the gauge theory: the $n$-fold ghost-of-ghost fields in the BRST complex correspond to the $n$-fold homotopies in $[X,B{G}_{\mathrm{conn}}]$.

In particular, in intensional type theory we find the gauge group of a homotopy type, as indicated in the following table.

Expressions in intensional type theory:

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
Given a type $X$, there is (the underlying space) of a group $G$ of ways that $X$ is equivalent to itself.	$X:\mathrm{Type}⊢(X\stackrel{\simeq }{\to }X):\mathrm{Type}$	A loop space object $\begin{array}{ccc}G& \to & *\\ \downarrow & ⇙& {\downarrow }^X\\ *& \stackrel{X}{\to }& \mathrm{Type}\end{array}$	A smooth ∞-group.	n-groups
Given a function between collections of things $X$ and $Y$, and given a thing $y$, there is its preimage-up-to-equivalence.	$(f:(X\to Y)),(y:Y)⊢{\sum }_{x:X}(f\left(x\right)\simeq y)$	A homotopy pullback $\begin{array}{ccc}X{\times }_Y\{y\}& \to & X\\ \downarrow & ⇙& {\downarrow }^f\\ *& \underset{y}{\to }& Y\end{array}$	The homotopy fiber of a homomorphism of smooth moduli stacks.

Suppose then that we have such a map between collections of gauge fields

$$f:[X,B{G}_{\mathrm{conn}}]\to [Y,B{H}_{\mathrm{conn}}]$$f \colon [X, \mathbf{B}G_{conn}] \to [Y, \mathbf{B}H_{conn}]

on two possibly different spacetimes with two possibly different gauge groups.

(For instance we might be looking at Montonen-Olive duality/_S-duality_ or Seiberg duality of super Yang-Mills theory.)

Then we should call $f$ an equivalence - in the physics literature often: a duality – if, while not necessarily being a “bijection”, it is such that the preimage ${\varphi }^{-1}(\nabla )\in [X,B{G}_{\mathrm{conn}}]$ of a gauge field $\nabla \in [Y,B{H}_{\mathrm{conn}}]$ consists of gauge fields that are all gauge equivalent to each other, with the gauge equivalences exhibiting this equivalence themselves all being gauge equivalent to each other, etc.

If this is the case one says that all homotopy fibers – all gauge pre-images – of $\varphi$ are contractible – are gauge equivalent to a single gauge field – and that $\varphi$ is a weak homotopy equivalence.

For consistency we should demand that the notion of equivalence is such that the space of direct equivalences $[X,B{G}_{\mathrm{conn}}]\simeq [Y,B{H}_{\mathrm{conn}}]$ is itself equivalent to the space of such weak homotopy equivalences (“dualities”) $[X,B{G}_{\mathrm{conn}}]\stackrel{\simeq }{\to }[Y,B{H}_{\mathrm{conn}}]$.

This requirement is called the univalence axiom. The intensional type theory-language considered so far equipped with this axiom is called homotopy type theory.

We indicate now some central judgements that are expressible in homotopy type theory. This involves fundamental judgements in group theory and in representation theory, two of the pillars of modern quantum theory/quantum field theory.

Structures expressible in homotopy type theory:

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
Given a type $X$, there is a group $G$ of ways that $X$ is equivalent to itself.	$X:\mathrm{Type}⊢(X\stackrel{\simeq }{\to }X):\mathrm{Type}$	A loop space object $\begin{array}{ccc}G& \to & *\\ \downarrow & ⇙& {\downarrow }^X\\ *& \stackrel{X}{\to }& \mathrm{Type}\end{array}$	A smooth automorphism ∞-group.	n-groups
Given a type $X$, there is the delooping $BG$ of $G$, which is the collection of things equipped with equivalences to $X$.	$X:\mathrm{Type}⊢BG≔{\sum }_{Y:\mathrm{Type}}[X\simeq Y]$	The looping and delooping relation $\begin{array}{cccc}G\simeq & \Omega BG& \to & *\\ & \downarrow & ⇙& {\downarrow }^{}\\ & *& \underset{}{\to }& BG\end{array}$	The smooth moduli stack of smooth $G$-principal ∞-bundles.	Principal n-bundles
Given a thing in $BG$, there is a thing $V$.	$\begin{array}{c}*:BG⊢V(*):\mathrm{Type}\\ \mathrm{or}\mathrm{with}\mathrm{more}\mathrm{emphasis}:\\ (*,*\prime ,g):{\sum }_{*,*\prime :BG}(*⇝*\prime )⊢V(*\stackrel{g}{⇝}*\prime ):\mathrm{Type}\end{array}$	A homotopy fiber sequence $\begin{array}{ccc}V& \to & V⫽G\\ & & {\downarrow }^{\overline{\rho }}\\ & & BG\end{array}$ with homotopy fiber $V$ over $BG$.	An ∞-action/∞-representation of $G$ on some $V$, together with its universal $\rho$-associated $V$-fiber ∞-bundle over the moduli stack $BG$ for $G$-principal ∞-bundles.	Higher actions
Given a function $g$ classifying a $G$-principal bundle and given a point in the delooping, there is the $G$-principal bundle $P$ itself, being the collection of identifications of the fiber $g(x)$ with $X$	$(g:X\to BG),(*:BG)⊢P≔{\sum }_{x:X}(g(x)\simeq *)$	$\begin{array}{cccc}P& \to & *& \simeq EG\\ \downarrow & ⇙& \downarrow \\ X& \stackrel{g}{\to }& BG\end{array}$	The principal ∞-bundle given as the homotopy pullback of the universal principal ∞-bundle.	Principal ∞-bundles
There is a $G$-equivariant map from the principal bundle to the representation space.	$⊢\sigma :{\prod }_{*:BG}(P\to V)$	An element $\begin{array}{ccccc}X& & \stackrel{\sigma }{\to }& & V\\ & \searrow & ⇙& \swarrow \\ & & BG\end{array}$ of $V$ in the slice topos $H_{/BG}$	A section of the $\rho$-associated $V$-fiber ∞-bundle.

In gauge theory physics, a representation $\rho$ of the gauge group $G$ encodes the particle-content of the model (in theoretical physics): a section of the $\rho$-associated bundle to the gauge bundle is a matter field in the model.

Therefore all the ingredients so far encode the kinematics of gauge theory, its setup before an actual dynamics is specified.

Dynamics in physics says how things move, hence how they trace out trajectories in a given spacetime or more generally in some phase space.

Our language for reasoning about physics should be able to express this. For $X$ a homotopy type that models spacetime (the collection of all points of spacetime) there should be a homotopy type $\Pi (X)$ whose homotopies and higher homotopies are the smooth trajectories, the smooth paths and higher paths in $X$.

In order to analyse the notion of smoothness here – we will say: the way that points hold together by cohesion – there should also be

• an expression $♭X$ for the discrete collection of points underlying $X$ – detaching all points;

• an expression $♯X$ which dissolves the cohesion and produces the codiscrete smooth structure on $X$.

There are some natural simple axioms on these constructions. For instance every smooth path in a discrete space $♭X$ should be constant: $\Pi (♭X)\simeq ♭X$.

With such natural axioms understood, these three constructions constitute an adjoint triple of modalities $(\Pi ⊣♭⊣♯)$ in our language. In particular $\Pi$ and $♭$ are a monad and comonad on the type system, in the sense of computer science and $♯$ is even an internal monad.

Equipping the above homotopy type theory with these modalities turns it into what we call cohesive homotopy type theory.

Structures expressible in cohesive homotopy type theory:

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
Given a cohesive homotopy type $X$, there is the dissolved homotopy type $♯X$ in which all separate points are collected to one cohesive blob.	$X:\mathrm{Type}⊢♯X:\mathrm{Type}$	The codiscrete object-monad on a (higher) local topos.	The codiscrete smooth structure on the points of $X$.	Locality of the topos of smooth spaces
Given a cohesive homotopy type, there is the map that dissolves the cohesion of the points.	$X:\mathrm{Type}⊢{\mathrm{DeCoh}}_X:X\to ♯X$	The unit of the codiscrete object monad.	The function that sends smooth families in a smooth moduli stack to families of points.
Given $X$ there is the collection $\Pi (X)$ of points in $X$ and smooth trajectories between points in $X$.	$(X:♯\mathrm{Type})⊢\Pi (X):♯\mathrm{Type}$	The construction of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos.	The smooth path ∞-groupoid of $X$.	The local ∞-connectedness of the (∞,1)-topos of smooth ∞-groupoids
Given $X$, there is a canonical map to $\Pi (X)$.	$(X:♯\mathrm{Type})⊢{\mathrm{ConstantPathInclusion}}_X:X\to \Pi (X)$.	The unit of the $\Pi$-monad on a locally ∞-connected (∞,1)-topos.	The inclusion of $X$ into its smooth path ∞-groupoid as the constant paths.
Given $X$, there is the result of detaching the points in $X$.	$(A:♯\mathrm{Type})⊢♭A:♯\mathrm{Type}$	The operation of the discrete object comonad on a (higher) local topos.	The moduli stack for flat ∞-connections.
Given $A$, there is a map from flat $A$-connections to the underlying $A$-bundles	$(A:♯\mathrm{Type})⊢{\mathrm{UnderlyingBundle}}_A:♭A\to A$	The counit of the discrete object-comonad on a (higher) local topos.	The function that sends a flat ∞-connection to its underlying principal ∞-bundle.	Flat connections

Adding the modalities $(\Pi ⊣♭⊣♯)$ to the above language of homotopy type theory yields a language that we call cohesive homotopy type theory (following a term introduced by Lawvere).

Fundamental judgements in cohesive homotopy type theory include those indicated in the following table, which capture central concepts of gauge theory and its (higher) geometric quantization.

Structures expressible in cohesive homotopy type theory:

Gauge fields, matter fields, and smooth action functionals on their moduli stacks…

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
A flat connection $\nabla$ on $X$ is a rule for sending paths $(x\stackrel{\gamma }{\to }y)\in \Pi X$ to group elements, respecting composition.	$\mathrm{transport}(\nabla ):\sum _{x,y:\Pi X}(x⇝y)\to \sum _{*,*\prime :BG}(*⇝*\prime )$	$\frac{\Pi (X)\stackrel{\mathrm{transport}(\nabla )}{\to }BG}{X\stackrel{\nabla }{\to }♭BG}$.	The higher parallel transport $\mathrm{trans}(\nabla )$ of a flat connection $\nabla$: a (higher) gauge field with vanishing field strength.	Flat connections
A closed differential form $\omega$ is a flat connection $\nabla$ and a trivialization of the underlying bundle.	$\begin{array}{rl}& {♭}_{\mathrm{dR}}BG≔\\ & {\sum }_{\nabla :♭BG}(\mathrm{UnderlyingBundle}(\nabla )\simeq *)\end{array}$	$\begin{array}{ccc}{♭}_{\mathrm{dR}}BG& \stackrel{\mathrm{UnderlyingConnection}}{}& ♭BG\\ & \begin{array}{cc}& \\ & ⇙\end{array}& {}^{\genfrac{}{}{0ex}{}{\mathrm{Underlying}}{\mathrm{Bundle}}}\\ *& \stackrel{}{}& BG\end{array}$	The coefficients for de Rham hypercohomology – flat ∞-Lie algebra valued differential forms.	de Rham coefficients
A general connection $\nabla$ is the equivalence between the curvature $\mathrm{curv}(c)$ of a bundle $c$ and a closed differential form $\omega$.	$\nabla :\sum _{\genfrac{}{}{0ex}{}{c:B^n𝔾}{\omega :{\Omega }_{\mathrm{cl}}^{n+1}}}(\mathrm{curv}\left(c\right)=\omega )$	$\begin{array}{ccc}{B}^n{𝔾}_{\mathrm{conn}}& \stackrel{F_{(-)}}{}& {\Omega }_{\mathrm{cl}}^{n+1}\\ & \begin{array}{cc}& \\ & ⇙\end{array}& \\ B^n𝔾& \stackrel{\mathrm{curv}}{}& {♭}_{\mathrm{dR}}{B}^{n+1}𝔾\end{array}$	The coefficients for smooth differential cohomology: abelian (higher) gauge fields.	Circle principal n-connections
There is a cohesive function from $G$-gauge fields to higher $𝔾$-gauge fields.	$⊢\exp (iS):B{G}_{\mathrm{conn}}\to B^n{𝔾}_{\mathrm{conn}}$	A differential universal characteristic class.	An extended action functional/prequantum n-bundle for extended higher Chern-Simons-type gauge theory.

… and their ∞-geometric prequantization (see there for a more comprehensive disctionary):

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
There is a $𝔾$-equivariant map $\psi$ from the prequantum bundle to the representation space.	$⊢\psi :\prod _{\nabla :B{𝔾}_{\mathrm{conn}}}(P(\nabla )\to V(\nabla ))$	$\begin{array}{ccccc}X& & \stackrel{\psi }{\to }& & V⫽{𝔾}_{\mathrm{conn}}\\ & {}_{\nabla }\searrow & ⇙& {\swarrow }_{\overline{\rho }}\\ & & B{𝔾}_{\mathrm{conn}}\end{array}$	A prequantum state.	Geometric quantization
There is a differentially $𝔾$-equivariant equivalence $\exp (\hat{O})$ from the prequantum bundle to itself.	$⊢\exp (\hat{O}):\prod _{\nabla :B{𝔾}_{\mathrm{conn}}}\left(P(\nabla )\stackrel{\simeq }{\to }P(\nabla )\right)$	$\begin{array}{ccccc}X& & \stackrel{\exp (\hat{O})}{\to }& & X\\ & {}_{\nabla }\searrow & ⇙& {\swarrow }_{\nabla }\\ & & B{𝔾}_{\mathrm{conn}}\end{array}$	A prequantum operator: an element of the quantomorphism group/Heisenberg group of the quantum system.	Geometric quantization

Finally, in order to be able to concretely speak about not just about any gauge field, but the concrete particular gauge fields in the observable universe, our language should be able to express the existence of the continuum real line.

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
There is the continuum line.	$\begin{array}{rl}⊢& ℝ:\mathrm{Type}\\ & i:\mathbb{Z}\to ℝ\\ & {\mathrm{GeometricallyContract}}_{ℝ}:(\Pi (ℝ)\simeq \mathrm{Point})\end{array}$	line object	real line	The continuum real worldline

This then induces the existence of the circle group $U(1)=ℝ/\mathbb{Z}$. The electromagnetic field is a gauge field for gauge group $U(1)$. Therefore in the language of cohesive homotopy type theory we can say

Let there be light.

ordinary language	syntax	semantics	model	chapter
	general abstract	general concrete	concrete particular
There is the collection of higher $U(1)$-principal connections.	$n:\mathbb{N}⊢B^{n}U(1{)}_{\mathrm{conn}}:\mathrm{Type}$	The coefficients for ordinary differential cohomology (with coefficients in an Eilenberg-MacLane object.)	The smooth higher moduli stack of smooth circle n-bundles with connection.	Circle-principal n-connections.
There is light.	$⊢{\nabla }_{\mathrm{em}}:[X,BU(1{)}_{\mathrm{conn}}]$	A cocycle in ordinary differential cohomology in degree-2.	A configuration of the electromagnetic field on spacetime $X$.	Circle principal connection

$$

There are of many more constructions in fundamental (quantum) physics that are naturally expressible in cohesive homotopy type theory, but the above should already give an idea and highlight the cornerstones of the following discussion.

$$

We now end this introduction and overview and turn to the in-depth account of geometry of physics.

Contents

I) Geometry

We begin by laying the foundations of differential geometry. Doing this in th natural abstract way seamlessly leads over to the foundations of higher differential geometry (see also motivation for higher differential geometry). Once this is set up, we discuss the fundamental constructions: groups, actions/representations, fiber bundles, connections, Chern-Weil theory.

1. Coordinate systems

2. Smooth spaces

3. Differential forms

4. Differentiation

5. Smooth homotopy types

6. Groups

7. Principal bundles

8. Manifolds and Orbifolds

9. Reduction of structure groups

10. Representations and Associated bundles

11. Flat connections

12. de Rham coefficients

13. Maurer-Cartan forms

14. Principal connections

15. Characteristic classes

16. Integration

17. Super-geometry

Coordinate systems

Every kind of geometry is modeled on a collection of archetypical basic spaces and geometric homomorphisms between them. In differential geometry the archetypical spaces are the abstract standard Cartesian coordinate systems, denoted ${ℝ}^n$, in every dimension $n\in \mathbb{N}$, and the geometric homomorphism between them are smooth functions ${ℝ}^{n_1}\to {ℝ}^{n_1}$, hence smooth (and possibly degenerate) coordinate transformations.

Here we discuss the central aspects of the nature of such abstract coordinate systems in themselves. At this point these are not yet coordinate systems on some other space. That is instead the topic of the next section Smooth spaces.

Model Layer

In this Mod Layer we discuss the concrete particulars of coordinate systems: the continuum real line $ℝ$, the Cartesian spaces ${ℝ}^n$ formed from it and the smooth functions between these.

The continuum real (world-)line

The fundamental premise of differential geometry as a model of geometry in physics is the following.

Premise. The abstract worldline of any particle is modeled by the continuum real line $ℝ$.

This comes down to the following sequence of premises.

1. There is a linear ordering of the points on a worldline: in particular if we pick points at some intervals on the worldline we may label these in an order-preserving way by integers

   $\mathbb{Z}$.

2. These intervals may each be subdivided into $n$ smaller intervals, for each natural number $n$. Hence we may label points on the worldline in an order-preserving way by the rational numbers

   $\mathbb{Q}$.

3. This labeling is dense: every point on the worldline is the supremum of an inhabited bounded subset of such labels. This means that a worldline is the real line, the continuum of real numbers

   $ℝ$.

The adjective “real” in “real number” is a historical shadow of the old idea that real numbers are related to observed reality, hence to physics in this way. The experimental success of this assumption shows that it is valid at least to very good approximation.

Speculations are common that in a fully exact theory of quantum gravity, currently unavailable, this assumption needs to be refined. For instance in p-adic physics one explores the hypothesis that the relevant completion of the rational numbers as above is not the reals, but p-adic numbers ${\mathbb{Q}}_p$ for some prime number $p\in \mathbb{N}$. Or for example in the study of QFT on non-commutative spacetime one explore the idea that at small scales the smooth continuum is to be replaced by an object in noncommutative geometry. Combining these two ideas leads to the notion of non-commutative analytic space as a potential model for space in physics. And so forth.

For the time being all this remains speculation and differential geometry based on the continuum real line remains the context of all fundamental model building in physics related to observed phenomenology. Often it is argued that these speculations are necessitated by the very nature of quantum theory applied to gravity. But, at least so far, such statements are not actually supported by the standard theory of quantization: we discuss below in Geometric quantization how not just classical physics but also quantum theory, in the best modern version available, is entirely rooted in differential geometry based on the continuum real line.

This is the motivation for studying models of physics in geometry modeled on the continuum real line. On the other hand, in all of what follows our discussion is set up such as to be maximally independent of this specific choice (this is what topos theory accomplishes for us, discussed below Smooth spaces – Semantic Layer). If we do desire to consider another choice of archetypical spaces for the geometry of physics we can simply “change the site”, as discussed below and many of the constructions, propositions and theorems in the following will continue to hold. This is notably what we do below in Supergeometric coordinate systems when we generalize the present discussion to a flavor of differential geometry that also formalizes the notion of fermion particles: “differential supergeometry”.

Cartesian spaces and smooth functions

Definition

A function of sets $f:ℝ\to ℝ$ is called a smooth function if, coinductively:

1. the derivative $\frac{df}{dx}:ℝ\to ℝ$ exists;

2. and is itself a smooth function.

We write $C^{\mathrm{\infty }}(ℝ)\in \mathrm{Set}$ for the set of all smooth functions on $ℝ$.

Remark

The superscript ”${}^{\mathrm{\infty }}$” in ”$C^{\mathrm{\infty }}(ℝ)$” refers to the order of the derivatives that exist for smooth functions. More generally for $k\in \mathbb{N}$ one writes $C^k(ℝ)$ for the set of $k$-fold differentiable functions on $ℝ$. These will however not play much of a role for our discussion here.

Definition

For $n\in \mathbb{N}$, the Cartesian space ${ℝ}^n$ is the set

$${ℝ}^n=\{(x^1,\cdots ,x^n)\mid x^i\in ℝ\}$$\mathbb{R}^n = \{ (x^1 , \cdots, x^{n}) | x^i \in \mathbb{R} \}

of $n$-tuples of real numbers. For $1\le k\le n$ write

$$i^k:ℝ\to {ℝ}^n$$i^k : \mathbb{R} \to \mathbb{R}^n

for the function such that $i^k(x)=(0,\cdots ,0,x,0,\cdots ,0)$ is the tuple whose $k$th entry is $x$ and all whose other entries are $0\in ℝ$; and write

$${𝕡}^k:{ℝ}^n\to ℝ$$\mathbb{p}^k : \mathbb{R}^n \to \mathbb{R}

for the function such that $p^k(x^1,\cdots ,x^n)=x^k$.

A homomorphism of Cartesian spaces is a smooth function

$$f:{ℝ}^{n_1}\to {ℝ}^{n_2},$$f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} \,,

hence a function $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ such that all partial derivatives exist and are continuous (…).

Example

Regarding ${ℝ}^n$ as an $ℝ$-vector space, every linear function ${ℝ}^{n_1}\to {ℝ}^{n_2}$ is in particular a smooth function.

Remark

But a homomorphism of Cartesian spaces in def. 2 is not required to be a linear map. We do not regard the Cartesian spaces here as vector spaces.

Definition

A smooth function $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ is called a diffeomorphism if there exists another smooth function ${ℝ}^{n_2}\to {ℝ}^{n_1}$ such that the underlying functions of sets are inverse to each other

$$f\circ g=\mathrm{id}$$f \circ g = id

and

$$g\circ f=\mathrm{id}.$$g \circ f = id \,.

Proposition

There exists a diffeomorphism ${ℝ}^{n_1}\to {ℝ}^{n_2}$ precisely if $n_1=n_2$.

Definition

We will also say equivalently that

1. a Cartesian space ${ℝ}^n$ is an abstract coordinate system;

2. a smooth function ${ℝ}^{n_1}\to {ℝ}^{n_2}$ is an abstract coordinate transformation;

3. the function $p^k:{ℝ}^n\to ℝ$ is the $k$th coordinate of the coordinate system ${ℝ}^n$. We will also write this function as $x^k:{ℝ}^n\to ℝ$.

4. for $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ a smooth function, and $1\le k\le n_2$ we write

   1. $f^k≔p^k\circ f$

   2. $(f^1,\cdots ,f^n)≔f$.

Remark

It follows with this notation that

$${\mathrm{id}}_{{ℝ}^n}=(x^1,\cdots ,x^n):{ℝ}^n\to {ℝ}^n.$$id_{\mathbb{R}^n} = (x^1, \cdots, x^n) : \mathbb{R}^n \to \mathbb{R}^n \,.

Hence an abstract coordinate transformation

$$f:{ℝ}^{n_1}\to {ℝ}^{n_2}$$f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}

may equivalently be written as the tuple

$$(f^1(x^1,\cdots ,x^{n_1}),\cdots ,f^{n_2}(x^1,\cdots ,x^{n_1})).$$\left( f^1 \left( x^1, \cdots, x^{n_1} \right) , \cdots, f^{n_2}\left( x^1, \cdots, x^{n_1} \right) \right) \,.

Propositions

Abstract coordinate systems according to prop. 4 form a category – to be denoted CartSp – whose

• objects are the abstract coordinate systems ${ℝ}^n$ (the class of objects is the set $\mathbb{N}$ of natural numbers $n$);

• morphisms $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ are the abstract coordinate transformations = smooth functions.

Composition of morphisms is given by composition of functions.

We have that

1. The identity morphisms are precisely the identity functions.

2. The isomorphisms are precisely the diffeomorphisms.

Definition

Write CartSp${}^{\mathrm{op}}$ for the opposite category of CartSp.

This is the category with the same objects as $\mathrm{CartSp}$, but where a morphism ${ℝ}^{n_1}\to {ℝ}^{n_2}$ in ${\mathrm{CartSp}}^{\mathrm{op}}$ is given by a morphism ${ℝ}^{n_1}\leftarrow {ℝ}^{n_2}$ in $\mathrm{CartSp}$.

We will be discussing below the idea of exploring smooth spaces by laying out abstract coordinate systems in them in all possible ways. The reader should begin to think of the sets that appear in the following definition as the set of ways of laying out a given abstract coordinate systems in a given space. This is discussed in more detail below in Smooth spaces.

Definition

A functor $X:{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Set}$ (a “presheaf”) is

1. for each abstract coordinate system $U$ a set $X(U)$

2. for each coordinate transformation $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ a function $X(f):X({ℝ}^{n_1})\to X({ℝ}^{n_2})$

such that

1. identity is respected $X({\mathrm{id}}_{{ℝ}^n})={\mathrm{id}}_{X({ℝ}^n)}$;

2. composition is respected $X(f_2)\circ X(f_1)=X(f_2\circ f_1)$

The fundamental theorems about smooth functions

The special properties smooth functions that make them play an important role different from other classes of functions are the following.

1. existence of bump functions and partitions of unity

2. the Hadamard lemma and Borel's theorem

Or maybe better put: what makes smooth functions special is that the first of these properties holds, while the second is still retained.

(…)

This ends the “Model layer”-part of the discussion of coordinate systems. In the Semantics layer below we continue with a more sophisticated perspective on this topic. The reader wishing to stick to more elementary discussion for the moment should skip ahead to the next “Model layer”-discussion of Smooth spaces below.

Semantic Layer

In this Sem Layer we discuss the concrete general aspects of abstract coordinate systems, def. 4: the fact that they naturally form:

1. an algebraic theory,

2. a site.

The algebraic theory of smooth algebras

Propositions

• The category CartSp has all finite products.

• Every object is a finite product of the object $ℝ$ (the real line itself).

• The terminal object is ${ℝ}^0$, the point.

Hence CartSp is (the syntactic category) of an algebraic theory (a Lawvere theory).

This is called the theory of smooth algebras.

Definition

A product-preserving functor

$$A:\mathrm{CartSp}\to \mathrm{Set}$$A : CartSp \to Set

is a smooth algebra. A homomorphism of smooth algebras is a natural transformation between the corresponding functors.

The basic example is:

Example

For $n\in \mathbb{N}$, the smooth algebra $C^{\mathrm{\infty }}({ℝ}^n)$ is the functor $\mathrm{CartSp}\to \mathrm{Set}$ which is functor corepresented by ${ℝ}^n\in$ CartSp. This means that to ${ℝ}^k\in \mathrm{CartSp}$ it assigns the set

$${\mathrm{Hom}}_{\mathrm{CartSp}}({ℝ}^n,{ℝ}^k)=C^{\mathrm{\infty }}({ℝ}^n,{ℝ}^k)$$Hom_{CartSp}(\mathbb{R}^n , \mathbb{R}^k) = C^\infty(\mathbb{R}^n, \mathbb{R}^k)

of smooth functions from ${ℝ}^n$ to ${ℝ}^k$, and to a smooth function $f:{ℝ}^{k_1}\to {ℝ}^{k_2}$ it assigns the function

$$f\circ (-):C^{\mathrm{\infty }}({ℝ}^n,{ℝ}^{k_1})\to C^{\mathrm{\infty }}({ℝ}^n,{ℝ}^{k_2})$$f\circ (-) \colon C^\infty(\mathbb{R}^n, \mathbb{R}^{k_1}) \to C^\infty(\mathbb{R}^n, \mathbb{R}^{k_2})

given by postcomposition with $f$.

Remark

Example 2 shows how we are to think of a functor $A:\mathrm{CartSp}\to \mathrm{Set}$ as encoding an algebra: such a functor assigns to ${ℝ}^n$ a set to be interpreted as a set of “smooth functions on something with values in ${ℝ}^n$”, only that the “something” here is not pre-defined, but is instead indirectly characterized by this assignment.

Due to this we will often denote smooth algebras as ”$C^{\mathrm{\infty }}(X)$”, even if ”$X$” is not a pre-defined object, and write their value on ${ℝ}^n$ as $C^{\mathrm{\infty }}(X,{ℝ}^n)$.

This is illustrated by the next example.

Example

The smooth algebra of dual numbers $C^{\mathrm{\infty }}(D)$ is the smooth algebra which assigns to ${ℝ}^n$ the Cartesian product

$$C^{\mathrm{\infty }}(D,{ℝ}^n)≔{ℝ}^n\times {ℝ}^n$$C^\infty(D,\mathbb{R}^n) \coloneqq \mathbb{R}^n \times \mathbb{R}^n

of two copies of ${ℝ}^n$, which we will write as

$$\{(ϵ\mapsto (\stackrel{\rightharpoonup }{x}+ϵ\stackrel{\rightharpoonup }{v}))\mid \stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{v}\in {ℝ}^n\}.$$\left\{ (\epsilon \mapsto (\vec x + \epsilon \vec v)) | \vec x , \vec v \in \mathbb{R}^n \right\} \,.

Moreover, a smooth function $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ is sent to the function

$$C^{\mathrm{\infty }}(D,f):C^{\mathrm{\infty }}(D,{ℝ}^{n_1})\to C^{\mathrm{\infty }}(D,{ℝ}^{n_2})$$C^\infty(D, f) \colon C^\infty(D, \mathbb{R}^{n_1}) \to C^\infty(D, \mathbb{R}^{n_2})

given by

$$\begin{array}{r}(ϵ\mapsto (\stackrel{\rightharpoonup }{x}+ϵ\stackrel{\rightharpoonup }{v}))\\ (ϵ\mapsto f(\stackrel{\rightharpoonup }{x})+(df)(\stackrel{\rightharpoonup }{v}))& \mapsto (ϵ\mapsto (f\left(\stackrel{\rightharpoonup }{x}\right)+\sum _{j=1}^{n_2}\left(\sum _{i=1}^{n_1}\frac{\partial f^j}{\partial x^i}{v}^i\right){\stackrel{\rightharpoonup }{e}}_j))\end{array}.$$\begin{aligned} \left(\epsilon \mapsto \left(\vec x + \epsilon \vec v\right)\right) \\ \left( \epsilon \mapsto f(\vec x) + (\mathbf{d}f)(\vec v) \right) &\mapsto \left( \epsilon \mapsto \left( f\left(\vec x\right) + \sum_{j = 1}^{n_2} \left(\sum_{i = 1}^{n_1}\frac{\partial f^j}{\partial x^i} v^i\right) \vec e_j \right) \right) \end{aligned} \,.

Remark

As the notation suggests, we may think of $C^{\mathrm{\infty }}(D)$ as the functions on a first order infinitesimal neighbourhood of the origin in ${ℝ}^n$.

The coverage of differentially good open covers

We discuss a standard structure of a site on the category CartSp. Following Johnstone -- Sketches of an Elephant, it will be useful and convenient to regard a site as a (small) category equipped with a coverage. This generates a genuine Grothendieck topology, but need not itself already be one.

Definition

For $n\in \mathbb{N}$ the standard open n-ball is the subset

$$D^n=\{(x_i{)}_{i=1}^n\in {ℝ}^n\mid \sum _{i=1}^n(x_i{)}^2<1\}\hookrightarrow {ℝ}^n.$$D^n = \{ (x_i)_{i =1}^n \in \mathbb{R}^n | \sum_{i = 1}^n (x_i)^2 \lt 1 \} \hookrightarrow \mathbb{R}^n \,.

Proposition

There is a diffeomorphism

$${ℝ}^n\stackrel{\simeq }{\to }{D}^n.$$\mathbb{R}^n \stackrel{\simeq}{\to} D^n \,.

Definition

A differentially good open cover of a Cartesian space ${ℝ}^n$ is a set $\{U_i\hookrightarrow {ℝ}^n\}$ of open subset inclusions of Cartesian spaces such that these cover ${ℝ}^n$ and such for each non-empty finite intersection there exists a diffeomorphism

$${ℝ}^n\stackrel{\simeq }{\to }{U}_{i_1}\cap \cdots \cap U_{i_k}$$\mathbb{R}^n \stackrel{\simeq}{\to} U_{i_1} \cap \cdots \cap U_{i_k}

that identifies the $k$-fold intersection with a Cartesian space itself.

Remark

Differentiably good covers are useful for computations. Their full impact is however on the homotopy theory of simplicial presheaves over CartSp. This we discuss further below, around prop. 39.

Proposition

Every open cover refines to a differentially good open cover, def. 9.

A proof is at good open cover.

Remark

Despite its appearance, this is not quite a classical statement. The classical statement is only that every open cover is refined by a topologically good open cover. See the comments here in the references-section at open ball for the situation concerning this statement in the literature.

Remark

The good open covers do not yet form a Grothendieck topology on CartSp. One of the axioms of a Grothendieck topology is that for every covering family also its pullback along any morphism in the category is a covering family. But while the pullback of every open cover is again an open cover, and hence open covers form a Grothendieck topology on CartSp, not every pullback of a good open cover is again good.

Example

Let $\{{ℝ}^2\stackrel{{\varphi }_i}{\hookrightarrow }{ℝ}^2{\}}_{i\in \{1,2\}}$ be the open cover of the plane by an open left half space

$${ℝ}^2\simeq \{(x_1,x_2)\in {ℝ}^2\mid x_1<1\}\stackrel{{\varphi }_1}{\hookrightarrow }{ℝ}^2$$\mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \lt 1 \} \stackrel{\phi_1}{\hookrightarrow} \mathbb{R}^2

and a right open half space

$${ℝ}^2\simeq \{(x_1,x_2)\in {ℝ}^2\mid x_1>-1\}\stackrel{{\varphi }_2}{\hookrightarrow }{ℝ}^2.$$\mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \gt -1 \} \stackrel{\phi_2}{\hookrightarrow} \mathbb{R}^2 \,.

The intersection of the two is the open strip

$${ℝ}^2\simeq \{(x_1,x_2)\in {ℝ}^2\mid -1<x_1<1\}\hookrightarrow {ℝ}^2.$$\mathbb{R}^2 \simeq \{ (x_1, x_2) \in \mathbb{R}^2 | -1 \lt x_1 \lt 1 \} \hookrightarrow \mathbb{R}^2 \,.

So this is a good open cover of ${ℝ}^2$.

But consider then the smooth function

$$2(\cos (2\pi (-)),\sin (2\pi (-))):{ℝ}^1\to {ℝ}^2$$2(\cos(2 \pi (-)), \sin(2 \pi (-))) \colon \mathbb{R}^1 \to \mathbb{R}^2

which sends the line to a curve in the plane that periodically goes around the circle of radius 2 in the plane.

Then the pullback of the above good open cover on ${ℝ}^2$ to ${ℝ}^1$ along this function is an open cover of $ℝ$ by two open subsets, each being a disjoint union of countably many open intervals in $ℝ$. Each of these open intervals is an open 1-ball hence diffeomorphic to ${ℝ}^1$, but their disjoint union is not contractible (it does not contract to the point, but to many points!).

So the pullback of the good open cover that we started with is an open cover which is not good anymore. But it has an evident refinement by a good open cover.

This is a special case of what the following statement says in generality.

Proposition

The differentially good open covers, def. 9, constitute a coverage on CartSp.

Hence CartSp equipped with that coverage is a site.

Proof

By definition of coverage we need to check that for $\{U_i\hookrightarrow {ℝ}^n{\}}_{i\in I}$ any good open cover and $f:{ℝ}^k\to {ℝ}^n$ any smooth function, we can find a good open cover $\{K_j\to {ℝ}^k{\}}_{j\in J}$ and a function $J\to I$ such that for each $j\in J$ there is a smooth function $\varphi :K_j\to U_{\rho (j)}$ that makes this diagram commute:

$$\begin{array}{ccc}{K}_j& \stackrel{\varphi }{\to }& U_{i(j)}\\ \downarrow & & \downarrow \\ {ℝ}^k& \stackrel{f}{\to }& {ℝ}^n\end{array}.$$\array{ K_j &\stackrel{\phi}{\to}& U_{i(j)} \\ \downarrow && \downarrow \\ \mathbb{R}^k &\stackrel{f}{\to}& \mathbb{R}^n } \,.

To obtain this, let $\{f^{*}{U}_i\to {ℝ}^k\}$ be the pullback of the original covering family, in that

$$f^{*}{U}_i≔\{x\in {ℝ}^k\mid f(x)\in U_i\}\hookrightarrow {ℝ}^k.$$f^* U_i \coloneqq \{ x \in \mathbb{R}^k | f(x) \in U_i \} \hookrightarrow \mathbb{R}^k \,.

This is evidently an open cover, albeit not necessarily a good open cover. But by prop. 5 there does exist a good open cover $\{{\tilde{K}}_{\tilde{j}}\hookrightarrow {ℝ}^k{\}}_{\tilde{j}\in \tilde{J}}$ refining it, which in turn means that for all $\tilde{j}$ there is

$$\begin{array}{ccc}{\tilde{K}}_{\tilde{j}}& \to & K_{j(\tilde{j})}\\ \downarrow & & \downarrow \\ {ℝ}^k& \stackrel{=}{\to }& {ℝ}^k\end{array}.$$\array{ \tilde K_{\tilde j} &\to& K_{j(\tilde j)} \\ \downarrow && \downarrow \\ \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k } \,.

Therefore then the pasting composite of these commuting squares

$$\begin{array}{ccccc}{\tilde{K}}_{\tilde{j}}& \to & K_{j(\tilde{j})}& \to & U_{i(j(\tilde{j}))}\\ \downarrow & & \downarrow & & \downarrow \\ {ℝ}^k& \stackrel{=}{\to }& {ℝ}^k& \stackrel{f}{\to }& {ℝ}^n\end{array}$$\array{ \tilde K_{\tilde j} &\to& K_{j(\tilde j)} &\to& U_{i(j(\tilde j))} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k &\stackrel{f}{\to}& \mathbb{R}^n }

solves the condition required in the definition of coverage.

By example 4 this good open cover coverage is not a Grothendieck topology. But as any coverage, it uniquely completes to one which has the same sheaves.

Proposition

The Grothendieck topology induced on CartSp by the differentially good open cover coverage of def. 6 has as covering families the ordinary open covers.

Remark

This means that for every sheaf-theoretic construction to follow we can just as well consider the Grothendieck topology of open covers on $\mathrm{CartSp}$. The sheaves of the open cover topology are the same as those of the good open cover coverage. But the latter is (more) useful for several computational purposes in the following. It is the good open cover coverage that makes manifest, below, that sheaves on $\mathrm{CartSp}$ form a locally connected topos and in consequence then a cohesive topos. This kind of argument becomes all the more pronounced as we pass further below to (∞,1)-sheaves on CartSp. This will be discussed in Smooth n-groupoids – Semantic Layer – Local Infinity-Connectedness below.

The slice category

(…)

• slice category ${\mathrm{CartSp}}_{{ℝ}^n}$

• subterminal object, open cover

(…)

Syntactic Layer

In this Syn Layer we discuss the abstract generals of abstract coordinate systems, def. 4: the internal language of a category with products, which is type theory with product types.

Judgments about types and terms

We now introduce a different notation for objects and morphisms in a category (such as the category CartSp of def. 2). This notation is designed to, eventually, make more transparent what exactly it is that happens when we reason deductively about objects and morphisms of a category.

But before we begin to make any actual deductions about objects and morphisms in a category below, in this section here we express the given objects and morphisms at hand in the first place. Such basic statements of the form “There is an object called $A$” are to be called judgments, in order not to confuse these with genuine propositions that we eventually formalize within this metalanguage.

To express that there is an object

$$X\in 𝒞$$X \in \mathcal{C}

in a category $𝒞$, we write now equivalently the string of symbols (called a sequent)

$$⊢X:\mathrm{Type}.$$\vdash \; X \colon Type \,.

We say that these symbols express the judgment that $X$ is a type. We also say that $⊢X:\mathrm{Type}$ is the syntax of which $X\in 𝒞$ is the categorical semantics.

For instance the terminal object $*\in 𝒞$ we call the categorical semantics of the unit type and write syntactically as

$$⊢\mathrm{Unit}:\mathrm{Type}.$$\vdash \; Unit \colon Type \,.

If we want to express that we do assume that a terminal object indeed exists, hence that we want to be able to deduce the existence of a terminal object from no hypothesis, then we write this judgment below a horizontal line

$$\frac{}{⊢\mathrm{Unit}:\mathrm{Type}}.$$\frac{}{\vdash \; Unit \colon Type} \,.

We will see more interesting such horizontal-line statements below.

Next, to express an element of the object $X$ in $𝒞$, hence a morphism

$$*\stackrel{x}{\to }X$$* \stackrel{x}{\to} X

in $𝒞$ we write equivalently the sequent

$$⊢x:X$$\vdash \; x \colon X

and call this the judgment that $x$ is a term of type $X$.

Notice that every object $X\in 𝒞$ becomes the terminal object in the slice category ${𝒞}_{/X}$. Let $A\to X$ be any morphism in $𝒞$, regarded as an object in the slice category

$$A\in {𝒞}_{/X}.$$A \in \mathcal{C}_{/X} \,.

We declare that the syntax of which this is the categorical semantics is given by the sequent

$$x:X⊢A(x):\mathrm{Type}.$$x \colon X \;\vdash \; A(x) \colon Type \,.

We say that this expresses the judgement that $A$ is an $X$-dependent type; or a type in the context of a free variable $x$ of type $X$.

Notice that an element of $A\in {𝒞}_{/X}$ is a generalized element of $A$ in $𝒞$, namely a morphism $X\to A$ which fits into a commuting diagram

$$\begin{array}{ccccc}X& & \stackrel{a}{\to }& & A\\ & {}_{{\mathrm{id}}_X}\searrow & & {\swarrow }_{}\\ & & X\end{array}$$\array{ X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{} \\ && X }

in $𝒞$.

We declare that the syntax of which such

$$a\in A(\mathrm{in}{𝒞}_{/X})$$a \in A \;\;\;\; (in \mathcal{C}_{/X})

is this the categorical semantics is the sequent

$$x:X⊢a(x):A(x).$$x\colon X \;\vdash\; a(x) : A(x) \,.

We say that this expresses the judgment that $a(x)$ is a term depending on the free variable $x$ of type $X$.

This completes the list of judgment syntax to be considered. Notice that if the category $𝒞$ has products then, even though it does not explicitly appear above, this is sufficient to express any morphism $X\stackrel{f}{\to }Y$ in $𝒞$ as the semantics of a term: we regard this morphism naturally as being the corresponding morphism in the slice category ${𝒞}_{/X}$ which as a commuting diagram in $𝒞$ itself is

$$\begin{array}{ccccc}X& & \stackrel{(f,{\mathrm{id}}_X)}{\to }& & Y\times X\\ & {}_{{\mathrm{id}}_X}\searrow & & {\swarrow }_{p_2}\\ & & X\end{array}.$$\array{ X && \stackrel{(f,id_X)}{\to} && Y\times X \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X } \,.

This is the categorical semantics for which the syntax is simply

$$x:X⊢y(x):Y,$$x \colon X \;\vdash\; y(x) \colon Y \,,

being the judgment which expresses that $y(x)$ is a term in context of an $X$-dependent type $Y$ in the special degenerate case that $Y$ does not actually vary with $x:X$.

Natural deduction rules for product types

With the above symbolic notation for making judgments about the presence of objects and morphisms in a category $𝒞$, we now consider a system of rule of deduction that tells us how we may process these symbols (how to do computations) such that the new symbols we obtain in turn express new objects and new morphisms in $𝒞$ that we can build out of the given ones by universal constructions in the sense of category theory.

This way of deducing new expressions from given ones is very basic as well as very natural and hence goes by the technical term natural deduction. For every kind of type (every universal construction in category theory) there is, in natural deduction, one set of rules for how to deductively reason about it. This set of rules, in turn, always consists of four kinds of rules, called the

1. type formation rule

2. term introduction rule

3. term elimination rule

4. computation rule.

These are going to be the syntax in type theory of which universal constructions in category theory is the categorical semantics.

In our running example where $𝒞=$ CartSp, the only universal construction available is that of forming products. We therefore introduce now the natural deduction rules by way of example of the special case of product types.

1. type formation rule Let

$$A,B\in 𝒞$$A , B \in \mathcal{C}

be two objects in a category with products. Then there exists the product object

$$A\times B\in 𝒞.$$A \times B \in \mathcal{C} \,.

We now declare that the syntax of which this state of affairs is the categorical semantics is the collection of symbols of the form

$$\frac{A:\mathrm{Type}B:\mathrm{Type}}{A\times B:\mathrm{Type}}.$$\frac{A \colon Type \;\;\;\;\; B \colon Type}{ A \times B \colon Type} \,.

Here on top of the horizontal line we have the two judgments which express that, syntactically, $A$ is a type and $B$ is a type, and semantically that $A\in 𝒞$ and $B\in 𝒞$. Below the horizontal line is, in turn, the judgment which expresses that there is, syntactically, a product type, which semantically is the product $A\times B\in 𝒞$. The horizontal line itself is to indicate that if we are given the (symbols of) the collection of judgments on top, then we are entitled to also obtain the judgment on the bottom.

Remark (Computation) All this may seem, on first sight, like being a lot of fuss about something of grandiose banality. To see what is gradually being accomplished here despite of this appearance, as we proceed in this discussion, the reader can and should think of this as the first steps in the definition of a programming language: the notion of judgment is a syntactic rule for strings of symbols that a computer is to read in, and a natural deduction-step as the type formation rule above is an operation that this computer validates as being an allowed step of transforming a memory state with a given collection of such strings into a new memory state to which the string below the horizontal line is added. As we add the remaining rules below, what looks like a grandiose banality so far will remain grandiose, but no longer be a banality. The reader feeling in need of more motivational remarks along these lines might want to take a break here and have a look at the entry computational trinitarianism first, that provides more pointers to the grandiose picture which we are approaching here.

Next, the second natural deduction rule for product types is the

2. term elimination rule. The fact that $A\times B\in 𝒞$ is equipped with two projection morphisms

$$\begin{array}{c}A\stackrel{p_1}{\leftarrow }A\times B\stackrel{p_2}{\to }B\end{array}$$\array{ A \stackrel{p_1}{\leftarrow} A \times B \stackrel{p_2}{\to} B }

means that from every element $t$ of $A\times B$ we may deduce the existence of elements $p_1(t)$ and $p_2(t)$ of $A$ and $B$, respectively. We declare now that this is the categorical semantics of which the natural deduction syntax is:

$$\frac{⊢t:A\times B}{⊢p_1(t):A}\frac{⊢t:A\times B}{⊢p_2(t):B}.$$\frac{\vdash \; t \colon A \times B}{\vdash \; p_1(t) \colon A} \;\;\;\;\;\;\;\;\; \frac{\vdash \; t \colon A \times B}{\vdash \; p_2(t) \colon B} \,.

As before, this is to say that if syntactically we are given strings of symbols expressing judgments as on the top of these horizontal lines, then we may “naturally deduce” also the judgment of the string of symbols on the bottom of this line.

3. term introduction rule. The first part of the universal property of the product in category theory is that for $Q\in 𝒞$ any other object equipped with morphisms

$$\begin{array}{ccc}& & Q\\ & {}^a\swarrow & & {\searrow }^b\\ A& & & & B\end{array}$$\array{ && Q \\ & {}^{\mathllap{a}}\swarrow && \searrow^{\mathrlap{b}} \\ A && && B }

in $𝒞$, we obtain a canonical morphism

$$Q\to A\times B$$Q \to A \times B

in $𝒞$. This is now declared to be the categorical semantics of which the natural deduction syntax is

$$\frac{⊢a:A⊢b:B}{⊢(a,b):A\times B}.$$\frac{ \vdash\; a \colon A \;\;\;\;\;\; \vdash\; b \colon B }{ \vdash (a,b) \colon A \times B } \,.

With the elements that are the semantics of the terms appearing here made explicit, this is the syntax for a diagram

$$\begin{array}{ccc}& & Q\\ & {}^a\swarrow & {\downarrow }^{(a,b)}& {\searrow }^b\\ A& & A\times B& & B\end{array}.$$\array{ && Q \\ & {}^{\mathllap{a}}\swarrow &\downarrow^{\mathrlap{(a,b)}}& \searrow^{\mathrlap{b}} \\ A && A \times B && B } \,.

4. computation rule. The next part of the universal property of the product in category theory is that the resulting diagram

$$\begin{array}{ccc}& & Q\\ & {}^a\swarrow & \downarrow & {\searrow }^b\\ A& \stackrel{p_1}{\leftarrow }& A\times B& \stackrel{p_2}{\to }& B\end{array}$$\array{ && Q \\ & {}^{\mathllap{a}}\swarrow &\downarrow& \searrow^{\mathrlap{b}} \\ A &\stackrel{p_1}{\leftarrow}& A \times B & \stackrel{p_2}{\to} & B }

is in fact a commuting diagram. Syntactically this is, clearly, the rule that the following identifications of strings of symbols are to be enforced

$$p_1(a,b)=a{p}_2(a,b)=b.$$p_1(a,b) = a \;\;\;\;\;\; p_2(a,b) = b \,.

This concluces the description of the natural deduction about objects, morphisms and products in a category using its type theory syntax. We summarize the dictionary between category theory and type theory discussed so far below.

In the next section we promote our running example category $𝒞$, which admits only very few universal constructions (just products), to a richer category, the sheaf topos over it. That richer category then accordingly comes with a richer syntax of natural deduction inside it, namely with full dependent type theory. This we discuss in the Syn Layer below.

Natural deduction rules for dependent sum types

(…) dependent sum type (…)

Dictionary: type theory / category theory

The dictionary between dependent type theory with product types and category theory of categories with products.

type theory	category theory
syntax	semantics
judgment	diagram
type	object in category
$⊢A:\mathrm{Type}$	$A\in 𝒞$
term	element
$⊢a:A$	$*\stackrel{a}{\to }A$
dependent type	object in slice category
$x:X⊢A(x):\mathrm{Type}$	$\begin{array}{c}A\\ \downarrow \\ X\end{array}\in {𝒞}_{/X}$
term in context	generalized elements/element in slice category
$x:X⊢a(x):A(x)$	$\begin{array}{ccccc}X& & \stackrel{a}{\to }& & A\\ & {}_{{\mathrm{id}}_X}\searrow & & {\swarrow }_{}\\ & & X\end{array}$
$x:X⊢a(x):A$	$\begin{array}{ccccc}X& & \stackrel{({\mathrm{id}}_X,a)}{\to }& & X\times A\\ & {}_{{\mathrm{id}}_X}\searrow & & {\swarrow }_{p_1}\\ & & X\end{array}$

$$

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	substitution…………………….	pullback
	$\frac{x_2:X_2⊢A(x_2):\mathrm{Type}{x}_1:X_1⊢f(x_1):X_2}{x_1:X_1⊢A(f(x_1)):\mathrm{Type}}$	$$ $\begin{array}{ccc}{f}^{*}A& \to & A\\ \downarrow & & \downarrow \\ X_1& \stackrel{f}{\to }& X_2\end{array}$

$$

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	product type	product
type formation	$\frac{⊢A:\mathrm{Type}⊢B:\mathrm{Type}}{⊢A\times B:\mathrm{Type}}$	$A,B\in 𝒞\Rightarrow A\times B\in 𝒞$
term introduction	$\frac{⊢a:A⊢b:B}{⊢(a,b):A\times B}$	$\begin{array}{ccc}& & Q\\ & {}^a\swarrow & {\downarrow }_{(a,b)}& {\searrow }^b\\ A& & A\times B& & B\end{array}$
term elimination	$\frac{⊢t:A\times B}{⊢p_1(t):A}\frac{⊢t:A\times B}{⊢p_2(t):B}$	$\begin{array}{ccc}& & Q\\ & & {\downarrow }^t& & \\ A& \stackrel{p_1}{\leftarrow }& A\times B& \stackrel{p_2}{\to }& B\end{array}$
computation rule	$p_1(a,b)=a{p}_2(a,b)=b$	$\begin{array}{ccc}& & Q\\ & {}^a\swarrow & {\downarrow }_{(a,b)}& {\searrow }^b\\ A& \stackrel{p_1}{\leftarrow }& A\times B& \stackrel{p_2}{\to }& B\end{array}$

$$

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	dependent sum type	dependent sum
type formation	$\frac{⊢X:\mathrm{Type}x:X⊢A(x):\mathrm{Type}}{⊢\left({\sum }_{x:X}A\left(x\right)\right):\mathrm{Type}}$
term introduction	$\frac{x:X⊢a:A(x)}{⊢(x,a):{\sum }_{x\prime :X}A(x\prime )}$
term elimination	$\frac{⊢t:\left({\sum }_{x:X}A\left(x\right)\right)}{⊢p_1(t):X⊢p_2(t):A(p_1(t))}$
computation rule	$p_1(x,a)=x{p}_2(x,a)=a$

Below in Smooth spaces - Syntactic Layer we complete this dictionary to one between dependent type theory with dependent products and toposes.

Smooth spaces

In the section Coordinate systems we have set up the archetypical spaces of differential geometry. Here we now define in terms of these the most general smooth spaces that differential geometry can deal with. We also discuss basic properties of these smooth spaces, all in the Mod Layer.

In the Sem Layer we discuss smooth spaces as a topos and in fact as a cohesive topos. This is essentially the stage on which all of the fellowing developments take place. Or rather, the refinement of this to a higher topos is, which we discuss further below in the chapter Smooth ∞-Groupoids.

Model Layer

In this Model Layer we discuss concretely the definition of smooth spaces and of homomorphisms between them, together with basic examples and properties.

Plots of smooth spaces and their gluing

The general kind of “smooth space” that we want to consider is a something that can be probed by laying out coordinate systems as in def. 4 inside it, and that can be obtained by gluing all the possible coordinate systems in it together.

At this point we want to impose no further conditions on a “space” than this. In particular we do not assume that we know beforehand a set of points underlying $X$. Instead, we define smooth spaces $X$ entirely operationally as something about which we can ask “Which ways are there to lay out ${ℝ}^n$ inside $X$?” and such that there is a self-consistent answer to this question. The following definitions make precise what we mean by this. The reader wishing to see more motivational discussion first might look at conceptual exposition.

For brevity we will refer “a way to lay out a coordinate system in $X$” as a plot of $X$. The first set of consistency conditions on plots of a space is that they respect coordinate transformations. This is what the following definition formalizes.

Definition

A smooth pre-space $X$ is

1. a collection of sets: for each Cartesian space ${ℝ}^n$ (hence for each natural number $n$) a set

   $$X({ℝ}^n)\in \mathrm{Set}$$X(\mathbb{R}^n) \in Set

   – to be thought of as the set of ways of laying out ${ℝ}^n$ inside $X$;

2. for each abstract coordinate transformation, hence for each smooth function $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ a function between the corresponding sets

   $$X(f):X({ℝ}^{n_2})\to X({ℝ}^{n_1})$$X(f) : X(\mathbb{R}^{n_2}) \to X(\mathbb{R}^{n_1})

   – to be thought of as the function that sends a plot of $X$ by ${ℝ}^{n_2}$ to the correspondingly transformed plot by ${ℝ}^{n_1}$ induced by laying out ${ℝ}^{n_1}$ inside ${ℝ}^{n_2}$.

such that this is compatible with coordinate transformations:

1. the identity coordinate transformation does not change the plots:

   $$X({\mathrm{id}}_{{ℝ}^n})={\mathrm{id}}_{X({ℝ}^n)},$$X(id_{\mathbb{R}^n}) = id_{X(\mathbb{R}^n)} \,,

2. changing plots along two consecutive coordinate transformations $f_1:{ℝ}^{n_1}\to {ℝ}^{n_2}$ and $f_2:{ℝ}^{n_2}\to {ℝ}^{n_3}$ is the same as changing them along the composite coordinate transformation $f_2\circ f_1$:

   $$X(f_1)\circ X(f_2)=X(f_2\circ f_1).$$X(f_1) \circ X(f_2) = X(f_2 \circ f_1) \,.

But there is one more consistency condition for a collection of plots to really be probes of some space: it must be true that if we glue small coordinate systems to larger ones, then the plots by the larger ones are the same as the plots by the collection of smaller ones that agree where they overlap. We first formalize this idea of “plots that agree where their coordinate systems overlap.”

Definition

Let $X$ be a smooth pre-space, def. 10. For $\{U_i\to {ℝ}^n{\}}_{i\in I}$ a differentially good open cover, def. 9, let

$$\mathrm{GluedPlots}(\{U_i\to {ℝ}^n\},X)\in \mathrm{Set}$$GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \in Set

be the set of $I$-tuples of $U_i$-plots of $X$ which coincide on all double intersections

$$\begin{array}{ccc}& & U_i\cap U_j\\ & {}^{{\iota }_i}\swarrow & & {\searrow }^{{\iota }_j}\\ U_i& & & & U_j\\ & \searrow & & \swarrow \\ & & {ℝ}^n\end{array}$$\array{ && U_i \cap U_j \\ & {}^{\mathllap{\iota_i}}\swarrow && \searrow^{\mathrlap{\iota_j}} \\ U_i &&&& U_j \\ & \searrow && \swarrow \\ && \mathbb{R}^n }

(also called the matching families of $X$ over the given cover):

$$\mathrm{GluedPlots}(\{U_i\to {ℝ}^n\},X)≔\{{(p_i\in X(U_i))}_{i\in I}\mid {\forall }_{i,j\in I}:X({\iota }_i)(p_i)=X({\iota }_j)(p_j)\}.$$GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \;\;\coloneqq\;\; \left\{ \; \left(p_i \in X(U_i)\right)_{i \in I} \;|\;\; \forall_{i,j \in I} \;:\; X(\iota_i)(p_i) = X(\iota_j)(p_j) \; \right\} \,.

Remark

In def. 11 the equation

$$X({\iota }_i)(p_i)=X({\iota }_j)(p_j)$$X(\iota_i)(p_i) = X(\iota_j)(p_j)

says in words:

The plot $p_i$ of $X$ by the coordinate system $U_i$ inside the bigger coordinate system ${ℝ}^n$ coincides with the plot $p_j$ of $X$ by the other coordinate system $U_j$ inside $X$ when both are restricted to the intersection $U_i\cap U_j$ of $U_i$ with $U_j$ inside ${ℝ}^n$.

Remark

For each differentially good open cover $\{U_i\to X{\}}_{i\in I}$ and each smooth pre-space $X$, def. 10, there is a canonical function

$$X({ℝ}^n)\to \mathrm{GluedPlots}(\{U_i\to {ℝ}^n\},X)$$X(\mathbb{R}^n) \to GluedPlots(\{U_i \to \mathbb{R}^n\}, X)

from the set of ${ℝ}^n$-plots of $X$ to the set of tuples of glued plots, which sends a plot $p\in X({ℝ}^n)$ to its restriction to all the ${\varphi }_i:U_i\hookrightarrow {ℝ}^n$:

$$p\mapsto (X({\varphi }_i)(p){)}_{i\in I}.$$p \mapsto (X(\phi_i)(p))_{i \in I} \,.

If $X$ is supposed to be consistently probable by coordinate systems, then it must be true that the set of ways of laying out a coordinate system ${ℝ}^n$ inside it coincides with the set of ways of laying out tuples of glued coordinate systems inside it, for each good cover $\{U_i\to {ℝ}^n\}$ as above. Therefore:

Definition

A smooth pre-space $X$, def. 10 is a smooth space if for all differentially good open covers $\{U_i\to {ℝ}^n\}$, def. 9, the canonical function of remark 11 from plots to glued plots is a bijection

$$X({ℝ}^n)\stackrel{\simeq }{\to }\mathrm{GluedPlots}(\{U_i\to {ℝ}^n\},X).$$X(\mathbb{R}^n) \stackrel{\simeq}{\to} GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \,.

Remark

We may think of a smooth space as being a kind of space whose local models (in the general sense discussed at geometry) are Cartesian spaces:

while definition 12 explicitly says that a smooth space is something that is consistently probeable by such local models; by a general abstract fact – which we discuss in more detail below in Smooth Spaces - Semantic Layer – that is sometimes called the co-Yoneda lemma it follows in fact that smooth spaces are precisely the objects that are obtained by gluing coordinate systems together.

For instance we will see that two open 2-balls ${ℝ}^2\simeq D^2$ along a common rim yields the smooth space version of the sphere $S^2$, a basic example of a smooth manifold. But before we examine such explicit constructions, we discuss here for the moment more general properties of smooth spaces. The reader instead wishing to see more of these concrete examples at this point should jump ahead to Smooth Spaces - Outlook.

But the following most basic example we consider right now:

Example

For $n\in {ℝ}^n$, there is a smooth space, def. 12, whose set of plots over the abstract coordinate systems ${ℝ}^k$ is the set

$$\mathrm{CartSp}({ℝ}^k,{ℝ}^n)\in \mathrm{Set}$$CartSp(\mathbb{R}^k, \mathbb{R}^n) \in Set

of smooth functions from ${ℝ}^k$ to ${ℝ}^n$.

Clearly this is the rule for plots that characterize ${ℝ}^n$ itself as a smooth space, and so we will just denote this smooth space by the same symbols ”${ℝ}^n$”:

$${ℝ}^n:{ℝ}^k\mapsto \mathrm{CartSp}({ℝ}^k,{ℝ}^n).$$\mathbb{R}^n \colon \mathbb{R}^k \mapsto CartSp(\mathbb{R}^k, \mathbb{R}^n) \,.

In particular the real line $ℝ$ is this way itself a smooth space.

In a moment we find a formal justification for this slight abuse of notation.

Another basic class of examples of smooth spaces are the discrete smooth spaces:

Definition

For $S\in$ Set a set, write

$$\mathrm{Disc}S\in \mathrm{Smooth}0\mathrm{Type}$$Disc S \in Smooth0Type

for the smooth space whose set of $U$-plots for every $U\in \mathrm{CartSp}$ is always $S$.

$$\mathrm{Disc}S:U\mapsto S$$Disc S \colon U \mapsto S

and which sends every coordinate transformation $f:{ℝ}^{n_1}\to {ℝ}^{n_2}$ to the identity function on $S$.

A smooth space of this form we call a discrete smooth space.

More examples of smooth spaces can be built notably by intersecting images of two smooth spaces inside a bigger one. In order to say this we first need a formalization of homomorphism of smooth spaces. This we turn to now.

Homomorphisms of smooth spaces

We discuss “functions” or “maps” between smooth spaces, def. 12, which preserve the smooth space structure in a suitable sense. As with any notion of function that preserves structure, we refer to them as homomorphisms.

The idea of the following definition is to say that whatever a homomorphism $f:X\to Y$ between two smooth spaces is, it has to take the plots of $X$ by ${ℝ}^n$ to a corresponding plot of $Y$, such that this respects coordinate transformations.

Definition

Let $X$ and $Y$ be two smooth spaces, def. 12. Then a homomorphism $f:X\to Y$ is

• for each abstract coordinate system ${ℝ}^n$ (hence for each $n\in \mathbb{N}$) a function

  $f_{{ℝ}^n}:X({ℝ}^n)\to Y({ℝ}^n)$

  that sends ${ℝ}^n$-plots of $X$ to ${ℝ}^n$-plots of $Y$

such that

• for each smooth function $\varphi :{ℝ}^{n_1}\to {ℝ}^{n_2}$ we have

  $$Y(\varphi )\circ f_{{ℝ}^{n_1}}=f_{{ℝ}^{n_2}}\circ X(\varphi )$$Y(\phi) \circ f_{\mathbb{R}^{n_1}} = f_{\mathbb{R}^{n_2}} \circ X(\phi)

  hence a commuting diagram

  $$\begin{array}{ccc}X({ℝ}^{n_1})& \stackrel{f_{{ℝ}^{n_1}}}{\to }& Y({ℝ}^{n_1})\\ {\downarrow }^{X(\varphi )}& & {\downarrow }^{Y(\varphi )}\\ X({ℝ}^{n_2})& \stackrel{f_{{ℝ}^{n_2}}}{\to }& Y({ℝ}^{n_1})\end{array}.$$\array{ X(\mathbb{R}^{n_1}) &\stackrel{f_{\mathbb{R}^{n_1}}}{\to}& Y(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{X(\phi)}} && \downarrow^{\mathrlap{Y(\phi)}} \\ X(\mathbb{R}^{n_2}) &\stackrel{f_{\mathbb{R}^{n_2}}}{\to}& Y(\mathbb{R}^{n_1}) } \,.

For $f_1:X\to Y$ and $f_2:X\to Y$ two homomorphisms of smooth spaces, their composition $f_2\circ f_1:X\to Y$ is defined to be the homomorphism whose component over ${ℝ}^n$ is the composite of functions of the components of $f_1$ and $f_2$:

$$(f_2\circ f_1{)}_{{ℝ}^n}≔{f_2}_{{ℝ}^n}\circ {f_1}_{{ℝ}^n}.$$(f_2\circ f_1)_{\mathbb{R}^n} \coloneqq {f_2}_{\mathbb{R}^n} \circ {f_1}_{\mathbb{R}^n} \,.

Definition

Write $\mathrm{Smooth}0\mathrm{Type}$ for the category whose objects are smooth spaces, def. 12, and whose morphisms are homomorphisms of smooth spaces, def. 14.

At this point it may seem that we have now two different notions for how to lay out a coordinate system in a smooth space $X$: on the hand, $X$ comes by definition with a rule for what the set $X({ℝ}^n)$ of its ${ℝ}^n$-plots is. On the other hand, we can now regard the abstract coordinate system ${ℝ}^n$ itself as a smooth space, by example 5, and then say that an ${ℝ}^n$-plot of $X$ should be a homomorphism of smooth spaces of the form ${ℝ}^n\to X$.

The following proposition says that these two superficially different notions actually naturally coincide.

Proposition

Let $X$ be any smooth space, def. 12, and regard the abstract coordinate system ${ℝ}^n$ as a smooth space, by example 5. There is a natural bijection

$$X({ℝ}^n)\simeq {\mathrm{Hom}}_{\mathrm{Smooth}0\mathrm{Type}}({ℝ}^n,X)$$X(\mathbb{R}^n) \simeq Hom_{Smooth0Type}(\mathbb{R}^n, X)

between the postulated ${ℝ}^n$-plots of $X$ and the actual ${ℝ}^n$-plots given by homomorphism of smooth spaces ${ℝ}^n\to X$.

Proof

This is a special case of the Yoneda lemma, as will be made more explicit below in The topos of smooth spaces. The reader unfamiliar with this should write out the simple proof explicitly: use the defining commuting diagrams in def. 14 to deduce that a homomorphism $f:{ℝ}^n\to X$ is uniquely fixed by the image of the identity element in ${ℝ}^n({ℝ}^n)≔\mathrm{CartSp}({ℝ}^n,{ℝ}^n)$ under the component function $f_{{ℝ}^n}:{ℝ}^n({ℝ}^n)\to X({ℝ}^n)$.

Example

Let $ℝ\in \mathrm{Smooth}0\mathrm{Type}$ denote the real line, regarded as a smooth space by def. 5. Then for $X\in \mathrm{Smooth}0\mathrm{Type}$ any smooth space, a homomorphism of smooth spaces

$$f:X\to ℝ$$f \colon X \to \mathbb{R}

is a smooth function on $X$. Prop. 8 says here that when $X$ happens to be an abstract coordinate system regarded as a smooth space by def. 5, then this general notion of smooth functions between smooth spaces reproduces the basic notion of def, 2.

Definition

The 0-dimensional abstract coordinate system ${ℝ}^0$ we also call the point and regarded as a smooth space we will often write it as

$$*\in \mathrm{Smooth}0\mathrm{Type}.$$* \in Smooth0Type \,.

For any $X\in \mathrm{Smooth}0\mathrm{Type}$, we say that a homomorphism

$$x:*\to X$$x \colon * \to X

is a point of $X$.

Remark

By prop. 8 the points of a smooth space $X$ are naturally identified with its 0-dimensional plots, hence with the “ways of laying out a 0-dimensional coordinate system” in $X$:

$$\mathrm{Hom}(*,X)\simeq X({ℝ}^0).$$Hom(*, X) \simeq X(\mathbb{R}^0) \,.

Products and fiber products of smooth spaces

Definition

Let $X,Y\in \mathrm{Smooth}0\mathrm{Type}$ by two smooth spaces. Their product is the smooth space $X\times Y\in \mathrm{Smooth}0\mathrm{Type}$ whose plots are pairs of plots of $X$ and $Y$:

$$X\times Y({ℝ}^n)≔X({ℝ}^n)\times Y({ℝ}^n)\in \mathrm{Set}.$$X\times Y (\mathbb{R}^n) \coloneqq X(\mathbb{R}^n) \times Y(\mathbb{R}^n) \;\; \in Set \,.

The projection on the first factor is the homomorphism

$$p_1:X\times Y\to X$$p_1 \colon X \times Y \to X

which sends ${ℝ}^n$-plots of $X\times Y$ to those of $X$ by forming the projection of the cartesian product of sets:

$${p_1}_{{ℝ}^n}:X({ℝ}^n)\times Y({ℝ}^n)\stackrel{p_1}{\to }X({ℝ}^n).$${p_1}_{\mathbb{R}^n} \colon X(\mathbb{R}^n) \times Y(\mathbb{R}^n) \stackrel{p_1}{\to} X(\mathbb{R}^n) \,.

Analogously for the projection to the second factor

$$p_2:X\times Y\to Y.$$p_2 \colon X \times Y \to Y \,.

Proposition

Let $*={ℝ}^0$ be the point, regarded as a smooth space, def. 16. Then for $X\in \mathrm{Smooth}0\mathrm{Type}$ any smooth space the canonical projection homomorphism

$$X\times *\to X$$X \times * \to X

is an isomorphism.

Definition

Let $f:X\to Z$ and $g:Y\to Z$ be two homomorphisms of smooth spaces, def. 14. There is then a new smooth space to be denoted

$$X{\times }_{Z}Y\in \mathrm{Smooth}0\mathrm{Type}$$X \times_Z Y \in Smooth0Type

(with $f$ and $g$ understood), called the fiber product of $X$ and $Y$ along $f$ and $g$, and defined as follows:

the set of ${ℝ}^n$-plots of $X{\times }_{Z}Y$ is the set of pairs of plots of $X$ and $Y$ which become the same plot of $Z$ under $f$ and $g$, respectively:

$$(X{\times }_{Z}Y)({ℝ}^n)=\{(p_X\in X({ℝ}^n),p_Y\in Y({ℝ}^n))\mid f_{{ℝ}^n}(p_X)=g_{{ℝ}^n}(p_Y)\}.$$(X \times_Z Y)(\mathbb{R}^n) = \left\{ (p_X \in X(\mathbb{R}^n), p_Y \in Y(\mathbb{R}^n)) \; |\; f_{\mathbb{R}^n}(p_X) = g_{\mathbb{R}^n}(p_Y) \right\} \,.

Smooth mapping spaces and smooth moduli spaces

Definition

Let $\Sigma ,X\in \mathrm{Smooth}0\mathrm{Type}$ be two smooth spaces, def. 12. Then the smooth mapping space

$$[\Sigma ,X]\in \mathrm{Smooth}0\mathrm{Type}$$[\Sigma,X] \in Smooth0Type

is the smooth space defined by saying that its set of ${ℝ}^n$-plots is

$$[\Sigma ,X]({ℝ}^n)≔\mathrm{Hom}(\Sigma \times {ℝ}^n,X).$$[\Sigma, X](\mathbb{R}^n) \coloneqq Hom(\Sigma \times \mathbb{R}^n, X) \,.

Here in $\Sigma \times {ℝ}^n$ we first regard the abstract coordinate system ${ℝ}^n$ as a smooth space by example 5 and then we form the product smooth space by def. 17.

Remark

This means in words that a ${ℝ}^n$-plot of the mapping space $[\Sigma ,X]$ is a smooth ${ℝ}^n$-parameterized family of homomorphisms $\Sigma \to X$.

Proposition

There is a natural bijection

$$\mathrm{Hom}(K,[\Sigma ,X])\simeq \mathrm{Hom}(K\times \Sigma ,X)$$Hom(K, [\Sigma, X]) \simeq Hom(K \times \Sigma, X)

for every smooth space $K$.

Proof

With a bit of work this is straightforward to check explicitly by unwinding the definitions. It follows however from general abstract results once we realize that $[-,-]$ is of course the internal hom of smooth spaces. This we come to below in Smooth spaces - Semantic Layer.

Remark

This says in words that a smooth function from any $K$ into the mapping space $[\Sigma ,X]$ is equivalently a smooth function from $K\times \Sigma$ to $X$. The latter we may regard as a $K$-parameterized smooth family of smooth functions $\Sigma \to X$. Therefore in view of the previous remark 14 this says that smooth mapping spaces have a universal property not just over abstract coordinate systems, but over all smooth spaces.

We will therefore also say that $[\Sigma ,X]$ is the smooth moduli space of smooth functions from $\Sigma \to X$, because it is such that smooth maps $K\to [\Sigma ,X]$ into it modulate, as we move around on $K$, a family of smooth functions $\Sigma \to X$, depending on $K$.

First interesting examples of such smooth moduli spaces are discussed in Differential forms – Model Layer below. Many more interesting examples follow once we pass from smooth 0-types to smooth $n$-types below in Smooth n-groupoids.

We will see many more examples of smooth moduli spaces, starting below in Differential forms - Model Layer.

Proposition

The set of points, def. 16, of a smooth mapping space $[\Sigma ,X]$ is the bare set of homomorphism $\Sigma \to X$: there is a natural isomorphism

$$\mathrm{Hom}(*,[\Sigma ,X])\simeq \mathrm{Hom}(\Sigma ,X).$$Hom(*, [\Sigma, X]) \simeq Hom(\Sigma, X) \,.

Proof

Combine prop. 10 with prop. 9.

Example

Given a smooth space $X\in \mathrm{Smooth}0\mathrm{Type}$, its smooth path space is the smooth mapping space

$$PX≔[{ℝ}^1,X].$$\mathbf{P}X \coloneqq [\mathbb{R}^1, X] \,.

By prop. 11 the points of $PX$ are indeed precisely the smooth trajectories ${ℝ}^1\to X$. But $PX$ also knows how to smoothly vary such smooth trajectories.

This is central for variational calculus which determines equations of motion in physics. This we turn to below in Variational calculus.

Remark

In physics, if $X$ is a model for spacetime, then $PX$ may notably be interpreted as the smooth space of worldlines in $X$, hence the smooth space of paths or trajectories of a particle in $X$.

Example

If in the above example 7 the path is constraind to be a loop in $X$, one obtains the smooth loop space

$$LX≔[S^1,X].$$\mathbf{L}X \coloneqq [S^1, X] \,.

The smooth moduli space of smooth functions

In example 6 we saw that a smooth function on a general smooth space $X$ is a homomorphism of smooth spaces, def. 14

$$f:X\to ℝ.$$f \colon X \to \mathbb{R} \,.

The collection of these forms the hom-set ${\mathrm{Hom}}_{\mathrm{Smooth}0\mathrm{Type}}(X,ℝ)$. But by the discussion in Smooth mapping spaces such hom-sets are naturally refined to smooth spaces themselves.

Definition

For $X\in \mathrm{Smooth}0\mathrm{Type}$ a smooth space, we say that the moduli space of smooth functions on $X$ is the smooth mapping space (def. 19), from $X$ into the standard real line $ℝ$

$$[X,ℝ]\in \mathrm{Smooth}0\mathrm{Type}.$$[X, \mathbb{R}] \in Smooth0Type \,.

We will also denote this by

$$C^{\mathrm{\infty }}(X)≔[X,ℝ],$$\mathbf{C}^\infty(X) \coloneqq [X, \mathbb{R}] \,,

since in the special case that $X$ is a Cartesian space this is the smooth refinement of the set $C^{\mathrm{\infty }}(X)$ of smooth functions, def. 1, on $X$.

Remark

We call this a moduli space because by prop. 10 above and in the sense of remark 15 it is such that smooth functions into it modulate smooth functions $X\to ℝ$.

By prop. 11 a point $*\to [X,{ℝ}^1]$ of the moduli space is equivalently a smooth function $X\to {ℝ}^1$.

Outlook

Remark

Later we define/see the following:

• A smooth manifold is a smooth space that is locally equivalent to a coordinate system;

• A diffeological space is a smooth space such that every coordinate labels a point in the space. In other words, a diffeological space is a smooth space that has an underlying set $X_s\in \mathrm{Set}$ of points such that the set of ${ℝ}^n$-plots is a subset of the set of all functions:

  $$X({ℝ}^n)\hookrightarrow \mathrm{Functions}({ℝ}^n,S_s).$$X(\mathbb{R}^n) \hookrightarrow Functions(\mathbb{R}^n, S_s) \,.

  (This need not be the case in a general smooth space, important counterexamples are the universal smooth moduli spaces of differential forms in Smooth moduli space of differential forms).

We will establish a long sequence of faithful inclusions

$\{$coordinate systems$\}$ $\hookrightarrow$ $\{$smooth manifolds$\}$ $\hookrightarrow$ $\{$diffeological spaces$\}$ $\hookrightarrow$ $\{$smooth spaces$\}$ $\hookrightarrow$ $\{$smooth groupoids$\}$ $\hookrightarrow \cdots$

This ends the Model-layer discussion of smooth spaces. We now pass to a more advanced discussion of this topic in the Semantics layer below. The reader wishing to stick to more elementary discussion for the time being should skip ahead to the Model-layer discussion of Differential forms below.

Semantic Layer

In this Semantic Layer we mention some basic definitions of topos theory and discuss the topos formed by the smooth spaces defined in Smooth Spaces - Mayer Mod.

Toposes

Definition

A sheaf topos is subtopos of presheaf topos

$$\mathrm{Sh}(C)\stackrel{\stackrel{\overline{(-)}}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C).$$Sh(C) \stackrel{\overset{\overline{(-)}}{\leftarrow}}{\hookrightarrow} PSh(C) \,.

Remark

Here the left adjoint $\overline{(-)}$, which is equivalently

• the inverse image of the geometric embedding geometric morphism

• the sheafification functor

is precisely such that for every covering $\{U_i\to U{\}}_{i\in I}$ in the site $C$ the sieve inclusion

$$S(\{U_i\})\hookrightarrow U\in \mathrm{PSh}(C)$$S(\{U_i\}) \hookrightarrow U \;\;\;\; \in PSh(C)

(a dense monomorphism) is sent to an isomorphism.

Hence the sheaf topos is the left exact localization at the covering sieve inclusions.

Remark

The presheaf topos $\mathrm{PSh}(C)$ is the free cocompletion of the category $C$, hence the category obtained from $C$ by freely forming colimits of its objects.

In contrast, the localization $\mathrm{Sh}(C)\hookrightarrow \mathrm{PSh}(C)$ enforces relations between these free colimits.

Therefore in total we may think of $\mathrm{Sh}(C)$ as obtained by generators and relations from the site $C$:

• the objects of $C$ are the generators;

• the coverings of $C$ are the relations.

Def. 21 is the “external” characterization of sheaf toposes.

More intrinsically we have Giraud's theorem:

Theorem

A sheaf topos is equivalently a presentable category with

1. universal colimits

2. effective quotients

3. disjoint coproducts

This characterization, or rather its refinement to (infinity,1)-categories of (infinity,1)-sheaves, is crucial, below, for the discussion of principal bundles and their associated fiber bundles.

For other general considerations we need rather that every sheaf topos is in particular an elementary topos

Definition

An elementary topos is

• a locally cartesian closed category

• with a subobject classifier.

The first of these says that the internal language is dependent type theory with dependent sum types and dependent product types, the second says that its has a type of propositions. This we turn to in Smooth Spaces - Syntactic Layer below.

Subobjects

• subobject, subobject classifier

• (-1)-truncation

Slice toposes

• slice topos

• etale geometric morphism

Local, connected and cohesive toposes

• local topos

  • concrete object

• locally connected topos, connected topos

  • connected object

• cohesive topos

The topos of smooth spaces

Remark

• A smooth pre-space, def. 10 is equivalently a presheaf on CartSp, hence a functor $X:{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Set}$;

• a smooth space, def. 12, is equivalently a presheaf on CartSp which is a sheaf with respect to the differentially good open cover-coverage:

  • the set of “glued plots”, def. 11 is the descent object;

  • the canonical morphism of remark 11 is the descent morphism;

  • the condition that it be an isomorphism (for each differentially good open cover) is the descent- or sheaf condition.

The sheaf topos, def. 21, of smooth spaces enjoys a few crucial properties: it is

• a locally connected and connected topos

• a local topos

• a cohesive topos. This we discuss in the remainder of this Semantic Layer.

Connectedness

The full sheaf topos $\mathrm{Sh}(\mathrm{CartSp})$ on CartSp is a locally connected topos in that the terminal global section geometric morphism to Set is an essential geometric morphism:

$$\mathrm{Sh}(\mathrm{CartSp})\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{L\mathrm{Const}}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{Set}$$Sh(CartSp) \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{L Const}{\leftarrow}}{\underset{\Gamma}{\to}}} Set

The extra left adjoint ${\Pi }_0:\mathrm{Sh}(\mathrm{CartSp})\to \mathrm{Set}$ sends diffeological spaces to the set of path-connected components of their underlying topological spaces.

Proposition

The sheaf topos $\mathrm{Sh}(\mathrm{CartSp})$ on CartSp is a locally connected topos.

Proof

The following argument works for every site $C$ which is such that constant presheaves on $C$ are already sheaves.

Notice that this is the case for $C=\mathrm{CartSp}$ because every Cartesian space is connected: for $S\in \mathrm{Set}$ a compatible family of elements of $\mathrm{Const}S$ on a cover $\{U_i\to {ℝ}^n\}$ of some ${ℝ}^n$ is an element of $S$ on each patch, such that their restriction maps to intersections of patches coincide. But the restriction maps are all identities, so this says that all these elements coincide. Therefore the set of compatible families is just the set $S$ itself, hence the presheaf $\mathrm{Const}S$ is a sheaf.

So with $L:\mathrm{PSh}(C)\to \mathrm{Sh}(C)$ the sheafification functor we have that $L\mathrm{Const}S\simeq \mathrm{Const}S$.

Whenever this is the case the left adjoint to the constant presheaf functor, which always exists for presheaves and is given by the colimit functor, is also left adjoint on the level of sheaves, because for each $X\in \mathrm{Sh}(C)$ and $S\in \mathrm{Set}$ we have natural bijections

$$\begin{array}{rl}{\mathrm{Hom}}_{\mathrm{Sh}(C)}(X,L\mathrm{Const}S)& ={\mathrm{Hom}}_{\mathrm{PSh}(C)}(X,L\mathrm{Const}S)\\ & \simeq {\mathrm{Hom}}_{\mathrm{PSh}(C)}(X,\mathrm{Const}S)\\ & \simeq {\mathrm{Hom}}_{\mathrm{Set}}(\underrightarrow{\lim }X,S)\end{array}.$$\begin{aligned} Hom_{Sh(C)}(X, L Const S) & = Hom_{PSh(C)}(X, L Const S) \\ & \simeq Hom_{PSh(C)}(X, Const S) \\ & \simeq Hom_{Set}(\lim_\to X, S) \end{aligned} \,.

Definition

Write ${\Pi }_0:={\lim }_{\to }:\mathrm{Sh}(\mathrm{CartSp})\to \mathrm{Set}$ for the left adjoint to $\mathrm{LConst}:\mathrm{Set}\stackrel{\mathrm{Const}}{\to }\mathrm{PSh}(C)\stackrel{L}{\to }\mathrm{Sh}(C)$.

Proposition

For $X\in \mathrm{Sh}(C)$ a diffeological space, ${\Pi }_0(X)$ is the set of path-connected components of the topological space underlying $X$.

Proof

By the co-Yoneda lemma we may write

$$X={\underrightarrow{\lim }}_{(U\to X)\in y/X}U$$X = {\lim_\to}_{(U \to X) \in y/X} U

and since ${\Pi }_0$ commutes with colimits we have

$${\Pi }_0(X)\simeq {\Pi }_0{\underrightarrow{\lim }}_{(U\to X)}U\simeq {\underrightarrow{\lim }}_{(U\to X)}{\Pi }_0(U).$$\Pi_0(X) \simeq \Pi_0 {\lim_\to}_{(U \to X)} U \simeq {\lim_\to}_{(U \to X)} \Pi_0(U) \,.

But also by the co-Yoneda lemma we have that the colimit over any representable is the singleton set, hence our expression

$$\cdots \simeq {\underrightarrow{\lim }}_{(U\to X)}*$$\cdots \simeq {\lim_\to}_{(U \to X)} *

is the colimit over the category of plots of $X$ of the functor that is constant on the point. This colimit is the coproduct of points over the connected components of the diagram category.

The connected components of the category of plots $y/X$ are the path-connected (or “plot-connected”) components of the underlying topological space of $X$.

Proposition

The sheaf topos $\mathrm{Sh}(\mathrm{CartSp})$ on CartSp is actually a connected topos.

Proof

Since $\mathrm{CartSp}$ is a connected category it is immediate that $\mathrm{Const}:\mathrm{Set}\to \mathrm{PSh}(\mathrm{CartSp})$ is a full and faithful functor. By the above this equals $L\mathrm{Const}$, which is hence also full and faithful.

By the discussion at connected topos we could equivalently convince ourselves that ${\Pi }_0$ preserves the terminal object. The terminal object of $\mathrm{Sh}(\mathrm{CartSp})$ is $y({ℝ}^0)$, hence representable. By the above, ${\Pi }_0$ sends all representable objects to the singleton set, which is the terminal object of $\mathrm{Set}$.

Locality

Proposition

The sheaf topos $\mathrm{Sh}(\mathrm{CartSp})$ is also a local topos

$$(L\mathrm{Const}⊣\Gamma ⊣\mathrm{CoDisc}):\mathrm{Sh}(\mathrm{CartSp})\stackrel{\stackrel{L\mathrm{Const}}{\leftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{CoDisc}}{\leftarrow }}}\mathrm{Set}$$(L Const \dashv \Gamma \dashv CoDisc) : Sh(CartSp) \stackrel{\overset{L Const}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{CoDisc}{\leftarrow}}} Set

Proof

The site CartSp is a local site: it has a terminal object and the only covering sieve of this object is the trivial one. This implies the claim, by the discussion at local site.

Concretely, the extra right adjoint $\mathrm{CoDisc}$ takes a set $S$ to the presheaf given by the assigmnent

$$\mathrm{CoDisc}(S):U\mapsto {\mathrm{Hom}}_{\mathrm{Set}}(\mathrm{CartSp}(*,U),S),$$CoDisc(S) : U \mapsto Hom_{Set}(CartSp(*,U), S) \,,

that takes a Cartesian space $U$ to the set of functions from its underlying set of points to $S$. This is clearly a sheaf (a function of sets from $U$ to $S$ is clearly fixed by all its restrictions to a collections of subsets of $U$ whose unition is $U$.)

Geometrically, the object $\mathrm{CoDisc}S\in \mathrm{Sh}(\mathrm{CartSp})$ is the diffeological space codiscrete (indiscrte) smooth structure.

Proposition

Every local topos comes with its notion of concrete sheaves that form a sub-quasitopos. For the local topos $\mathrm{Sh}(\mathrm{CartSp})$ these are precisely the diffeological spaces.

$$\mathrm{Set}\overleftarrow{\underset{\mathrm{CoDisc}}{\hookrightarrow }}\mathrm{DiffologicalSp}\overleftarrow{\hookrightarrow }\mathrm{Sh}(\mathrm{CartSp})$$Set \stackrel{\leftarrow}{\underset{CoDisc}{\hookrightarrow}} DiffologicalSp \stackrel{\leftarrow}{\hookrightarrow} Sh(CartSp)

Proof

The concrete sheaves for the local topos $\mathrm{Sh}(\mathrm{CartSp})$ are by definition those objects $X$ for which the $(\Gamma ⊣\mathrm{CoDisc})$-unit

$$X\to \mathrm{CoDisc}\Gamma X$$X \to CoDisc \Gamma X

is a monomorphism. Monomorphisms of sheaves are tested objectwise, so that means equivalently that for every $U\in \mathrm{CartSp}$ we have that

$$X(U)\simeq {\mathrm{Hom}}_{\mathrm{Sh}}(U,X)\to {\mathrm{Hom}}_{\mathrm{Sh}}(U,\mathrm{Codisc}\Gamma X)\simeq {\mathrm{Hom}}_{\mathrm{Set}}(\Gamma U,\Gamma X)$$X(U) \simeq Hom_{Sh}(U,X) \to Hom_{Sh}(U, Codisc \Gamma X) \simeq Hom_{Set}(\Gamma U, \Gamma X)

is a monomorphism. This is precisely the condition on a sheaf to be a diffeological space.

Cohesion

Proposition

The sheaf topos $\mathrm{Sh}(\mathrm{CartSp})$ is even a cohesive topos in which the axiom pieces have points holds.

Proof

The site CartSp is a cohesive site (see there for detail). This implies the statement.

This implies that $\mathrm{Sh}(\mathrm{CartSp})$ is a locally connected topos, connected topos, local topos. It means in addition that it is also a strongly connected topos.

This means that there is a homotopy category or concordance category of smooth spaces, with the same objects as $\mathrm{Sh}(\mathrm{CartSp})$, but with hom-sets given by

$$\mathrm{Conc}(X,Y):={\Pi }_0[X,Y{]}_{\mathrm{Sh}(\mathrm{CartSp})},$$Conc(X,Y) := \Pi_0 [X,Y]_{Sh(CartSp)} \,,

where $[X,Y{]}_{\mathrm{Sh}(\mathrm{CartSp})}$ is the internal hom in the cartesian closed category $\mathrm{Sh}(\mathrm{CartSp})$.

Syntactic Layer

In this Syntactic Layer we discuss the two further aspects that the internal language of a topos adds to the internal language of a just a category with finite products (which is the dependent type theory with unit type and product type discussed in Coordinate systems - Syntactic Layer): dependent product types and a type of propositions.

dependent type theory $\leftrightarrow$ locally cartesian closed category

type of propositions $\leftrightarrow$ subobject classifier

(…)

Type theory of a topos

Natural deduction rules for dependent product types

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	dependent product type	dependent product
type formation	$\frac{⊢X:\mathrm{Type}x:X⊢A(x):\mathrm{Type}}{⊢\left({\prod }_{x:X}A\left(x\right)\right):\mathrm{Type}}$
term introduction	$\frac{x:X⊢a\left(x\right):A\left(x\right)}{⊢(x\mapsto a\left(x\right)):{\prod }_{x\prime :X}A(x\prime )}$
term elimination	$\frac{⊢f:\left({\prod }_{x:X}A\left(x\right)\right)⊢x:X}{x:X⊢f(x):A(x)}$
computation rule	$(y\mapsto a(y))(x)=a(x)$

In the special case that $A$ does not actually deopend on $X$:

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	function type	internal hom
type formation	$\frac{⊢X:\mathrm{Type}⊢A:\mathrm{Type}}{⊢(X\to A):\mathrm{Type}}$
term introduction	$\frac{x:X⊢a(x):A}{⊢(x\mapsto a\left(x\right)):(X\to A)}$
term elimination	$\frac{⊢f:(X\to A)⊢x:X}{x:X⊢f(x):A}$
computation rule	$(y\mapsto a(y))(x)=a(x)$

Internal logic of a topos

What is called logic is the syntax for (-1)-truncated objects in slice categories, hence of monomorphisms regarded as objects of slice categories.

(…)

• propositions as types, programs as proofs

• computational trinitarianism

• type of propositions

Modal type theory

Modalities

• modal logic, modal type theory

• monad (in computer science)

Sharp modality and Codiscrete types

Definition

Let $H$ be a local topos

$$H\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{CoDisc}}{\leftarrow }}}.$$\mathbf{H} \stackrel{\stackrel{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{CoDisc}{\leftarrow}}} \,.

Write

$$(♭⊣♯):H\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\underset{\Gamma }{\to }}.\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{coDisc}}{\to }}.$$(\flat \dashv \sharp) \; \colon \; \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} \, . \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \, .

for the induced adjoint monad $♯$ and comonad $♭$. We

• call $♯$ the sharp modality;

• call $♭$ the flat modality.

The term modality refers to modal logic. (…)

Remark

We refer to $♯A$ as the sharp type of $A$. This may be thought of as referring to the fact that by adjunction a homomorphism $X\to ♯A$ is equivalently a function $\Gamma X\to \Gamma A$ of the underlying sets. This means that smooth maps $X\to ♯A$ are like maps into $A$ that do not have to respect the cohesive structure on $A$, but instead can be arbitrarily “kinky” (“sharp”).

This motivates the following terminology.

Definition

We write

$${\mathrm{DeCoh}}_X:X\to ♯X$$DeCoh_X \colon X \to \sharp X

For the unit of the $(\Gamma ⊢\mathrm{coDisc})$-adjunction.

(…)

Differential forms

We introduce the standard concept of differential forms in Model Layer, adding to the traditional discussion a precise version of the statement that differential forms are equivalently “incremental smooth $n$-dimensional measures”, which accurately captures the role that they play in physics, notably in local action functionals.

We define differential forms on general smooth spaces seamlessly in terms of the smooth moduli space ${\Omega }^{\in }\in \mathrm{Smooth}0\mathrm{Type}$ of differential forms. This has the special property that it is, for $n\ge 1$, a non-concrete smooth space. In Semantic Layer below we take this as occasion to discuss the notion of concrete objects in a local topos, such as the topos of smooth spaces. We show how the concretification of the smooth mapping space $[X,{\Omega }^n]$ for any smooth space $X$ is the smooth (moduli) space of differential forms on $X$. Below in Action functionals for Chern-Simons type gauge theories the theory of concretification in a local topos is a central ingredient in the canonical existence of certain action functionals.

The process of concretification involves the general abstract notion of images. The type-theory of this notion we discuss in Syntactic Layer here.

Model Layer

We have seen above in The continuum real (world-)line that that real line $ℝ$ is the basic kinematical structure in the differential geometry of physics. Notably the smooth path spaces $[ℝ,X]$ from example 7 are to be thought of as the smooth spaces of trajectories (for instance of some particle) in a smooth space $X$, hence of smooth maps $ℝ\to X$.

But moreover, dynamics in physics is encoded by functionals on such trajectories: by “action functionals”. In the simplest case these are for instance homomorphisms of smooth spaces

$$S:[I,X]\to ℝ,$$S \colon \left[I, X\right] \to \mathbb{R} \,,

where $I\hookrightarrow ℝ$ is the standard unit interval.

Such action functionals we discuss in their own right in Variational calculus below. Here we first examine in detail a fundamental property they all have: they are supposed to be local.

Foremost this means that the value associated to a trajectory is built up incrementally from small contributions associated to small sub-trajectories: if a rajectory $\gamma$ is decomposed as a trajectory ${\gamma }_1$ followed by a trajectory ${\gamma }_2$, then the action functional is additive

$$S(\gamma )=S({\gamma }_1)+S({\gamma }_2).$$S(\gamma) = S(\gamma_1) + S(\gamma_2) \,.

As one takes this property to the limit of iterative subdivision, one finds that action functionals are entirely determined by their value on infinitesimal displacements along the worldline. If $\gamma :ℝ\to X$ denotes a path and ”$\dot{\gamma }(x)$” denotes the corresponding “infinitesimal path” at worldline parameter $x$, then the value of the action functional on such an infinitesimal path is traditionally written as

$$dS(\dot{\gamma }{)}_x\in ℝ,$$\mathbf{d}S(\dot \gamma)_x \in \mathbb{R} \,,

to be read as “the small change $dS$ of $S$ along the infinitesimal path ${\dot{\gamma }}_x$”.

This function $dS$ that assigns numbers to infinitesimal paths is called a differential form. Etymologically this originates in the use of “form” as in bilinear form: something that is evaluated. Here it is evaluated on infinitesimal differences, referred to as differentials.

We define smooth differential forms on Cartesian spaces in

• Differential forms on abstract coordinate sysetms

Then we discuss how this induces a notion of smooth differential forms on general smooth spaces in

• Smooth universal moduli space of differential forms.

Further below we provide a precise version of the statement that “Differential 1-forms are differential measures along paths.” in

• Differential 1-forms are smooth incremental path measures.

Differential forms on abstract coordinate systems

We introduce the basic concept of a smooth differential form on a Cartesian space ${ℝ}^n$. Below in Differential forms on smooth spaces we use this to define differential forms on any smooth space.

Definition

For $n\in \mathbb{N}$ a smooth differential 1-form $\omega$ on a Cartesian space ${ℝ}^n$ is an $n$-tuple

$${({\omega }_i\in \mathrm{CartSp}({ℝ}^n,ℝ))}_{i=1}^n$$\left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n

of smooth functions, which we think of equivalently as the coefficients of a formal linear combination

$$\omega =\sum _{i=1}^n{f}_{i}d{x}^i$$\omega = \sum_{i = 1}^n f_i \mathbf{d}x^i

on a set $\{d{x}^1,d{x}^2,\cdots ,d{x}^n\}$ of cardinality $n$.

Write

$${\Omega }^1({ℝ}^k)\simeq \mathrm{CartSp}({ℝ}^k,ℝ{)}^{\times k}\in \mathrm{Set}$$\Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set

for the set of smooth differential 1-forms on ${ℝ}^k$.

Remark

We think of $d{x}^i$ as a measure for infinitesimal displacements along the $x^i$-coordinate of a Cartesian space. This idea is made precise below in Differential 1-forms are smooth increnemental path measures.

If we have a measure of infintesimal displacement on some ${ℝ}^n$ and a smooth function $f:{ℝ}^{\tilde{n}}\to {ℝ}^n$, then this induces a measure for infinitesimal displacement on ${ℝ}^{\tilde{n}}$ by sending whatever happens there first with $f$ to ${ℝ}^n$ and then applying the given measure there. This is captured by the following definition.

Definition

For $\varphi :{ℝ}^{\tilde{k}}\to {ℝ}^k$ a smooth function, the pullback of differential 1-forms along $\varphi$ is the function

$${\varphi }^{*}:{\Omega }^1({ℝ}^k)\to {\Omega }^1({ℝ}^{\tilde{k}})$$\phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})

between sets of differential 1-forms, def. 26, which is defined on basis-elements by

$${\varphi }^{*}d{x}^i≔\sum _{j=1}^{\tilde{k}}\frac{\partial {\varphi }^i}{\partial {\tilde{x}}^j}d{\tilde{x}}^j$$\phi^* \mathbf{d} x^i \coloneqq \sum_{j = 1}^{\tilde k} \frac{\partial \phi^i}{\partial \tilde x^j} \mathbf{d}\tilde x^j

and then extended linearly by

$$\begin{array}{rl}{\varphi }^{*}\omega & ={\varphi }^{*}\left(\sum _i{\omega }_{i}d{x}^i\right)\\ & ≔\sum _{i=1}^k{\left({\varphi }^{*}\omega \right)}_i\sum _{j=1}^{\tilde{k}}\frac{\partial {\varphi }^i}{\partial {\tilde{x}}^j}d{\tilde{x}}^j\\ & =\sum _{i=1}^k\sum _{j=1}^{\tilde{k}}({\omega }_i\circ \varphi )\cdot \frac{\partial {\varphi }^i}{\partial {\tilde{x}}^j}d{\tilde{x}}^j\end{array}.$$\begin{aligned} \phi^* \omega & = \phi^* \left( \sum_{i} \omega_i \mathbf{d}x^i \right) \\ & \coloneqq \sum_{i = 1}^k \left(\phi^* \omega\right)_i \sum_{j = 1}^{\tilde k} \frac{\partial \phi^i }{\partial \tilde x^j} \mathbf{d} \tilde x^j \\ & = \sum_{i = 1}^k \sum_{j = 1}^{\tilde k} (\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j} \mathbf{d} \tilde x^j \end{aligned} \,.

Remark

The term “pullback” in pullback of differential forms is not really related, certainly not historically, to the term pullback in category theory. One can relate the pullback of differential forms to categorical pullbacks, but this is not really essential here. The most immediate property that both concepts share is that they take a morphism going in one direction to a map between structures over domain and codomain of that morphism which goes in the other direction, and in this sense one is “pulling back structure along a morphism” in both cases.

Even if in the above definition we speak only about the set ${\Omega }^1({ℝ}^k)$ of differential 1-forms, this set naturally carries further structure.

Definition

1. The set ${\Omega }^1({ℝ}^k)$ is naturally an abelian group with addition given by componentwise addition

   $$\begin{array}{rl}\omega +\lambda & =\sum _{i=1}^k{\omega }_{i}d{x}^i+\sum _{j=1}^k{\lambda }_{j}d{x}^j\\ & =\sum _{i=1}^k({\omega }_i+{\lambda }_i)d{x}^j\end{array},$$\begin{aligned} \omega + \lambda & = \sum_{i = 1}^k \omega_i \mathbf{d}x^i + \sum_{j = 1}^k \lambda_j \mathbf{d}x^j \\ & = \sum_{i = 1}^k(\omega_i + \lambda_i) \mathbf{d}x^j \end{aligned} \,,

2. The abelian group ${\Omega }^1({ℝ}^k)$ is naturally equipped with the structure of a module over the ring $C^{\mathrm{\infty }}({ℝ}^k,ℝ)=\mathrm{CartSp}({ℝ}^k,ℝ)$ of smooth functions, where the action $C^{\mathrm{\infty }}({ℝ}^k,ℝ)\times {\Omega }^1({ℝ}^k)\to {\Omega }^1({ℝ}^k)$ is given by componentwise multiplication

   $$f\cdot \omega =\sum _{i=1}^k(f\cdot {\omega }_i)d{x}^i.$$f \cdot \omega = \sum_{i = 1}^k( f \cdot \omega_i) \mathbf{d}x^i \,.

Remark

More abstractly, this just says that ${\Omega }^1({ℝ}^k)$ is the free module over $C^{\mathrm{\infty }}({ℝ}^k)$ on the set $\{d{x}^i{\}}_{i=1}^k$.

The following definition captures the idea that if $d{x}^i$ is a measure for displacement along the $x^i$-coordinate, and $d{x}^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way te get a measure, to be called $d{x}^i\wedge d{x}^j$, for infinitesimal surfaces (squares) in the $x^i$-$x^j$-plane. And this should keep track of the orientation of these squares, whith

$$d{x}^j\wedge d{x}^i=-d{x}^i\wedge d{x}^j$$\mathbf{d}x^j \wedge \mathbf{d}x^i = - \mathbf{d}x^i \wedge \mathbf{d} x^j

being the same infinitesimal measure with orientation reversed.

Definition

For $k,n\in \mathbb{N}$, the smooth differential forms on ${ℝ}^k$ is the exterior algebra

$${\Omega }^{•}({ℝ}^k)≔{\wedge }_{C^{\mathrm{\infty }}({ℝ}^k)}^{•}{\Omega }^1({ℝ}^k)$$\Omega^\bullet(\mathbb{R}^k) \coloneqq \wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k)

over the ring $C^{\mathrm{\infty }}({ℝ}^k)$ of smooth functions of the module ${\Omega }^1({ℝ}^k)$ of smooth 1-forms, prop. 28.

We write ${\Omega }^n({ℝ}^k)$ for the sub-module of degree $n$ and call its elements the smooth differential n-forms.

Remark

Explicitly this means that a differential n-form $\omega \in {\Omega }^n({ℝ}^k)$ on ${ℝ}^k$ is a formal linear combination over $C^{\mathrm{\infty }}({ℝ}^k)$ of basis elements of the form $d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_n}$ for $i_1<i_2<\cdots <i_n$:

$$\omega =\sum _{1\le i_1<i_2<\cdots <i_n<k}{\omega }_{i_1,\cdots ,i_n}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_n}.$$\omega = \sum_{1 \leq i_1 \lt i_2 \lt \cdots \lt i_n \lt k} \omega_{i_1, \cdots, i_n} \mathbf{d}x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n} \,.

Definition

The pullback of differential 1-forms of def. 26 extends as an $C^{\mathrm{\infty }}({ℝ}^k)$-algebra homomorphism to ${\Omega }^n(-)$, given for a smooth function $f:{ℝ}^{\tilde{k}}\to {ℝ}^k$ on basis elements by

$$f^{*}(d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_n})=(f^{*}d{x}^{i_1}\wedge \cdots \wedge f^{*}d{x}^{i_n}).$$f^* \left( \mathbf{d}x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n} \right) = \left(f^* \mathbf{d}x^{i_1} \wedge \cdots \wedge f^* \mathbf{d}x^{i_n} \right) \,.

Differential forms on smooth spaces

Above we have defined differential $n$-form on abstract coordinate systems. Here we extend this definition to one of differential $n$-forms on arbitrary smooth spaces. We start by observing that the space of all differential $n$-forms on cordinate systems themselves naturally is a smooth space.

Proposition

The assignment of differential $n$-forms

$${\Omega }^n(-):{ℝ}^k\mapsto {\Omega }^n({ℝ}^k)$$\Omega^n(-) \colon \mathbb{R}^k \mapsto \Omega^n(\mathbb{R}^k)

of def. 30 together with the pullback of differential forms-functions of def. 31

$$\begin{array}{ccc}{ℝ}^{k_1}& \mapsto & {\Omega }^1({ℝ}^{k_1})\\ {\uparrow }^f& & {\downarrow }^{f^{*}}\\ {ℝ}^{k_2}& \mapsto & {\Omega }^1({ℝ}^{k_2})\end{array}$$\array{ \mathbb{R}^{k_1} &\mapsto & \Omega^1(\mathbb{R}^{k_1}) \\ \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f^*}} \\ \mathbb{R}^{k_2} &\mapsto& \Omega^1(\mathbb{R}^{k_2}) }

defines a smooth space in the sense of def. 12:

$${\Omega }^n(-)\in \mathrm{Smooth}0\mathrm{Type}.$$\Omega^n(-) \in Smooth0Type \,.

Definition

We call this

$${\Omega }^n:\mathrm{Smooth}0\mathrm{Type}$$\Omega^n \colon Smooth0Type

the universal smooth moduli space of differential $n$-forms.

The reason for this terminology is that homomorphisms of smooth spaces into ${\Omega }^1$ modulate differential $n$-forms on their domain, by prop. 8 (and hence by the Yoneda lemma):

Example

For the Cartesian space ${ℝ}^k$ regarded as a smooth space by example 5, there is a natural bijection

$${\Omega }^n({ℝ}^k)\simeq \mathrm{Hom}({ℝ}^k,{\Omega }^1)$$\Omega^n(\mathbb{R}^k) \simeq Hom(\mathbb{R}^k, \Omega^1)

between the set of smooth $n$-forms on ${ℝ}^n$ according to def. 26 and the set of homomorphism of smooth spaces, ${ℝ}^k\to {\Omega }^1$, according to def. 14.

In view of this we have the following elegant definition of smooth $n$-forms on an arbitrary smooth space.

Definition

For $X\in \mathrm{Smooth}0\mathrm{Type}$ a smooth space, def. 12, a differential n-form on $X$ is a homomorphism of smooth spaces of the form

$$\omega :X\to {\Omega }^n(-).$$\omega \colon X \to \Omega^n(-) \,.

Accordingly we write

$${\Omega }^n(X)≔\mathrm{Smooth}0\mathrm{Type}(X,{\Omega }^n)$$\Omega^n(X) \coloneqq Smooth0Type(X,\Omega^n)

for the set of smooth $n$-forms on $X$.

We may unwind this definition to a very explicit description of differential forms on smooth spaces. This we do in a moment in remark 26.

Notice that differential 0-forms are equivalently smooth $ℝ$-valued functions.

Proposition

${\Omega }^0\simeq ℝ$

Definition

For $f:X\to Y$ a homomorphism of smooth spaces, def. 14, the pullback of differential forms along $f$ is the function

$$f^{*}:{\Omega }^n(Y)\to {\Omega }^n(X)$$f^* \colon \Omega^n(Y) \to \Omega^n(X)

given by the hom-functor into the smooth space ${\Omega }^n$ of def. 32:

$$f^{*}≔\mathrm{Hom}(-,{\Omega }^n).$$f^* \coloneqq Hom(-, \Omega^n) \,.

This means that it sends an $n$-form $\omega \in {\Omega }^n(Y)$ which is modulated by a homomorphism $Y\to {\Omega }^n$ to the $n$-form $f^{*}\omega \in {\Omega }^n(X)$ which is modulated by the composite $X\stackrel{f}{\to }Y\to {\Omega }^n$.

Proposition

For $X={ℝ}^{\tilde{k}}$ and $Y={ℝ}^k$ definition 34 reproduces def. 31.

Proof

Again by the Yoneda lemma.

Remark

Using def. 34

Unwinding def. 33 yields the following explicit description:

a differential $n$-form $\omega \in {\Omega }^n(X)$ on a smooth space $X$ is

1. for each way $\varphi :{ℝ}^k\to X$ of laying out a coordinate system ${ℝ}^k$ in $X$ a differential $n$-form

   $${\varphi }^{*}\omega \in {\Omega }^n({ℝ}^k)$$\phi^* \omega \in \Omega^n(\mathbb{R}^k)

   on the abstract coordinate system, as given by def. 30;

2. for each abstract coordinate transformation $f:{ℝ}^{k_2}\to {ℝ}^{k_1}$ a corresponding compatibility condition between local differential forms ${\varphi }_1:{ℝ}^{k_1}\to X$ and ${\varphi }_2:{ℝ}^{k_2}\to X$ of the form

   $$f^{*}{\varphi }_1^{*}\omega ={\varphi }_2^{*}\omega .$$f^* \phi_1^* \omega = \phi_2^* \omega \,.

Hence a differential form on a smooth space is simply a collection of differential forms on all its coordinate systems such that these glue along all possible coordinate transformations.

The following adds further explanation to the role of ${\Omega }^n\in \mathrm{Smooth}0\mathrm{Tye}$ as a moduli space. Notice that since ${\Omega }^n$ is itself a smooth space, we may speak about differential $n$-forms on ${\Omega }^n$ itsefl.

Definition

The universal differential $n$-forms is the differential $n$-form

$${\omega }_{\mathrm{univ}}^n\in {\Omega }^n({\Omega }^n)$$\omega^n_{univ} \in \Omega^n(\Omega^n)

which is modulated by the identity homomorphism $\mathrm{id}:{\Omega }^n\to {\Omega }^n$.

With this definition we have:

Proposition

For $X\in \mathrm{Smooth}0\mathrm{Type}$ any smooth space, every differential $n$-form on $X$, $\omega \in {\Omega }^n(X)$ is the pullback of differential forms, def. 34, of the universal differential $n$-form, def. 36, along a homomorphism $f$ from $X$ into the moduli space ${\Omega }^n$ of differential $n$-forms:

$$\omega =f^{*}{\omega }_{\mathrm{univ}}^n.$$\omega = f^* \omega^n_{univ} \,.

Remark

This statement is of course in a way a big tautology. Nevertheless it is a very useful tautology to make explicit. The whole concept of differential forms on smooth spaces here may be thought of as simply a variation of the theme of the Yoneda lemma.

This ends the Model-layer discussion of differential forms. We now pass to a more advanced discussion of this topic in the Semantics layer below. The reader wishing to stick to more elementary discussion for the time being should skip ahead to the Model-layer discussion of differentiation below.

Semantic Layer

Concrete smooth spaces

The smooth universal moduli space of differential forms ${\Omega }^n(-)$ from def. 32 is noteworthy in that it has a property not shared by many smooth spaces that one might think of more naively: while evidently being “large” (the space of all differential forms!) it has “very few points” and “very few $k$-dimensional subspaces” for low $k$. In fact

Proposition

For $k<n$ the smooth space ${\Omega }^n$ admits only a unique probe by ${ℝ}^k$:

$$\mathrm{Hom}({ℝ}^k,{\Omega }^n(-))\simeq {\Omega }^n({ℝ}^k)=\{0\}.$$Hom(\mathbb{R}^k, \Omega^n(-)) \simeq \Omega^n(\mathbb{R}^k) = \{0\} \,.

Proof

By the Yoneda lemma a smooth morphism ${ℝ}^k\to {\Omega }^n$ is a differential n-form $\omega \in {\Omega }^n({ℝ}^k)$. But for $n>k$ there is only the 0 element.

So while ${\Omega }^n()$ is a large smooth space, it is “not supported on probes” in low dimensions in as much as one might expect, from more naive notions of smooth spaces.

We now formalize this. The formal notion of an smooth space which is supported on its probes is that of a concrete object. There is a univeral map that sends any smooth space to its concretification. The universal moduli spaces of differential forms turn out to be non-concrete in that their concetrification is the point.

Definition

Let $H$ be a local topos. Write $♯:H\to H$ for the corresponding sharp modality, def. 24. Then.

1. An object $X\in H$ is called a concrete object if

   $${\mathrm{DeCoh}}_X:X\to ♯X$$DeCoh_X \colon X \to \sharp X

   is a monomorphism.

2. For $X\in H$ any object, its concretification $\mathrm{Conc}(X)\in H$ is the image factorization of ${\mathrm{DeCoh}}_X$, hence the factorization into an epimorphism followed by a monomorphism

   $${\mathrm{DeCoh}}_X:X\to \mathrm{Conc}(X)\hookrightarrow ♯X.$$DeCoh_X : X \to Conc(X) \hookrightarrow \sharp X \,.

Remark

Hence the concretification $\mathrm{Conc}(X)$ of an object $X$ is itself a concrete object and it is universal with this property. (…)

Proposition

Let $C$ be a site of definition for the local topos $H$, with terminal object $*$. Then for $X:C^{\mathrm{op}}\to \mathrm{Set}$ a sheaf, ${\mathrm{DeCoh}}_X$ is given over $U\in C$ by

$$X(U)\underset{\mathrm{Yoneda}}{\overset{\simeq }{\to }}H(U,X)\stackrel{{\Gamma }_{U,X}}{\to }\mathrm{Set}(\Gamma (U),\Gamma (X)).$$X(U) \underoverset{Yoneda}{\simeq}{\to} \mathbf{H}(U, X) \stackrel{\Gamma_{U,X}}{\to} Set(\Gamma(U),\Gamma(X)) \,.

Proposition

For $n\ge 1$ we have

$$\mathrm{Conc}({\Omega }^n)\simeq *.$$Conc(\Omega^n) \simeq * \,.

In this sense the smooth moduli space of differential $n$-forms is maximally non-concrete.

Smooth moduli space of differential forms on a smooth space

We discuss the smooth space of differential forms on a fixed smooth space $X$.

Remark

For $X$ a smooth space, the smooth mapping space $[X,{\Omega }^n]\in \mathrm{Smooth}0\mathrm{Type}$ is the smooth space whose ${ℝ}^k$-plots are differential $n$-forms on the product $X\times {ℝ}^k$

$$[X,{\Omega }^n]:{ℝ}^k\mapsto {\Omega }^n(X\times {ℝ}^k).$$[X, \Omega^n] \colon \mathbb{R}^k \mapsto \Omega^n(X \times \mathbb{R}^k) \,.

This is not quite what one usually wants to regard as an ${ℝ}^k$-parameterized of differential forms on $X$. That is instead usually meant to be a differential form $\omega$ on $X\times {ℝ}^k$ which has “no leg along ${ℝ}^k$”. Another way to say this is that the family of forms on $X$ that is represented by some $\omega$ on $X\times {ℝ}^k$ is that which over a point $v:*\to {\mathrm{ℝℝ}}^k$ has the value $({\mathrm{id}}_X,v{)}^{*}\omega$. Under this pullback of differential forms any components of $\omega$ with “legs along ${ℝ}^k$” are identified with the 0 differential form

This is captured by the following definition.

Definition

For $X\in \mathrm{Smooth}0\mathrm{Type}$ and $n\in \mathbb{N}$, the smooth space of differential $n$-forms ${\Omega }^n(X)$ on $X$ is the concretification, def. 37, of the smooth mapping space $[X,{\Omega }^n]$, def. 19, into the smooth moduli space of differential $n$-forms, def. 32:

$${\Omega }^n(X)≔\mathrm{Conc}([X,{\Omega }^n]).$$\mathbf{\Omega}^n(X) \coloneqq Conc([X, \Omega^n]) \,.

Proposition

The ${ℝ}^k$-plots of ${\Omega }^n({ℝ}^k)$ are indeed smooth differential $n$-forms on $X\times {ℝ}^k$ which are such that their evaluation on vector fields tangent to ${ℝ}^k$ vanish.

Proof (sketch)

By def. 25, def. 37 and prop. 22 the set of plots of ${\Omega }^n(X)$ over ${ℝ}^k$ is the image of the function

$${\Omega }^n(X\times {ℝ}^k)\simeq {\mathrm{Hom}}_{\mathrm{Smooth}0\mathrm{Type}}({ℝ}^k,[X,{\Omega }^n])\stackrel{{\Gamma }_{{ℝ}^k,[X,{\Omega }^n]}}{\to }{\mathrm{Hom}}_{\mathrm{Set}}(\Gamma ({ℝ}^k),\Gamma [X,{\Omega }^n])\simeq {\mathrm{Hom}}_{\mathrm{Set}}({ℝ}_s^k,{\Omega }^n(X)),$$\Omega^n(X \times \mathbb{R}^k) \simeq Hom_{Smooth0Type}(\mathbb{R}^k, [X,\Omega^n]) \stackrel{\Gamma_{ \mathbb{R}^k, [X,\Omega^n] }}{\to} Hom_{Set}(\Gamma(\mathbb{R}^k), \Gamma [X, \Omega^n]) \simeq Hom_{Set}(\mathbb{R}^k_s, \Omega^n(X)) \,,

where on the right ${ℝ}_s^k$ denotes, just for emphasis, the underlying set of ${ℝ}_s^k$. This function manifestly sends a smooth differential form $\omega \in {\Omega }^n(X\times {ℝ}^k)$ to the function from points $v$ of ${ℝ}^k$ to differential forms on $X$ given by

$$\omega \mapsto (v\mapsto ({\mathrm{id}}_X,v{)}^{*}\omega ).$$\omega \mapsto \left(v \mapsto (id_X, v)^* \omega \right) \,.

Under this function all components of differential forms with a “leg along” ${ℝ}^k$ are sent to the 0-form. Hence the image of this function is the collection of smooth forms on $X\times {ℝ}^k$ with “no leg along ${ℝ}^k$”.

Remark

For $n=0$ we have (for any $X\in \mathrm{Smooth}0\mathrm{Type}$)

$$\begin{array}{rl}{\Omega }^0(X)& ≔\mathrm{Conc}[X,{\Omega }^1]\\ & \simeq \mathrm{Conc}[X,ℝ]\\ & \simeq [X,ℝ]\end{array},$$\begin{aligned} \mathbf{\Omega}^0(X) & \coloneqq Conc [X, \Omega^1] \\ & \simeq Conc [X, \mathbb{R}] \\ & \simeq [X, \mathbb{R}] \end{aligned} \,,

by prop. 19.

Syntactic Layer

Images

Proposition

Let a morphism $f:X\to Y$ in $H$ be the categorical semantics of the syntax

$$x:X⊢f(x):Y.$$x \colon X \; \vdash \; f(x) \colon Y \,.

Then the syntax for the image $i_f:\mathrm{im}(f)\hookrightarrow Y$ is

$$⊢\left(\sum _{y:Y}{\exists }_{x:X}(f\left(x\right)=y)\right):\mathrm{Type}.$$\vdash\; \left( \sum_{y \colon Y} \exists_{x \colon X} \left(f\left(x\right) = y\right) \right) \colon Type \,.

Here ${\exists }_{\cdots }\cdots {=}_{\mathrm{def}}[{\sum }_{\cdots }\cdot ]$ is the bracket type of the dependent sum type.

Accordingly the syntax for the smooth moduli space of differential $n$-forms, def. 38, on a smooth space $X$,

$$\sum _{\omega :♯[X,{\Omega }^n]}{\exists }_{\lambda :[X,{\Omega }^n]}(\omega ={\mathrm{DeCoh}}_X(\lambda )).$$\sum_{\omega \colon \sharp [X, \Omega^n]} \exists_{\lambda \colon [X,\Omega^n]} (\omega = DeCoh_X(\lambda)) \,.

(…)

Differentiation

So far we have dealt with cohesive structures for which there is a notion of smooth variation, say of the position of a particle along a trajectory in spacetime. The idea of differentiation is to measure the difference between the position of two points on a cohesive trajectory in space as the difference between their worldline coordinates “tends to 0” without actually becoming 0. One also says that differentiation is forming “infinitesimal differences” of a cohesive process – and we will make precise here what this means.

There are two stages to the theory of differentiation:

1. We may think of differentiation as just a means to analyze more in detail the cohesive structure already given, without adding new structure, hence without a priori refining our notion of what a “cohesive trajectory” is. Indeed, given any line object $ℝ$ in a cohesive ∞-topos, there is a canonically a homomorphism of cohesive spaces

   $$d:ℝ\to {\Omega }_{\mathrm{cl}}^1(-,ℝ)$$\mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})

   from the line to the cohesive moduli space of closed differential 1-forms, which is such that it sends a cohesive curve on the line to the differential form on this curve whose value at each point is the differential of the curve, its rate of infinitesimal change at that point.

   Below in Differentiation of smooth functions and differetial forms we discuss this construction in the standard model of smooth cohesion for smooth spaces, where it reproduces what traditionally is called the de Rham differential $d$.

   Further below in Maurer-Cartan forms – Cohesive differentiation we show how $d$ comes out from just the abstract axioms of cohesionn.

2. We may think of differentiation as reflecting a refinement of smooth cohesion such that infinitesimal cohesive trajectories actually exist. Here, on top of having a measure for how a cohesive trajectory changes infinitesimally at a given point, it makes sense to ask concretely if two points on a trajectory are infinitesimally close to each other. In this approach the very notion of cohesion is refined to include explicitly a way to speak not just about a “cohesive blob of points”, but to decide whether it is in fact just an “infinitesimal cohesive blob of points” – an infinitesimally thickened point.

   Differential geometry with such an explicit notion of infinitesimals is known as synthetic differential geometry: the axioms here allow one to synthesize an infinitesimally thickened point and not just to reason about it as if it existed.

   In such a synthetic differential context then the differential $d$ from above not just exists as a whole, but we can “take it apart and re-synthesize it” by realizing its value at each point literally as the ordinary difference between two infinitesimally close points. Similarly, various other fundamental constructions in differential geometry, such as that of tangent bundles and jet bundles have a usefully transparent axiomatic characterization in the presence of synthetic infinitesimals. (Sophus Lie, one of the founding fathers of differential geometry famously said that he indeed found his theorems using such synthetic reasoning intuitively, and just did not publish them this way due to a lack of formalization of this language – at his time. ) This we discuss in the Mod Layer in D-geometry below.

In the Differentiation semantic layer below we formalize differentiation, and these two aspects of it, by adding to the notion of cohesive topos that of an infinitesimal cohesive neighbourhood.

Recalling that a cohesive topos is an abstract cohesive blob, an infinitesimal cohesive neighbourhood is accordingly an infinitesimally thicked cohesive blob (which is itself again a cohesive blob):

$$\begin{array}{ccccc}\mathrm{CohesiveBlob}& & \hookrightarrow & & \mathrm{InfinitesimallyThickenedCohesiveBlob}\\ & \searrow & & \swarrow \\ & & \mathrm{Point}\end{array}\begin{array}{ccccc}H& & \stackrel{}{\hookrightarrow }& & H_{\mathrm{th}}\\ & \searrow & & \swarrow \\ & & \mathrm{DiscSpaces}\end{array}.$$\array{ CohesiveBlob &&\hookrightarrow&& InfinitesimallyThickenedCohesiveBlob \\ & \searrow && \swarrow \\ && Point } \;\;\;\;\;\;\;\; \array{ \mathbf{H} &&\stackrel{}{\hookrightarrow}&& \mathbf{H}_{th} \\ & \searrow && \swarrow \\ && DiscSpaces } \,.

Model Layer

We discuss

• Differentiation of smooth functions and differential forms

  • first just on coordinate patches

  • and then on general smooth spaces.

By considering fiber products of smooth mapping spaces with discrete spaces of boundary configurations, we obtain from this the differentiation theory called

• Variational calculus

  • with the notion of Smooth functional

  • and Variational derivative of smooth functionals.

  • The central class of examples of this of interest in physics is the variation of action functionals that yields the Euler-Lagrange equations of motion in classical field theory. This we discuss in Euler-Lagrange equations.

Differentiation of smooth functions and differential forms

Differentiation on coordinate patches

By definition to smooth function $f:ℝ\to ℝ$ is associated its derivative, a smooth function $f\prime :ℝ\to ℝ$. And more generally to a smooth function $f:{ℝ}^n\to ℝ$ are associated its partial derivatives, smooth functions

$$\frac{\partial f}{\partial x^i}:{ℝ}^n\to ℝ$$\frac{\partial f}{\partial x^i} \colon \mathbb{R}^n \to \mathbb{R}

for $1\le i\le n$.

The de Rham differential collects all partial derivatives of a function into a single differential 1-form

Definition

For $n\in \mathbb{N}$, The de Rham differential on smooth functions in $C^{\mathrm{\infty }}({ℝ}^n)$ is the function

$$d:C^{\mathrm{\infty }}({ℝ}^n)\to {\Omega }^1({ℝ}^n)$$\mathbf{d} \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)

which takes $f\in C^{\mathrm{\infty }}({ℝ}^n)$ to

$$df≔\sum _{i=1}^n\frac{\partial f}{\partial x^i}d{x}^i.$$\mathbf{d}f \coloneqq \sum_{i = 1}^n \frac{\partial f}{\partial x^i} \mathbf{d} x^i \,.

Example

For $x^i:{ℝ}^n\to ℝ$ one of the coordinate functions, the de Rham differential $d{x}^i$ indeed coincides with the basis element of the same name according to def. 26, using that

$$\frac{\partial x^i}{\partial x^j}=\{\begin{array}{cc}1& \mid i=j\\ 0& \mid \mathrm{otherwise}\end{array}.$$\frac{\partial x^i}{\partial x^{j}} = \left\{ \array{ 1 & | i = j \\ 0 & | otherwise } \right. \,.

Proposition

The de Rham differentials $d:C^{\mathrm{\infty }}({ℝ}^n)\to {\Omega }^1({ℝ}^n)$ for all $n\in \mathbb{N}$ are compatible with pullback of differential 1-forms, def. 27, in that for each coordinate transformation $\varphi :{ℝ}^{\tilde{k}}\to {ℝ}^k$ the diagram

$$\begin{array}{ccc}{C}^{\mathrm{\infty }}({ℝ}^k)& \stackrel{d}{\to }& {\Omega }^1({ℝ}^k)\\ {\downarrow }^{{\varphi }^{*}}& & {\downarrow }^{{\varphi }^{*}}\\ C^{\mathrm{\infty }}({ℝ}^{\tilde{k}})& \stackrel{d}{\to }& {\Omega }^1({ℝ}^{\tilde{k}})\end{array}$$\array{ C^\infty(\mathbb{R}^k) &\stackrel{\mathbf{d}}{\to}& \Omega^1(\mathbb{R}^k) \\ \downarrow^{\mathrlap{\phi^\ast}} && \downarrow^{\mathrlap{\phi^\ast}} \\ C^\infty(\mathbb{R}^{\tilde k}) &\stackrel{\mathbf{d}}{\to}& \Omega^1(\mathbb{R}^{\tilde k}) }

commutes.

Proof

This is equivalently the statement of the chain rule of differentiation: For any $f\in C^{\mathrm{\infty }}({ℝ}^k)$ we have on the one hand, by def. 27 and def. 39

$$\begin{array}{rl}d{\varphi }^{*}f& =d(f\circ \varphi )\\ & =\sum _{j=1}^{\tilde{k}}\frac{\partial (f\circ \varphi )}{\partial x^j}d{x}^j\end{array}$$\begin{aligned} \mathbf{d} \phi^\ast f & = \mathbf{d} (f \circ \phi) \\ & = \sum_{j = 1}^{\tilde k} \frac{\partial (f \circ \phi)}{\partial x^j}\mathbf{d} x^j \end{aligned}

and on the other hand, applying the definition in the other order,

$$\begin{array}{rl}{\varphi }^{*}df& ={\varphi }^{*}\sum _{i=1}^k\frac{\partial f}{\partial x^i}d{x}^i\\ & =\sum _{i=1}^k\sum _{j=1}^{\tilde{k}}(\frac{\partial f}{\partial x^i}\circ \varphi )\cdot \left(\frac{\partial {\varphi }^i}{\partial x^j}\right)d{x}^j\end{array}.$$\begin{aligned} \phi^* \mathbf{d}f & = \phi^* \sum_{i = 1}^k \frac{\partial f}{\partial x^i} \mathbf{d}x^i \\ & = \sum_{i = 1}^k \sum_{j = 1}^{\tilde k} \left(\frac{\partial f}{\partial x^i}\circ \phi \right) \cdot \left(\frac{\partial \phi^i}{\partial x^j}\right) \mathbf{d}x^j \end{aligned} \,.

Both expressions agree precisely if for all $j$ we have

$$\frac{\partial (f\circ \varphi )}{\partial x^j}=\sum _{i=1}^k(\frac{\partial f}{\partial x^i}\circ \varphi )\cdot \left(\frac{\partial {\varphi }^i}{\partial x^j}\right)\in C^{\mathrm{\infty }}({ℝ}^{\tilde{k}}).$$\frac{\partial (f \circ \phi)}{\partial x^j} = \sum_{i = 1}^k \left(\frac{\partial f}{\partial x^i}\circ \phi\right) \cdot \left(\frac{\partial \phi^i}{\partial x^j}\right) \;\;\; \in C^\infty(\mathbb{R}^{\tilde k}) \,.

This is precisely the statement of the chain rule for differentiation.

Notice that as smooth spaces $ℝ={\Omega }^0=C^{\mathrm{\infty }}(-)$, by prop. 19. Therefore the above says that

Proposition

The de Rham differential, def. 39, constitutes a homomorphism of smooth spaces, def. 14

$$d:ℝ\to {\Omega }^1$$\mathbf{d} \colon \mathbb{R} \to \Omega^1

from the real line to the universal smooth moduli space of differential 1-forms, def. 32.

Remark

Below in Maurer-Cartan form on a Lie group we discuss a more general abstract origin of $d:ℝ\to {\Omega }_{\mathrm{cl}}^1$.

We now extend the de Rham differential to differential forms of higher degree.

Definition

For all $n\in \mathbb{N}$ let

$$d:{\Omega }^{•}({ℝ}^n)\to {\Omega }^{•}({ℝ}^n)$$\mathbf{d} \colon \Omega^\bullet(\mathbb{R}^n) \to \Omega^\bullet(\mathbb{R}^n)

be the unique extension of $d:C^{\mathrm{\infty }}(-)\to {\Omega }^1(-)$ to a degree-1 derivation with

$$dd{x}^i=0.$$\mathbf{d}\mathbf{d}x^i = 0 \,.

(…)

Proposition

For each $n\in \mathbb{N}$ the de Rham differential of def. 40 constitutes a homomorphism of smooth spaces

$$d:{\Omega }^n\to {\Omega }^{n+1}$$\mathbf{d} \colon \Omega^n \to \Omega^{n+1}

form the universal smooth moduli space of differental $n$-forms to that of differential $n+1$-forms.

Differentiation on smooth spaces

We now extend the notion of derivatives and de Rham differentials from smooth functions on Cartesian spaces to smooth functions on general smooth spaces.

Recall from def. 33 that the set of differential $n$-forms on a smooth space $X$ is ${\Omega }^n(X)≔\mathrm{Hom}(X,{\Omega }^n)$.

Definition

For $X\in \mathrm{Smooth}0\mathrm{Type}$ a smooth space and $n\in \mathbb{N}$, the de Rham differential on $n$-forms over $X$ is the function

$$d:{\Omega }^n(X)\to {\Omega }^{n+1}(X)$$\mathbf{d} \colon \Omega^n(X) \to \Omega^{n+1}(X)

which is the postcomposition with the homomorphism of smooth spaces of prop. 28:

$$\mathrm{Hom}(X,d):\mathrm{Hom}(X,{\Omega }^n)\to \mathrm{Hom}(X,{\Omega }^{n+1}).$$Hom(X,\mathbf{d}) \colon Hom(X,\Omega^n) \to Hom(X,\Omega^{n+1}) \,.

In particular the derivative of a smooth function $f:X\to ℝ$ is the composite

$$df:X\stackrel{f}{\to }ℝ\stackrel{d}{\to }{\Omega }_{\mathrm{cl}}^1\hookrightarrow {\Omega }^1.$$\mathbf{d}f \colon X \stackrel{f}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1_{cl} \hookrightarrow \Omega^1 \,.

Below in Variation is differentiation on smooth spaces we find that this notion of differentiation of smooth functions on smooth spaces subsumes what traditionally is called variational calculus of functionals on mapping spaces.

Example: The electromagnetic field strength

for instance electromagnetic potential

$$A=\varphi dt+A_{1}d{x}^1+A_{2}d{x}^2+A_{3}d{x}^3$$A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3

then the electromagnetic field strength is

$$F≔dA=E_{1}d{x}^1\wedge dt+E_{2}d{x}^2\wedge dt+E_{3}d{x}^3\wedge dt+B_{1}d{x}^2\wedge d{x}^3+B_{2}d{x}^3\wedge d{x}^1+B_{3}d{x}^1\wedge d{x}^2$$F \coloneqq \mathbf{d}A = E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_2 \mathbf{d}x^2 \wedge \mathbf{d}t + E_3 \mathbf{d}x^3 \wedge \mathbf{d}t + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2

with

$$E_i=\frac{\partial \varphi }{\partial x^i}-\frac{\partial A_i}{\partial t}$$E_i = \frac{\partial \phi}{\partial x^i} - \frac{\partial A_i}{\partial t}

and

$$B_1=\frac{\partial A_2}{\partial x^3}-\frac{\partial A_3}{\partial x^2}$$B_1 = \frac{\partial A_2}{\partial x^3} - \frac{\partial A_3}{\partial x^2}

etc

This are the first 2 of 4 Maxwell equations: $dF=0$

(the other 2 are discussed below in Riemannian geometry)

for

$$A,A\prime :X\to {\Omega }^1(-)$$A,A' : X \to \Omega^1(-)

a gauge transformation $A\to A\prime$ is $\lambda :X\to ℝ$ with

$$A\prime =A+d\lambda$$A' = A + \mathbf{d} \lambda

Variational calculus

Traditionally a functional is a function which is sufficiently like a smooth function, but defined not on a manifold, but on a mapping space between manifolds. Also traditionally, a variational derivative of such a functional is something aking to a derivative, generalized to this context, and subject to the condition that all variations preserve some boundary conditions.

We formulate this classical theory in the context of smooth spaces. Here a functional is simply a homomorphism of smooth spaces out of a smooth mapping space, as in def. 19. We may impose respect for boundary conditions by forming the fiber product of this mapping space with a discrete smooth space inclusion, given in def. 43 below. Then the variational derivative is simply the ordinary derivative of def. 41.

Discrete points of a smooth space

Definition

For $X\in \mathrm{Smooth}0\mathrm{Type}$ a smooth space, write

$$\Gamma X≔\mathrm{Hom}(*,X)\in \mathrm{Set}$$\Gamma X \coloneqq Hom(*,X) \in Set

for its set of points, the set of homomorphisms, def. 14, from the point to $X$.

Write

$$♭X≔\mathrm{Disc}(\Gamma (X))$$\flat X \coloneqq Disc (\Gamma(X))

for the discrete smooth space, def. 13, on this set of points.

Definition

For every smooth space $X$ there is a canonical homomorphism of smooth spaces

$$♭X\to X.$$\flat X \to X \,.

This sends a plot $U\to ♭X$, which by definition of $\mathrm{Disc}(-)$ is a point in $\Gamma X$, hence a homomorphism $x:*\to X$, to the plot $U\to *\stackrel{x}{\to }X$ of $X$.

Smooth functionals

Let $X$ be a smooth manifold. Let $\Sigma$ be a smooth manifold with boundary $\partial \Sigma \hookrightarrow \Sigma$.

Write

$$[\Sigma ,X]\in \mathrm{Smooth}0\mathrm{Type}$$[\Sigma, X] \in Smooth0Type

for the smooth space (a diffeological space) which is the mapping space from $\Sigma$ to $X$, hence such that for each $U\in$ CartSp we have

$$[\Sigma ,X](U)=C^{\mathrm{\infty }}(U\times \Sigma ,X).$$[\Sigma, X](U) = C^\infty(U \times \Sigma, X) \,.

Definition

Write

$$[\Sigma ,X{]}_{\partial \Sigma }≔[\Sigma ,X]{\times }_{[\partial \Sigma ,X]}♭[\partial \Sigma ,X]$$[\Sigma, X]_{\partial \Sigma} \coloneqq [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]

for the pullback in smooth spaces

$$\begin{array}{ccc}[\Sigma ,X{]}_{\partial \Sigma }& \to & ♭[\partial \Sigma ,X]\\ \downarrow & & \downarrow \\ [\Sigma ,X]& \stackrel{(-){\mid }_{\partial \Sigma }}{\to }& [\partial \Sigma ,X]\end{array},$$\array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,,

where

• the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma ,X]$ of configurations to the boundary;

• the right vertical morphism is the homomorphism from def. 43.

Proposition

The smooth space $[\Sigma ,X{]}_{\partial \Sigma }$ is a diffeological space whose underlying set is $C^{\mathrm{\infty }}(\Sigma ,X)$ and whose $U$-plots for $U\in$ CartSp are smooth functions $\varphi :U\times \Sigma \to X$ such that $\varphi (-,s):U\to X$ is the constant function for all $s\in \partial \Sigma \hookrightarrow \Sigma$.

Definition

A functional on the mapping space $[\Sigma ,X]$ is a homomorphism of smooth spaces

$$S:[\Sigma ,X{]}_{\partial \Sigma }\to ℝ.$$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R} \,.

Functional derivative / variational derivative

Write

$$d:ℝ\to {\Omega }^1$$\mathbf{d} \colon \mathbb{R} \to \Omega^1

for the de Rham differential incarnated as a homomorphism of smooth spaces from the real line to the smooth moduli space of differential 1-forms.

Definition

The functional derivative

$$dS\in {\Omega }^1([\Sigma ,X{]}_{\partial \Sigma })$$\mathbf{d}S \in \Omega^1([\Sigma,X]_{\partial \Sigma})

of a functional $S$, def. 45, is simply its de Rham differential as a homomorphism of smooth spaces, hence the composite

$$dS:[\Sigma ,X{]}_{\partial \Sigma }\stackrel{S}{\to }ℝ\stackrel{d}{\to }{\Omega }^1.$$\mathbf{d} S \colon [ \Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.

Definition

This means that for each smooth parameter space $U\in$ CartSp and for each smooth $U$-parameterized collection

$$\varphi :U\times \Sigma \to X$$\phi \colon U \times \Sigma \to X

of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,

$$S_{\varphi }:[\Sigma ,X{]}_{\partial \Sigma }(U)\to C^{\mathrm{\infty }}(U,ℝ)$$S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})

is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then

$$(dS{)}_{\varphi }\in {\Omega }^1(U)$$(\mathbf{d}S)_\phi \in \Omega^1(U)

is the smooth differential 1-form

$$(dS{)}_{\varphi }=d(S(\varphi )).$$(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi)) \,.

Example

Let $\Sigma =[0,1]\hookrightarrow ℝ$ be the standard interval. Let

$$L(-,-)dt\in {\Omega }^1([0,1],C^{\mathrm{\infty }}({ℝ}^2))$$L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))

and consider the functional

$$S:([0,1]\stackrel{\gamma }{\to }X)\mapsto {\int }_0^{1}L(\gamma (t),\dot{\gamma }(t))dt.$$S \colon ([0,1] \stackrel{\gamma}{\to} X) \mapsto \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t \,.

Then for $U=ℝ$ and any smooth $U$-parameterized collection $\{{\gamma }_u:\Sigma \to X{\}}_{u\in U}$ the functional derivative takes the value

$$\begin{array}{rl}d{S}_{{\gamma }_{(-)}}& =(\frac{d}{du}{\int }_0^{1}L({\gamma }_u(t),{\dot{\gamma }}_u(t))\mathrm{dt})du\\ & ={\int }_0^1(\frac{\partial L}{\partial \gamma }({\gamma }_u(t),{\dot{\gamma }}_u(t))\frac{d{\gamma }_u(t)}{du}+\frac{\partial L}{\partial \dot{\gamma }}({\gamma }_u(t),{\dot{\gamma }}_u(t))\frac{\partial {\dot{\gamma }}_u(t)}{\partial u})\mathrm{dt}du\\ & ={\int }_0^1(\frac{\partial L}{\partial \gamma }({\gamma }_u(t),{\dot{\gamma }}_u(t))\frac{d{\gamma }_u(t)}{du}+\frac{\partial L}{\partial \dot{\gamma }}({\gamma }_u(t),{\dot{\gamma }}_u(t))\frac{\partial }{\partial t}\frac{\partial {\gamma }_u(t)}{\partial u})\mathrm{dt}du\\ & ={\int }_0^1(\frac{\partial L}{\partial \gamma }({\gamma }_u(t),{\dot{\gamma }}_u(t))-\frac{\partial }{\partial t}\frac{\partial L}{\partial \dot{\gamma }}({\gamma }_u(t),{\dot{\gamma }}_u(t)))\frac{\partial {\gamma }_u(t)}{\partial u}\mathrm{dt}du\end{array}.$$\begin{aligned} \mathbf{d}S_{\gamma_{(-)}} & = \left( \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt \right) \mathbf{d}u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial \dot \gamma_u(t)}{\partial u} \right) dt \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u} \right) dt \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) - \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \right) \frac{\partial \gamma_u(t)}{\partial u} dt \mathbf{d}u \end{aligned} \,.

Here we used integration by parts? and used that the boundary term vanishes

$$\begin{array}{rl}{\int }_0^1\frac{\partial }{\partial t}(\frac{\partial }{\partial \dot{\gamma }}L({\gamma }_u(s),{\dot{\gamma }}_u(s))\frac{\partial {\gamma }_u(s)}{\partial u})ds& =(\frac{\partial }{\partial \dot{\gamma }}L({\gamma }_u(1),{\dot{\gamma }}_u(1))\frac{\partial {\gamma }_u(1)}{\partial u})-(\frac{\partial }{\partial \dot{\gamma }}L({\gamma }_u(0),{\dot{\gamma }}_u(0))\frac{\partial {\gamma }_u(0)}{\partial u})\\ & =0\end{array}$$\begin{aligned} \int_{0}^1 \frac{\partial}{\partial t} \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s)) \frac{\partial \gamma_u(s)}{\partial u} \right) d s & = \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1)) \frac{\partial \gamma_u(1)}{\partial u} \right) - \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0)) \frac{\partial \gamma_u(0)}{\partial u} \right) \\ & = 0 \end{aligned}

since by prop. 29 ${\gamma }_{(-)}(1)$ and ${\gamma }_{(-)}(0)$ are constant.

Euler-Lagrange equations

• critical locus

• Euler-Lagrange equation

• equations of motion

$𝒟$-geometry

• D-geometry

Infinitesimal smooth loci

Definition

Write

$$\mathrm{SmoothLoci}≔{\mathrm{SmoothAlgebras}}^{\mathrm{op}}≔{\mathrm{Func}}^{\times }(\mathrm{CartSp},\mathrm{Set})$$SmoothLoci \coloneqq SmoothAlgebras^{op} \coloneqq Func^\times(CartSp,Set)

for the category of spaces which are formally dual to smooth algebras: the opposite category of that of smooth algebras. This is called the category of smooth loci.

Definition

A smooth Artin algebra (also called a “Weil algebra” in the synthetic differential geometry-literature) is a smooth algebra $A$ whose underlying $ℝ$-vector space is a direct sum of the form

$$A=ℝ\oplus V,$$A = \mathbb{R} \oplus V \,,

where $V$ is of finite dimension and such that every element $v\in V\subset A$ is nilpotent, in that there is $n\in \mathbb{N}$ such that the $n$-fold product of $v$ with itself in $A$ vanishes: $v^n=0$.

Example

The smallest smooth Artin algebra is the ring of dual numbers, def. 3, for which $V=ℝ$.

Definition

Write

$$\mathrm{InfPoint}\hookrightarrow \mathrm{SmoothLoci}$$InfPoint \hookrightarrow SmoothLoci

for the full subcategory of $\mathrm{SmoothLoci}$, def. 48, of those that are duals of Artin algebras, def. 49.

We call this the category of infinitesimally thickened points.

de Rham space

• de Rham space

Jet bundles

• jet bundle

Semantic Layer

Synthetic differential geometry

• synthetic differential geometry

We have two full and faithful functors

$$\mathrm{CartSp}\hookrightarrow \mathrm{SmoothLoci}$$CartSp \hookrightarrow SmoothLoci

$$\mathrm{InfPoint}\hookrightarrow \mathrm{SmoothLoci}.$$InfPoint \hookrightarrow SmoothLoci \,.

Definition

Write CartSp${}_{\mathrm{th}}\hookrightarrow \mathrm{SmoothLocus}$ for the full subcategory of that of smooth loci, def. 48, on those of the form

$$U=U\times D$$\mathbf{U} = U \times D

with $U\in \mathrm{CartSp}\hookrightarrow \mathrm{SmoothLoci}$ and $D\in \mathrm{InfPoint}\hookrightarrow \mathrm{SmoothLoci}$.

We may call this the category of infinitesimally thickened Cartesian spaces or or formal smooth Cartesian spaces.

The category ${\mathrm{CartSp}}_{\mathrm{th}}$ carries several coverages of interest. One is this:

Definition

For $U=U\times D\in {\mathrm{CartSp}}_{\mathrm{th}}$ say that a covering family is a set of morphisms in ${\mathrm{CartSp}}_{\mathrm{th}}$ of the form $\{U_i\times D\stackrel{({\varphi }_i,{\mathrm{id}}_D)}{\to }U\times D{\}}_i$ such that $\{U_i\stackrel{{\varphi }_i}{\to }U{\}}_i$ is a covering family in CartSp.

The corresponding sheaf topos $\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$ is known as the Cahiers topos.

Proposition

The Cahiers topos $\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$, def. 52,
is a cohesive topos.

Definition

We write

$$\mathrm{SynthDiff}0\mathrm{Type}≔\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}}).$$SynthDiff0Type \coloneqq Sh(CartSp_{th}) \,.

• synthetic differential infinity-groupoid

Tangent bundle

Write $D\in \mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$ for the smooth locus formally dual to the ring of dual numbers, def. \ref{}. Write

$$i:*\to D$$i \colon * \to D

for the unique point inclusion.

Definition

For $X\in \mathrm{SynthDiff}0\mathrm{Type}$, the internal hom

$$TX≔[D,X]\in \mathrm{SynthDiff}0\mathrm{Type}$$T X \coloneqq [D,X] \in SynthDiff0Type

equipped with the morphism

$$[i,X]:[D,X]\to [*,X]\simeq X$$[i,X] \colon [D,X] \to [*,X] \simeq X

is the tangent bundle of $X$.

Definition

For $X\in \mathrm{SynthDiff}0\mathrm{Type}$, a vector field $v$ on $X$ is a section

$$\begin{array}{ccccc}X& & \stackrel{v}{\to }& & TX\\ & {}_{\mathrm{id}}\searrow & & {\swarrow }_{[i,D]}\\ & & X\end{array}.$$\array{ X &&\stackrel{v}{\to}&& T X \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{[i,D]}} \\ && X } \,.

The smooth space of vector fields is the internal hom in the slice topos over $X$

$${\Gamma }_X(TX)≔[X,TX{]}_X.$$\mathbf{\Gamma}_X(T X ) \coloneqq [X,T X]_X \,.

Differential equations

• equation

• differential equation

Differential cohesion of the topos of smooth spaces

Recall the sites

• CartSp, def. 6;

• InfPoint, def. 50;

• CartSp${}_{\mathrm{th}}$, def. 51.

Definition

Define functors

$$\mathrm{CartSp}\stackrel{\stackrel{i}{\hookrightarrow }}{\underset{p}{\leftarrow }}{\mathrm{CartSp}}_{\mathrm{th}}\stackrel{\iota }{\leftarrow }\mathrm{InfPoint}$$CartSp \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}} CartSp_{th} \stackrel{\iota}{\leftarrow} InfPoint

by

• $i:U\mapsto U\times *$;

• $p:U\times D\mapsto U$

• $\iota :D\mapsto *\times D$

Proposition

All three functors in def. 56 are morphisms of sites. The induced geometric morphism of sheaf toposes is of the form

$$\mathrm{Sh}(\mathrm{CartSp})\stackrel{}{\stackrel{\stackrel{{\mathrm{Lan}}_i}{\hookrightarrow }}{\stackrel{\stackrel{(-)\circ i}{\leftarrow }}{\stackrel{\stackrel{(-)\circ p}{\hookrightarrow }}{\underset{{\mathrm{Ran}}_p}{\leftarrow }}}}}\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})\stackrel{\stackrel{{\mathrm{Lan}}_{\iota }}{\leftarrow }}{\underset{(-)\circ \iota }{\to }}\mathrm{Sh}(\mathrm{InfPoint})$$Sh(CartSp) \stackrel{}{\stackrel{\overset{Lan_i}{\hookrightarrow}}{\stackrel{\overset{(-)\circ i}{\leftarrow}}{\stackrel{\overset{(-)\circ p}{\hookrightarrow}}{\underset{Ran_p}{\leftarrow}}}}} Sh(CartSp_{th}) \stackrel{\overset{Lan_\iota}{\leftarrow}}{\underset{(-)\circ \iota}{\to}} Sh(InfPoint)

where hence the morphism on the left is in particular an essential geometric embedding.

Proposition

The sequence of geometric morphisms

$$\mathrm{Sh}(\mathrm{CartSp})\stackrel{p^{*}}{\to }\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})\stackrel{{\iota }^{*}}{\to }\mathrm{Sh}(\mathrm{InfPoint})$$Sh(CartSp) \stackrel{p^*}{\to} Sh(CartSp_{th}) \stackrel{\iota^*}{\to} Sh(InfPoint)

exhibit a homotopy cofiber sequence in the (2,1)-category Topos.

Proof

By the discussion at (∞,1)Topos – Existence of limits and colimits the statement is equivalently that the inverse image functors

$$\mathrm{Sh}(\mathrm{CartSp})\stackrel{{\mathrm{Lan}}_p}{\leftarrow }\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})\stackrel{{\mathrm{Lan}}_{\iota }}{\leftarrow }\mathrm{Sh}(\mathrm{InfPoint})$$Sh(CartSp) \stackrel{Lan_p}{\leftarrow} Sh(CartSp_{th}) \stackrel{Lan_\iota}{\leftarrow} Sh(InfPoint)

form a homotopy fiber sequence in (∞,1)Cat. Computing this in the model structure for quasi-categories after passing to nerves, the morphism $N({\mathrm{Lan}}_p)$ is clearly an inner Kan fibration because of the subcategory inclusion $\mathrm{Sh}(\mathrm{CartSp})\hookrightarrow \mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$. So by the general discussion at homotopy pullback the homotopy fiber is given by the 1-categorical fiber of $N({\mathrm{Lan}}_p)$ in sSet. By the discussion at Left Kan extension - on representables ${\mathrm{Lan}}_p$ acts as $p$ on representables. The 1-categorical fiber of $N(p):N({\mathrm{CartSp}}_{\mathrm{th}})\to N(\mathrm{CartSp})$ is evidently $N(\mathrm{InfPoint})$. Since ${\mathrm{Lan}}_{\iota }$ is a left adjoint it preserves colimits and since ever sheaf is a colimit of representables, this is sufficient to imply the claim.

Differential cohesion

We axiomatize the existence of infinitesimals by further modalities on a cohesive topos.

Definition

Given a cohesive topos $H=\mathrm{Cohesive}0\mathrm{Type}$ over a base topos Discrete0Type, a structure of differential cohesion on $H$ is an geometric embedding into a cohesive topos $H_{\mathrm{th}}=\mathrm{InfThickenedCohesive}0\mathrm{Type}$ with an extra left eadjoint? that preserves the terminal object:

$$\begin{array}{ccccc}\mathrm{Cohesive}0\mathrm{Type}& & \stackrel{\stackrel{i_{!}}{\hookrightarrow }}{\stackrel{\stackrel{i^{*}}{\leftarrow }}{\underset{i_{*}}{\hookrightarrow }}}& & \mathrm{InfThickenedCohesive}0\mathrm{Type}\\ & {}_{\Gamma }\searrow {\nwarrow }^{\mathrm{Disc}}& & {}^{{\Gamma }_{\mathrm{th}}}\swarrow {\nearrow }_{{\mathrm{Disc}}_{\mathrm{th}}}\\ & & \mathrm{Discrete}0\mathrm{Type}\end{array}$$\array{ Cohesive0Type &&\stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}}&& InfThickenedCohesive0Type \\ & {}_{\mathllap{\Gamma}}\searrow \nwarrow^{\mathrlap{Disc}} && {}^{\mathllap{\Gamma_{th}}}\swarrow \nearrow_{\mathrlap{Disc_{th}}} \\ && Discrete0Type }

Definition

Given differential cohesion, def. 57, define the monad/comonad adjunction

$$(\mathrm{Red}⊣{\Pi }_{\inf }):H_{\mathrm{th}}\stackrel{\stackrel{i_{*}}{\leftarrow }}{\underset{i^{*}}{\to }}H\stackrel{\stackrel{i_{!}}{\leftarrow }}{\underset{i_{*}}{\to }}H.$$(Red \dashv \Pi_{inf}) \colon \mathbf{H}_{th} \stackrel{\overset{i_*}{\leftarrow}}{\underset{i^*}{\to}} \mathbf{H} \stackrel{\overset{i_!}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H} \,.

We call $\mathrm{Red}(X)$ the reduced type of $X$ and ${\Pi }_{\inf }(X)$ the infinitesimal path ∞-groupoid of $X$.

For the $(i_{*}⊣i^{*})$-unit we write

$${\mathrm{InfinitesimalPathInclusion}}_X:X\to {\Pi }_{\inf }(X)$$InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)

and call it the constant infinitesimal path inclusion on $X$.

The $(i_{*}⊣i^{*})$-counit

$$\mathrm{Red}X\to X$$Red X \to X

we call the inclusion of the reduced part of $X$.

Definition

Given a geometric embedding of ∞-toposes

$$\mathrm{CohesiveType}\stackrel{i}{\hookrightarrow }\mathrm{InfThickenedCohesiveType}$$CohesiveType \stackrel{i}{\hookrightarrow} InfThickenedCohesiveType

exhibiting differential cohesion, write

$$\mathrm{CohesiveType}\stackrel{i}{\hookrightarrow }\mathrm{InfThickenedCohesiveType}\to \mathrm{InfinitesimalType}$$CohesiveType \stackrel{i}{\hookrightarrow} InfThickenedCohesiveType \to InfinitesimalType

for the corresponding homotopy cofiber sequence in (∞,1)-topos. The full sub-(∞,1)-category that is the kernel of the global section geometric morphism of $\mathrm{InfininitesimalType}$ we call the (∞,1)-category of synthetic ∞-Lie algebras

$$L_{\mathrm{\infty }}\mathrm{Algebra}≔\ker (\Gamma )\hookrightarrow \mathrm{InfinitesimalType}stackrerl\Gamma \to \mathrm{DiscreteType}.$$L_\infty Algebra \coloneqq ker(\Gamma) \hookrightarrow InfinitesimalType \stackrerl{\Gamma}{\to} DiscreteType \,.

Example

For the moment see at Synthetic differential infinity-groupoid – Lie differentiation.

Proposition

Setting

• $H_{\mathrm{th}}≔\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$

• $H≔\mathrm{Sh}(\mathrm{CartSp})$

• $H_{\inf }≔\mathrm{Sh}(\mathrm{InfPoint})$

makes prop. 31 exhibit differential cohesive structure.

de Rham space

Definition

For $X\in H_{\mathrm{th}}$ we call ${\Pi }_{\inf }(X)\in H$ the de Rham space object of $X$.

Jet bundle

Definition

For $X\in H$

$${\mathrm{Jet}}_X≔({\mathrm{InfinitesimalPathInclusion}}_X{)}_{*}:H_{\mathrm{th}}/X\to H_{\mathrm{th}}/{\Pi }_{\inf }(X).$$Jet_X \coloneqq (InfinitesimalPathInclusion_X)_* \colon \mathbf{H}_{th}/X \to \mathbf{H}_{th}/\Pi_{inf}(X) \,.

For $(E\to X)\in H_{\mathrm{th}}/X$, ${\mathrm{Jet}}_X(E)$ is the jet bundle of $E$.

Formally étale / formally unramified / formally smooth

Definition

A morphism $f:X\to Y$ in $H$ is called a formally étale morphism if the naturality square of the ${\Pi }_{\inf }$-unit

\array{ X &\stackrel{}{\to}& \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\athrlap{\mathbf{\Pi}_{inf}(f)}} } \\ Y &\to& \mathbf{\Pi}_{inf}(Y) \array{ X &\stackrel{}{\to}& \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\athrlap{\mathbf{\Pi}_{inf}(f)}} } \\ Y &\to& \mathbf{\Pi}_{inf}(Y)

is an (∞,1)-pullback.

Proposition

If $X,Y\in$ SmoothMfd $\hookrightarrow$ $H\stackrel{i_{!}}{\to }{H}_{\mathrm{th}}$ then for $f:X\to Y$ any morphism

1. $f$ is formally étale morphism precisely if $f$ is a submersion of smooth manifolds;

2. $f$ is a formally unramified morphism precisely if it is an immersion of smooth manifolds;

3. $f$ is a formally smooth morphism precisely if it is a diffeomorphism.

Syntactic Layer

Differential homotopy type theory

• differential homotopy type theory

(…)

Smooth homotopy types

Fundamental physics is all based on the gauge principle. This says in particular that it is wrong to think of two different gauge field configurations (discussed in detail below in Fields) as being equal or not. Instead it makes sense to ask if there is (or not) a gauge transformation from one to the other that exibits an gauge equivalence between the two fields. The simplest example of this is described in detail below in Gauge transformations in electromagnetism.

But this means that the collection of gauge fields on a spacetime $X$, which we will write as a mapping space $[X,B{G}_{\mathrm{conn}}]$, cannot be a smooth space as considered above, for if it were such a smooth space, then we could ask if two gauge fields ${\nabla }_1,{\nabla }_2:*\to [X,B{G}_{\mathrm{conn}}]$ were equal or not.

Notice that this already applies to a single gauge field: given any $\nabla :*\to [X,B{G}_{\mathrm{conn}}]$ it is certainly equal to itself, but is nevertheless also gauge equivalent to itself, but the latter it may be in several non-equivalent ways: there may be non-trivial auto-gauge transformations $\nabla \to \nabla$. Since these can be composed, are, by definition, invertible, and contain the trivial gauge transformaiton, these form a group, the group of auto-gauge equivalences of $\nabla$. (Groups are discussed in detail below in Groups) If that gauge field $\nabla$ is itself the trivial gauge field, $\nabla ={\nabla }_0$, then this group of auto-gauge equivalences is the gauge group of the given gauge theory:

$$G\simeq \mathrm{Aut}({\nabla }_0)\simeq \{{\nabla }_0\stackrel{\simeq }{\to }{\nabla }_0\}.$$G \simeq Aut(\nabla_0) \simeq \{\nabla_0 \stackrel{\simeq}{\to} \nabla_0\} \,.

For this reason, the collection of all gauge fields and all gauge transformations between them form something that is a group over ever fixed element, but which generalizes the notion of a group in that there are not only auto-equivalences, but also equivalences going from one element to another. Such a structure is called a groupoid. Gauge fields form not a set with smooth structure, a smooth space, but a groupoid with smooth structure: a smooth groupoid.

At least ordinary gauge fields do. More generally, the gauge princple goes further: it is in general also a mistake to assume that given two gauge transformations ${\lambda }_1,{\lambda }_2:{\nabla }_1\to {\nabla }_2$ between two gauge fields, it makes good sense to ask whether they are equal or not. Again, the gauge prinicle says that we should instead ask if there is a gauge-of-gauge transformation between them, that exhibits a gauge equivalence $\rho :{\lambda }_1\simeq {\lambda }_2$. If these gauge-of-gauge equivalences are nontrivial it would seem that gauge fields form a generalization of the notion of a groupoid called a 2-groupoid. But in general the gauge principle goes on: we can in general never decide if two $n$-fold gauge-of-gauge transformations are actually equal, all we have is, possibly, an $(n+1)$-fold gauge transformation going between them which exhibits their gauge equivalence, this being so for all $0<n<\mathrm{\infty }$. One therefore says that gauge fields in general form an ∞-groupoid whose n-morphisms are $n$-fold gauge-of-gauge transformations. These $\mathrm{\infty }$-groupoids are also called homotopy types. And since at the same time there is still a notion of gauge fields varying smoothly, these are smooth ∞-groupoids or smooth homotopy types. This is what we discuss here.

Remarkably, the same concept appears in constructive mathematics: there it is in general wrong to consider an equality between two terms ${\nabla }_1,{\nabla }_2:[X,B{G}_{\mathrm{conn}}]$, instead one is to consider an explicit proof that they are equal, provide an explicit equivalence between them. Such a proof $\lambda$ is itself a term, of identity type, written just as before, $\lambda :({\nabla }_1\simeq {\nabla }_2)$. And, in turn, the same applies to these proofs of equivalence themselves: in constructive mathematics one demands a proof $\rho$ that two equivalences ${\lambda }_1,{\lambda }_2:({\nabla }_1\simeq {\nabla }_2)$ are equivalent, and hence a term $\rho :({\lambda }_1\simeq {\lambda }_2)$ of a higher identity type. If this goes ever on, and one speak of intensional type theory or homotopy type theory.

This remarkable matching of higher gauge theory and homotopy type theory is what drives the discussion here.

Contents:

1. Model layer

   This introduces groupoids, then smooth groupoids and eventually ∞-groupoids modeled as Kan complexes and then smooth ∞-groupoids.

2. Semantics layer

   This introduces the general abstract framework for smooth ∞-groupoids which is cohesion of (∞,1)-toposes.

3. Syntactic layer

   This introduces the internal language of cohesive (∞,1)-toposes, called cohesive homotopy type theory.

Remark The reader who is after elementary exposition may want to skip over this discussion of homotopy theory on first reading to the next chapterGroups_ and only come back here for reference as need arises.

Model Layer

Groupoids

For the content of this section currently see at Essence of gauge theory: Groupoids and basic homotopy 1-type theory below.

Gauge transformations in electromagnetism

The mathematical notion of groupoid is well familiar, in slight disguise, in basic gauge theory. Here we make this explicit for basic electromagnetism.

We have seen in The electromagnetic field strength that a configuration of the electromagnetic field on ${ℝ}^4$ is given by a differential 1-form $A\in {\Omega }^1({ℝ}^4)$, the “electromagnetic potential”. It describes a field configuration with field strength Lorentz tensor (with respect to the canonical coordinates $(t,x^1,x^2,x^3)≔(x^i{)}_{i=1}^4=$ on ${ℝ}^4$)

$$\begin{array}{rl}F& =dA\\ & =E_{1}d{x}^1\wedge dt+E_{1}d{x}^1\wedge dt+E_{1}d{x}^1\wedge dt+\\ & +B_{1}d{x}^2\wedge d{x}^3+B_{2}d{x}^3\wedge d{x}^1+B_{3}d{x}^1\wedge d{x}^2\end{array}$$\begin{aligned} F & = \mathbf{d}A \\ & = E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + \\ & + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2 \end{aligned}

with electric field strength $(E_i\in C^{\mathrm{\infty }}({ℝ}^4){)}_{i=1}^3$ and magnetic field strength $(B_i\in C^{\mathrm{\infty }}({ℝ}^4){)}_{i=1}^3$ given that is given by the de Rham differential of $A$:

$$\begin{array}{r}F=dA\end{array}.$$\begin{aligned} F = \mathbf{d}A \end{aligned} \,.

After Maxwell it was thought that $F\in {\Omega }^2({ℝ}^4)$ alone genuinely reflects the configuration of the electromagnetic field. But with the discovery of quantum mechanics it became clear that it is indeed the potential $A$ itself that reflects the configuration of the electromagnetic field: in the presence of magnetic flux or other topoligical constraints, there can be different $A,A\prime$ with the same $F=dA=dA\prime$ which nevertheless describe experimentally distinguishable electromagnetic field configurations. (Distinguishable by the Aharonov-Bohm effect and also to some extent by Dirac charge quantization; this is discussed at Circle-principal connections below.)

However, not all different gauge potentials describe different physics. The actual configuration space of electromagnetism on a spacetime $X$ is finer than ${\Omega }_{\mathrm{cl}}^2(X)$ but coarser than ${\Omega }^1(X)$. And it is not quite a smooth space itself, but a smooth groupoid:

one finds that two electromagnetic potentials $A,A\prime \in {\Omega }^1({ℝ}^4)$ for which there is a function $\lambda \in C^{\mathrm{\infty }}({ℝ}^4)$ such that

$$A\prime =A+d\lambda$$A' = A + \mathbf{d}\lambda

represent different but equivalent field configurations. One says that $\lambda$ induces a gauge transformation from $A$ to $A\prime$. We write $\lambda :A\stackrel{\simeq }{\to }A\prime$ to reflect this.

So the configuration space of electromagnetism does not just have points and coordinate systems. But it is also equipped with the information of a space of gauge transformations between any two coordinate systems laid out in it (which may be empty).

To see what the structure of such a smooth gauge groupoid should be, notice that the above defines an action of smooth functions $\lambda$ on smooth $1$-forms $A$,

Definition

For any $U\in$ CartSp, Write ${\Omega }_{\mathrm{vert}}^1(X\times U)$ for the set of differential 1-forms on $X\times U$ with no components along $U$, and write $C^{\mathrm{\infty }}(X\times U,U(1))$ for the group of circle group valued smooth functions. There is an action of this group on the 1-forms

$${\rho }_U:{\Omega }_{\mathrm{vert}}^1(X\times U)\times C^{\mathrm{\infty }}(X\times U,U(1))\to {\Omega }_{\mathrm{vert}}^1(X\times U)$$\rho_U \colon \Omega^1_{vert}(X \times U) \times C^\infty(X \times U, U(1)) \to \Omega^1_{vert}(X \times U)

given by

$${\rho }_U:(A,\lambda )\mapsto A+d_X\lambda .$$\rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}_X \lambda \,.

Given such an action of a discrete group on a set, we might be demoted to form the quotient set ${\Omega }_{\mathrm{vert}}^1(X\times U)/C^{\mathrm{\infty }}(X\times U,U(1))$. This set contains the gauge equivalence classes of $U$-parameterized collections of electromagnetic gauge fields on $X$. But it turns out that this is too little information to correctly capture physics. For that instead we need to remember not just that two gauge fields are equivalent, but how they are equivalent. That is, we for $\lambda$ a gauge transformation from $A_1$ to $A_2$, we should have an equivalence $\lambda :A_1\stackrel{\simeq }{\to }{A}_2$.

Definition

The action groupoid

$${\Omega }_{\mathrm{vert}}^1(X\times U)⫽C^{\mathrm{\infty }}(X\times U,U(1))≔({\Omega }_{\mathrm{vert}}^1(X\times U)\times C^{\mathrm{\infty }}(X\times U,U(1))\stackrel{\stackrel{t=\rho }{\to }}{\underset{s=p_1}{\to }}{\Omega }_{\mathrm{vert}}^1(X\times U))$$\Omega^1_{vert}(X\times U)\sslash C^\infty(X \times U, U(1)) \coloneqq ( \Omega^1_{vert}(X\times U) \times C^\infty(X \times U, U(1)) \stackrel{\overset{t = \rho}{\to}}{\underset{s = p_1}{\to}} \Omega^1_{vert}(X\times U) )

is the groupoid whose

• objects are $U$-parameterized collections of gauge potentials $A\in {\Omega }_{\mathrm{vert}}^1(X\times U)$;

• morphisms are pairs $(A,\lambda )$ with $A$ an object and $\lambda \in C^{\mathrm{\infty }}(X\times U,U(1))$, with domain $A$ and codomain $A+d_X\lambda$;

• composition is given by multiplication of the $\lambda$-labels in the group $C^{\mathrm{\infty }}(X\times U,U(1))$.

This is the discrete gauge groupoid for $U$-parameterized collections of fields. It refines the gauge group, which is recoverd as its fundamental group:

Proposition

Let $A_0≔0\in {\Omega }_{\mathrm{vert}}^1(X\times U)$ be the trivial gauge field. Then its automorphism group in the gauge groupoid of def. 64 is the group of circle-group valued functions on $U$:

$${\pi }_1({\Omega }_{\mathrm{vert}}^1(X\times U)⫽C^{\mathrm{\infty }}(X\times U,U(1)),A_0)=\mathrm{Aut}(A_0)\simeq C^{\mathrm{\infty }}(U,U(1)).$$\pi_1(\Omega^1_{vert}(X\times U)\sslash C^\infty(X \times U, U(1)), A_0) = Aut(A_0) \simeq C^\infty(U,U(1)) \,.

Proof

By definition, an automorphism of $A_0$ is given by a function $\lambda \in C^{\mathrm{\infty }}(X\times U,U(1))$ such that $A_0=A_0+d_X\lambda$. This is the case precisely if $d_X\lambda =0$, hence if $\lambda$ is contant along $X$. But a function on $X\times U$ which is constant along $X$ is canonically identified with a function on just $U$.

All this data in in fact natural in $U$. Recall that ${\Omega }_{\mathrm{vert}}^1(X\times U)={\Omega }^1(X)(U)$ is the set of $U$-charts of the smooth moduli space ${\Omega }^1(X)$ of smooth 1-forms on $X$. Similarly $C^{\mathrm{\infty }}(X\times U)=C^{\mathrm{\infty }}(X)(U)$.

Proposition

There is a homomorphism of smooth spaces (def. 14)

$$\rho :{\Omega }^1(X)\times C^{\mathrm{\infty }}(X)\to {\Omega }^1(X)$$\rho \colon \mathbf{\Omega}^1(X)\times \mathbf{C}^\infty(X) \to \mathbf{\Omega}^1(X)

from the product smooth space, def. 17, of the smooth moduli spaces of 1-forms and 0-forms on $X$, def. 38, to that of smooth functions, def. 20, whose component over $U\in$ CartSp is the action

$${\rho }_U:(A,\lambda )\mapsto A+d\lambda$$\rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}\lambda

of def. 63.

We then also want to consider a smooth action groupoid.

Definition

Write

$${\Omega }^1(X)⫽C^{\mathrm{\infty }}(X,U(1)):{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Grpd}$$\mathbf{\Omega}^1(X) \sslash \mathbf{C}^\infty(X, U(1)) : CartSp^{op} \to Grpd

for the contravariant functor from coordinate systems to the category of groupoids, which assigns to $U\in \mathrm{CartSp}$ the discrete action groupoid of $U$-collections of gauge fields of def. 64.

$$U\mapsto {\Omega }_{\mathrm{vert}}^1(X\times U)⫽C^{\mathrm{\infty }}(X\times U,U(1)).$$U \mapsto \Omega^1_{vert}(X \times U)\sslash C^\infty(X\times U, U(1)) \,.

Such a structure presheaf of groupoids is a common joint generalization of the notion of discrete groupoids and smooth spaces. We call them smooth groupoids. This is what we turn to in Smooth groupoids

Smooth groupoids

Definition

Write Grpd${}_1$ for the category of groupoids and functors between them. Write then

$$\mathrm{gPsh}(\mathrm{CartSp})≔\mathrm{Funct}({\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Grpd})$$gPsh(CartSp) \coloneqq Funct(CartSp^{op}, Grpd)

for the category of groupoid-valued presheaves.

Definition

For $n\in \mathbb{N}$, and $0<r<1\in ℝ$ let

$${ℝ}^n\simeq D_r^n\hookrightarrow {ℝ}^n$$\mathbb{R}^n \simeq D^n_r \hookrightarrow \mathbb{R}^n

Be the smooth function that regards ${ℝ}^n$ as the standard $n$-disk of radius $r$ around the origin in ${ℝ}^n$.

For $X\in \mathrm{gPSh}(\mathrm{CartSp})$ we write

$$(D^n{)}^{*}X≔{\underrightarrow{\lim }}_{r}X(D_r^n)\in \mathrm{Grpd}$$(D^n)^* X \coloneqq {\lim_\to}_r X(D^n_r) \in Grpd

for the stalk of $X$ at the origin of ${ℝ}^n$. This is a functor

$$(D^n{)}^{*}:\mathrm{gPSh}(\mathrm{CartSp})\to \mathrm{Grpd}.$$(D^n)^* \colon gPSh(CartSp) \to Grpd \,.

Definition

A morphism $f:X\to Y$ in $\mathrm{gPSh}(\mathrm{CartSp})$ we call a local weak equivalence if for every $n\in \mathbb{N}$ the stalk

$$(D^n{)}^{*}f:(D^n{)}^{*}X\to (D^n{)}^{*}Y$$(D^n)^* f \colon (D^n)^* X \to (D^n)^* Y

is an equivalence of groupoids.

Definition

Write

$$\mathrm{Smooth}1\mathrm{Type}≔L_{\mathrm{lwe}}\mathrm{gPSh}(\mathrm{CartSp})$$Smooth1Type \coloneqq L_{lwe} gPSh(CartSp)

for the (2,1)-category which is the simplicial localization of groupoid-valued presheaves at the local weak equivalences.

An object $X\in \mathrm{Smooth}1\mathrm{Type}$ we call a smooth groupoid or smooth homotopy 1-type.

Example

Every smooth space is canonically a smooth groupoid with only identity morphisms.

Proposition

The canonical identification yields a full subcategory

$$\mathrm{Smooth}0\mathrm{Type}\hookrightarrow \mathrm{Smooth}1\mathrm{Type}.$$Smooth0Type \hookrightarrow Smooth1Type \,.

Example

For $X$ a smooth space and $G$ a smooth group and

$$\rho :X\times G\to X$$\rho : X \times G \to X

an action then the action groupoid

$$X⫽G\in \mathrm{Smooth}1\mathrm{Type}$$X \sslash G \in Smooth1Type

$$X⫽G=(X\times G\stackrel{\stackrel{\rho }{\to }}{\underset{p_1}{\to }}X)$$X \sslash G = \left( X \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to}} X \right)

is a smooth groupoid.

Smooth path groupoid

• smooth path groupoid

Differential 1-forms are smooth incremental path measures

We give a more intrinsic characterization of differential 1-forms.

Definition

A smooth path with sitting instants in ${ℝ}^k$ is a smooth function $\gamma :ℝ\to {ℝ}^k$ such

1. the value of $\gamma$ converges to two points $x=\underset{t\to -\mathrm{\infty }}{\lim }\gamma \in {ℝ}^k$ and $y=\underset{t\to \mathrm{\infty }}{\lim }\gamma \in {ℝ}^k$;

2. the value of all derivatives of $\gamma$ converges to 0 at $t\to \pm \mathrm{\infty }$.

A localized diffeomorphism $\varphi :ℝ\to ℝ$ is a diffeomorphism such that there is an an open ball in $ℝ$ outside of which $\varphi$ restricts to the identity.

Write

$$[ℝ,{ℝ}^k]⫽\mathrm{Diff}(ℝ)\in \mathrm{Smooth}0\mathrm{Type}$$[\mathbb{R}, \mathbb{R}^k]\sslash Diff(\mathbb{R}) \in Smooth0Type

for the smooth quotient space of smooth paths modulo localized diffeomorphism.

Definition

Say that a smooth functor $P_1({ℝ}^k)\to Bℝ$ from the smooth path groupoid of ${ℝ}^k$ is

• a homomorphism of smooth space

  $$S:[ℝ,{ℝ}^k]\to ℝ$$S \colon [\mathbb{R}, \mathbb{R}^k] \to \mathbb{R}

• such that

  1. composition of paths ${\gamma }_1,{\gamma }_2$ is sent to addition in $ℝ$

     $$S({\gamma }_2\circ {\gamma }_1)=S({\gamma }_1)+S({\gamma }_2).$$S(\gamma_2\circ \gamma_1) = S(\gamma_1) + S(\gamma_2) \,.

  2. the constant path is sent to 0.

Proposition

There is an equivalence

$$[P_1(-),Bℝ]\stackrel{\stackrel{{\int }_{(-)}(-)}{\leftarrow }}{\underset{}{\to }}\mathrm{DK}(C^{\mathrm{\infty }}(-)\stackrel{d}{\to }{\Omega }^1(-).$$[\mathbf{P}_1(-), \mathbf{B}\mathbb{R}] \stackrel{\overset{\int_{(-)}(-)}{\leftarrow}}{\underset{}{\to}} DK(C^\infty(-) \stackrel{\mathbf{d}}{\to} \Omega^1(-) \,.

(SchreiberWaldorf).

$\mathrm{\infty }$-Groupoids and Kan complexes

An ∞-groupoid is first of all supposed to be a structure that has k-morphisms for all $k\in \mathbb{N}$, which for $k\ge 1$ go between $(k-1)$-morphisms. A useful tool for organizing such collections of morphisms is the notion of a simplicial set. This is a functor on the opposite category of the simplex category $\Delta$, whose objects are the abstract cellular $k$-simplices, denoted $[k]$ or $\Delta [k]$ for all $k\in \mathbb{N}$, and whose morphisms $\Delta [k_1]\to \Delta [k_2]$ are all ways of mapping these into each other. So we think of such a simplicial set given by a functor

$$K:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$$K : \Delta^{op} \to Set

as specifying

• a set $[0]\mapsto K_0$ of objects;

• a set $[1]\mapsto K_1$ of morphism;

• a set $[2]\mapsto K_2$ of 2-morphism;

• a set $[3]\mapsto K_3$ of 3-morphism;

and generally

• a set $[k]\mapsto K_k$ of k-morphisms

as well as specifying

• functions $([n]\hookrightarrow [n+1])\mapsto (K_{n+1}\to K_n)$ that send $n+1$-morphisms to their boundary $n$-morphisms;

• functions $([n+1]\to [n])\mapsto (K_n\to K_{n+1})$ that send $n$-morphisms to identity $(n+1)$-morphisms on them.

The fact that $K$ is supposed to be a functor enforces that these assignments of sets and functions satisfy conditions that make consistent our interpretation of them as sets of $k$-morphisms and source and target maps between these. These are called the simplicial identities.

But apart from this source-target matching, a generic simplicial set does not yet encode a notion of composition of these morphisms.

For instance for ${\Lambda }^1[2]$ the simplicial set consisting of two attached 1-cells

$${\Lambda }^1[2]=\left\{\begin{array}{ccc}& & 1\\ & \nearrow & & \searrow \\ 0& & & & 2\end{array}\right\}$$\Lambda^1[2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}

and for $(f,g):{\Lambda }^1[2]\to K$ an image of this situation in $K$, hence a pair $x_0\stackrel{f}{\to }{x}_1\stackrel{g}{\to }{x}_2$ of two composable 1-morphisms in $K$, we want to demand that there exists a third 1-morphisms in $K$ that may be thought of as the composition $x_0\stackrel{h}{\to }{x}_2$ of $f$ and $g$. But since we are working in higher category theory (and not to be evil), we want to identify this composite only up to a 2-morphism equivalence

$$\begin{array}{ccc}& & x_1\\ & {}^f\nearrow & {\Downarrow }^{\simeq }& {\searrow }^g\\ x_0& & \stackrel{h}{\to }& & x_2\end{array}.$$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\mathrlap{\simeq}}& \searrow^{\mathrlap{g}} \\ x_0 &&\stackrel{h}{\to}&& x_2 } \,.

From the picture it is clear that this is equivalent to demanding that for ${\Lambda }^1[2]\hookrightarrow \Delta [2]$ the obvious inclusion of the two abstract composable 1-morphisms into the 2-simplex we have a diagram of morphisms of simplicial sets

$$\begin{array}{ccc}{\Lambda }^1[2]& \stackrel{(f,g)}{\to }& K\\ \downarrow & {\nearrow }_{\exists h}\\ \Delta [2]\end{array}.$$\array{ \Lambda^1[2] &\stackrel{(f,g)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists h}} \\ \Delta[2] } \,.

A simplicial set where for all such $(f,g)$ a corresponding such $h$ exists may be thought of as a collection of higher morphisms that is equipped with a notion of composition of adjacent 1-morphisms.

For the purpose of describing groupoidal composition, we now want that this composition operation has all inverses. For that purpose, notice that for

$${\Lambda }^2[2]=\left\{\begin{array}{ccc}& & 1\\ & & & \searrow \\ 0& & \to & & 2\end{array}\right\}$$\Lambda^2[2] = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}

the simplicial set consisting of two 1-morphisms that touch at their end, hence for

$$(g,h):{\Lambda }^2[2]\to K$$(g,h) : \Lambda^2[2] \to K

two such 1-morphisms in $K$, then if $g$ had an inverse $g^{-1}$ we could use the above composition operation to compose that with $h$ and thereby find a morphism $f$ connecting the sources of $h$ and $g$. This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form

$$\begin{array}{ccc}{\Lambda }^2[2]& \stackrel{(g,h)}{\to }& K\\ \downarrow & {\nearrow }_{\exists f}\\ \Delta [2]\end{array}.$$\array{ \Lambda^2[2] &\stackrel{(g,h)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists f}} \\ \Delta[2] } \,.

Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in $K$.

In order for this to qualify as an $\mathrm{\infty }$-groupoid, this composition operation needs to satisfy an associativity law up to coherent 2-morphisms, which means that we can find the relevant tetrahedras in $K$. These in turn need to be connected by pentagonators and ever so on. It is a nontrivial but true and powerful fact, that all these coherence conditions are captured by generalizing the above conditions to all dimensions in the evident way:

let ${\Lambda }^i[n]\hookrightarrow \Delta [n]$ be the simplicial set – called the $i$th $n$-horn – that consists of all cells of the $n$-simplex $\Delta [n]$ except the interior $n$-morphism and the $i$th $(n-1)$-morphism.

Then a simplicial set is called a Kan complex, if for all images $f:{\Lambda }^i[n]\to K$ of such horns in $K$, the missing two cells can be found in $K$- in that we can always find a horn filler $\sigma$ in the diagram

$$\begin{array}{ccc}{\Lambda }^i[n]& \stackrel{f}{\to }& K\\ \downarrow & {\nearrow }_{\sigma }\\ \Delta [n]\end{array}.$$\array{ \Lambda^i[n] &\stackrel{f}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\sigma}} \\ \Delta[n] } \,.

The basic example is the nerve $N(C)\in \mathrm{sSet}$ of an ordinary groupoid $C$, which is the simplicial set with $N(C{)}_k$ being the set of sequences of $k$ composable morphisms in $C$. The nerve operation is a full and faithful functor from 1-groupoids into Kan complexes and hence may be thought of as embedding 1-groupoids in the context of general ∞-groupoids.

Model categories

• model structure on simplicial sets

• model structure on simplicial presheaves

Cohesive $\mathrm{\infty }$-Groupoids and locally Kan simplicial presheaves

But we need a bit more than just bare ∞-groupoids. In generalization to Lie groupoids, we need ∞-Lie groupoids. A useful way to encode that an $\mathrm{\infty }$-groupoid has extra structure modeled on geometric test objects that themselves form a category $C$ is to remember the rule which for each test space $U$ in $C$ produces the $\mathrm{\infty }$-groupoid of $U$-parameterized families of $k$-morphisms in $K$. For instance for an ∞-Lie groupoid we could test with each Cartesian space $U={ℝ}^n$ and find the $\mathrm{\infty }$-groupoids $K(U)$ of smooth $n$-parameter families of $k$-morphisms in $K$.

This data of $U$-families arranges itself into a presheaf with values in Kan complexes

$$K:C^{\mathrm{op}}\to \mathrm{KanCplx}\hookrightarrow \mathrm{sSet}$$K : C^{op} \to KanCplx \hookrightarrow sSet

hence with values in simplicial sets. This is equivalently a simplicial presheaf of sets. The functor category $[C^{\mathrm{op}},\mathrm{sSet}]$ on the opposite category of the category of test objects $C$ serves as a model for the (∞,1)-category of $\mathrm{\infty }$-groupoids with $C$-structure.

While there are no higher morphisms in this functor 1-category that could for instance witness that two $\mathrm{\infty }$-groupoids are not isomorphic, but still equivalent, it turns out that all one needs in order to reconstruct all these higher morphisms (up to equivalence!) is just the information of which morphisms of simplicial presheaves would become invertible if we were keeping track of higher morphism. These would-be invertible morphisms are called weak equivalences and denoted $K_1\stackrel{\simeq }{\to }{K}_2$.

For common choices of $C$ there is a well-understood way to define the weak equivalences $W\subset \mathrm{mor}[C^{\mathrm{op}},\mathrm{sSet}]$, and equipped with this information the category of simplicial presheaves becomes a category with weak equivalences . There is a well-developed but somewhat intricate theory of how exactly this 1-cagtegorical data models the full higher category of structured groupoids that we are after, but for our purposes we essentially only need to work inside the category of fibrant objects of a model category structure on simplicial presheaves, which in practice amounts to the fact that we use the following three basic constructions:

1. ∞-anafunctors – A morphisms $X\to Y$ between $\mathrm{\infty }$-groupoids with $C$-structure is not just a morphism $X\to Y$ in $[C^{\mathrm{op}},\mathrm{sSet}]$, but is a span of such ordinary morphisms

   $$\begin{array}{ccc}\hat{X}& \to & Y\\ {\downarrow }^{\simeq }\\ X\end{array}$$\array{ \hat X &\to& Y \\ \downarrow^{\mathrlap{\simeq}} \\ X }

   where the left leg is a weak equivalence. This is sometimes called an $\mathrm{\infty }$-anafunctor from $X$ to $Y$.

2. homotopy pullback – For $A\to B\stackrel{p}{\leftarrow }C$ a diagram, the (∞,1)-pullback of it is the ordinary pullback in $[C^{\mathrm{op}},\mathrm{sSet}]$ of a replacement diagram $A\to B\stackrel{\hat{p}}{\leftarrow }\hat{C}$, where $\hat{p}$ is a good replacement of $p$ in the sense of the following factorization lemma.

3. factorization lemma – For $p:C\to B$ a morphism in $[C^{\mathrm{op}},\mathrm{sSet}]$, a good replacement $\hat{p}:\hat{C}\to B$ is given by the composite vertical morphism in the ordinary pullback diagram

   $$\begin{array}{ccc}\hat{C}& \to & C\\ \downarrow & & {\downarrow }^p\\ B^{\Delta [1]}& \to & B\\ \downarrow \\ B\end{array},$$\array{ \hat C &\to& C \\ \downarrow && \downarrow^{\mathrlap{p}} \\ B^{\Delta[1]} &\to& B \\ \downarrow \\ B } \,,

where $B^{\Delta [1]}$ is the path object of $B$: the simplicial presheaf that is over each $U\in C$ the simplicial path space $B(U{)}^{\Delta [1]}$.

• simplicial presheaf

• model structure on simplicial presheaves

Good covers and split hypercovers

• Cech nerve

• good open cover

• split hypercover

Proposition

For $X$ a smooth manifold let $\{U_i\hookrightarrow X{\}}_i$ be an open cover. This is a differentiably good open cover, def. 9, precisely if the corresponding Cech nerve $C(\{U_i\})\to X$ in $\mathrm{sPh}(\mathrm{CartSp}{)}_{\mathrm{proj},\mathrm{loc}}$ is a split hypercover.

Semantic Layer

$\mathrm{\infty }$-Toposes

• (∞,1)-topos

Slice $\mathrm{\infty }$-toposes

• slice (∞,1)-topos

Local, $\mathrm{\infty }$-connected and cohesive $\mathrm{\infty }$-toposes

• local (∞,1)-topos $(♭⊣♯)$

• locally ∞-connected (∞,1)-topos $(\Pi ⊣♭)$

• cohesive (∞,1)-topos $(\Pi ⊣♭⊣♯)$

cohesion

• (shape modality $⊣$ flat modality $⊣$ sharp modality)

  $(ʃ⊣♭⊣♯)$

  • discrete object, codiscrete object, concrete object

  • structures in cohesion

differential cohesion

• (reduction modality $⊣$ infinitesimal shape modality $⊣$ infinitesimal flat modality)

  $(\mathrm{Red}⊣{ʃ}_{\inf }⊣{♭}_{\inf })$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

Some terminology:

Definition

For $X\in H$ locally $\mathrm{\infty }$-connected, we say $\Pi (X)$ is the shape? of $X$. If $H$ is in addition cohesive we also say that $\Pi (X)$ is the geometric realization of $X$ or the fundamental ∞-groupoid of $X$.

If $\Pi (X)\simeq *$ we say that $X$ is geometrically contractible.

The $\mathrm{\infty }$-topos of smooth $\mathrm{\infty }$-groupoids

• Lie groupoid, differentiable stack

• smooth ∞-groupoid

Local $\mathrm{\infty }$-Connectedness

(…)

Locality

(…)

Cohesion

(…)

Syntactic Layer

Identity types

• identity type

Homotopy type theory

• h-level

• homotopy type theory

Cohesive modality II: flat types

flat type $♭A$

maps $X\to ♭BG$ are flat $G$-connections

$$BG:\mathrm{Type}⊢{\mathrm{UnderlyingBundle}}_{BG}:♭BG\to BG$$\mathbf{B}G \colon Type \;\vdash\; UnderlyingBundle_{\mathbf{B}G} \colon \flat \mathbf{B}G \to \mathbf{B}G

Groups

Model Layer

1-Groups

Discrete

• group

• delooping homotopy 1-type

Topological and Lie groups

• topological group

• Lie group

Simplicial groups

Discrete

• simplicial group

Abelian

• abelian group

Topological and Lie simplicial groups

• simplicial topological space

• geometric realization of simplicial topological spaces

Connected groupoids

any Lie group $G$ induces its delooping Lie groupoid

$$BG=*⫽G=(G\overrightarrow{\to }*).$$\mathbf{B}G = * \sslash G = \left( G \stackrel{\to}{\to} * \right) \,.

2-Groups

Discrete

• 2-group

• crossed module

Braided and abelian

• braided 2-group

• symmetric 2-group

Abelian $\mathrm{\infty }$-groups

Chain complex

• chain complex

Spectrum

• spectrum, infinite loop space

Dold-Kan correspondence

• simplicial group, Dold-Kan correspondence, chain complex

• stable Dold-Kan correspondence

Definition

Write

$$\mathrm{DK}:{\mathrm{Ch}}_{•}(\mathrm{Ab}(\mathrm{Smooth}0\mathrm{Type}))\stackrel{\Xi }{\to }\mathrm{sAb}(\mathrm{Smooth}0\mathrm{Type})\to \mathrm{Smooth}0{\mathrm{Type}}^{{\Delta }^{\mathrm{op}}}\to L_{\mathrm{whe}}\mathrm{Smooth}0{\mathrm{Type}}^{{\Delta }^{\mathrm{op}}}\simeq \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$$DK \colon Ch_\bullet(Ab(Smooth0Type)) \stackrel{\Xi}{\to} sAb(Smooth0Type) \to Smooth0Type^{\Delta^{op}} \to L_{whe} Smooth0Type^{\Delta^{op}} \simeq Smooth \infty Grpd

for the functor that sends a chain complex of abelian group objects in smooth spaces first to the simplicial abelian group in smooth spaces given by the Dold-Kan correspondence, then forgets the abelian group structure and finally regards the resulting simplicial smooth space as a smooth ∞-groupoid under simplicial localization.

Cohomology

• cohomology

Nonabelian cohomology

• nonabelian cohomology

Abelian sheaf cohomology

• abelian sheaf cohomology

• Cech cohomology

Semantic Layer

$A_{\mathrm{\infty }}$-types

• A-infinity space

$\mathrm{\infty }$-Groups

• infinity-group

• looping and delooping

Syntactic Layer

Pointed connected types

• connected type?

Identity types of connected types

(…)

Principal bundles

Model Layer

Principal 1-Bundles

Let $G$ be a Lie group and $X$ a smooth manifold (all our smooth manifolds are assumed to be finite dimensional and paracompact).

We give a discussion of smooth $G$-principal bundles on $X$ in a manner that paves the way to a straightforward generalization to a description of principal ∞-bundles.

From the group $G$ we canonically obtain a groupoid that we write $BG$ and call the delooping groupoid of $G$. Formally this groupoid is

$$BG=(G\overrightarrow{\to }*)$$\mathbf{B}G = (G \stackrel{\to}{\to} *)

with composition induced from the product in $G$. A useful cartoon of this groupoid is

$$BG=\left\{\begin{array}{ccc}& & •\\ & {}^{g_1}\nearrow & =& {\searrow }^{g_2}\\ •& & \stackrel{g_2\cdot g_1}{\to }& & •\end{array}\right\}$$\mathbf{B}G = \left\{ \array{ && \bullet \\ & {}^{\mathllap{g_1}}\nearrow &=& \searrow^{\mathrlap{g_2}} \\ \bullet &&\stackrel{g_2 \cdot g_1 }{\to}&& \bullet } \right\}

where the $g_i\in G$ are elements in the group, and the bottom morphism is labeled by forming the product in the group. (The order of the factors here is a convention whose choice, once and for all, does not matter up to equivalence.)

But we get a bit more, even. Since $G$ is a Lie group, there is smooth structure on $BG$ that makes it a Lie groupoid, an internal groupoid in the category Diff of smooth manifolds: its collections of objects (trivially) and of morphisms each form a smooth manifold, and all structure maps (source, target, identity, composition) are smooth functions. We shall write

$$BG\in \mathrm{LieGrpd}$$\mathbf{B}G \in LieGrpd

for $BG$ regarded as equipped with this smooth structure. Here and in the following the boldface is to indicate that we have an object equipped with a bit more structure – here: smooth structure – than present on the object denoted by the same symbols, but without the boldface. Eventually we will make this precise by having the boldface symbols denote objects in the (∞,1)-topos Smooth∞Grpd which are taken by forgetful functors to objects in ∞Grpd denoted by the corresponding non-boldface symbols.¹

Also the smooth manifold $X$ may be regarded as a Lie groupoid – a groupoid with only identity morphisms. Its cartoon description is simply

$$X=\{x\stackrel{\mathrm{id}}{\to }x\}.$$X = \{x \stackrel{id}{\to} x \} \,.

But there are other groupoids associated with $X$:

Let $\{U_i\to X{\}}_{i\in I}$ be an open cover of $X$. To this is canonically associated the Cech groupoid $C(\{U_i\})$. Formally we may write this groupoid as

$$C(\{U_i\})=(\coprod _{i,j}{U}_i\cap U_j\stackrel{\stackrel{p_1}{\to }}{\underset{p_2}{\to }}\coprod _i{U}_i).$$C(\{U_i\}) = \left( \coprod_{i,j} U_i \cap U_j \stackrel{\overset{p_1}{\to}}{\underset{p_2}{\to}} \coprod_i U_i \right) \,.

A useful cartoon description of this groupoid is

$$C(\{U_i\})=\left\{\begin{array}{ccc}& & (x,j)\\ & \nearrow & =& \searrow \\ (x,i)& & \to & & (x,k)\end{array}\right\}.$$C(\{U_i\}) = \left\{ \array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) &&\to&& (x,k) } \right\} \,.

This indicates that the objects of this groupoid are pairs $(x,i)$ consisting of a point $x\in X$ and a patch $U_i\subset X$ that contains $x$, and a morphism is a triple $(x,i,j)$ consisting of a point and two patches, that both contain the point, in that $x\in U_i\cap U_j$. The triangle in the above cartoon symbolizes the evident way in which these morphisms compose. All this inherits a smooth structure from the fact that the $U_i$ are smooth manifolds and the inclusions $U_i\to X$ are smooth functions. hence also $C(U)$ becomes a Lie groupoid.

There is a canonical functor

$$C(\{U_i\})\to X:(x,i)\mapsto x.$$C(\{U_i\}) \to X \;\; :\;\; (x,i) \mapsto x \,.

This functor is an internal functor in Diff and moreover it is evidently essentially surjective and full and faithful.

However, while essential surjectivity and full-and-faithfulness implies that the underlying bare functor has a homotopy-inverse, that homotopy-inverse never has itself smooth component maps, unless $X$ itself is a Cartesian space and the chosen cover is trivial.

We do however want to think of $C(\{U_i\})$ as being equivalent to $X$ even as a Lie groupoid. One says that a smooth functor whose underlying bare functor is an equivalence of groupoids is a weak equivalence of Lie groupoids, which we write as $C(\{U_i\})\stackrel{\simeq }{\to }X$. Moreover, we shall think of $C(U)$ as a good equivalent replacement of $X$ if it comes from a cover that is in fact a good open cover in that all its non-empty finite intersections $U_{i_0\cdots i_k}:=U_{i_0}\cap \cdots \cap U_{i_k}$ are diffeomorphic to the Cartesian space ${ℝ}^{\dim X}$.

We shall discuss later in which precise sense this condition makes $C(U)$ good in the sense that smooth functors out of $C(U)$ model the correct notion of morphism out of $X$ in the context of smooth groupoids (namely it will mean that $C(U)$ is cofibrant in a suitable model category structure on the category of Lie groupoids). The formalization of this statement is what (∞,1)-topos theory is all about, to which we will come. For the moment we shall be content with accepting this as an ad hoc statement.

Observe that a functor

$$g:C(U)\to BG$$g : C(U) \to \mathbf{B}G

is given in components precisely by a collection of functions

$$\{g_{ij}:U_{ij}\to G{\}}_{i,j\in I}$$\{g_{i j} : U_{i j} \to G \}_{i,j \in I}

such that on each $U_i\cap U_k\cap U_j$ the equality $g_{jk}{g}_{ij}=g_{ik}$ of smooth functions holds:

$$\left(\begin{array}{ccc}& & (x,j)\\ & \nearrow & & \searrow \\ (x,i)& & \to & & (x,k)\end{array}\right)\mapsto \left(\begin{array}{ccc}& & •\\ & {}^{g_{ij}(x)}\nearrow & & {\searrow }^{g_{jk}(x)}\\ •& & \stackrel{g_{ik}(x)}{\to }& & •\end{array}\right).$$\left( \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) &&\to&& (x,k) } \right) \mapsto \left( \array{ && \bullet \\ & {}^{\mathllap{g_{i j}(x)}}\nearrow && \searrow^{\mathrlap{g_{j k}(x)}} \\ \bullet &&\stackrel{g_{i k}(x)}{\to}&& \bullet } \right) \,.

It is well known that such collections of functions characterize $G$-principal bundles on $X$. While this is a classical fact, we shall now describe a way to derive it that is true to the Lie-groupoid-context and that will make clear how smooth principal $\mathrm{\infty }$-bundles work.

First observe that in total we have discussed so far spans of smooth functors of the form

$$\begin{array}{ccc}C(U)& \stackrel{g}{\to }& BG\\ {\downarrow }^{\simeq }\\ X\end{array}.$$\array{ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

Such spans of functors, whose left leg is a weak equivalence, are sometimes known, essentially equivalently, as Morita morphisms or generalized morphisms of Lie groupoids, as Hilsum-Skandalis morphisms or groupoid bibundles, or as anafunctors. We are to think of these as concrete models for more intrinsically defined direct morphisms $X\to BG$ in the $(\mathrm{\infty },1)$-topos of $\mathrm{\infty }$-Lie groupoids.

Now consider yet another Lie groupoid canonically associated with $G$: we shall write $EG$ for the groupoid whose formal description is

$$EG=(G\times G\stackrel{\stackrel{\cdot }{\to }}{\underset{p_1}{\to }}G)$$\mathbf{E}G = \left( G \times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}} G \right)

with the evident composition operation. The cartoon description of this groupoid is

$$EG=\left\{\begin{array}{ccc}& & g_2\\ & {}^{g_2{g}_1^{-1}}\nearrow & =& {\searrow }^{g_3{g}_2^{-1}}\\ g_1& & \stackrel{g_3{g}_1^{-1}}{\to }& & g_3\end{array}\right\},$$\mathbf{E}G = \left\{ \array{ && g_2 \\ & {}^{\mathllap{g_2 g_1^{-1}}}\nearrow &=& \searrow^{\mathrlap{g_3 g_2^{-1}}} \\ g_1 &&\stackrel{ g_3 g_1^{-1}}{\to}&& g_3 } \right\} \,,

This again inherits an evident smooth structure from the smooth structure of $G$ and hence becomes a Lie groupoid.

There is an evident forgetful functor

$$EG\to BG$$\mathbf{E}G \to \mathbf{B}G

which sends

$$(g_1\to g_2)\mapsto (•\stackrel{g_2{g}_1^{-1}}{\to }•).$$(g_1 \to g_2) \mapsto (\bullet \stackrel{g_2 g_1^{-1}}{\to} \bullet) \,.

Consider then the pullback diagram

$$\begin{array}{ccc}\tilde{P}& \to & EG\\ \downarrow & & \downarrow \\ C(U)& \stackrel{g}{\to }& BG\\ {\downarrow }^{\simeq }\\ X\end{array}$$\array{ \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }

in the category $\mathrm{Grpd}(\mathrm{Diff})$. The object $\tilde{P}$ is the Lie groupoid whose cartoon description is

$$\begin{array}{c}\tilde{P}=\left\{\begin{array}{ccccc}(x,i,g_1)& & \stackrel{}{\to }& & (x,j,g_2=g_{ij}(x)g_1)\end{array}\right\}\end{array},$$\array{ \tilde P = \left\{ \array{ (x,i,g_1) &&\stackrel{}{\to}&& (x,j,g_2 = g_{i j}(x) g_1 ) } \right\} } \,,

where there is a unique morphism as indicated, whenever the group labels match as indicated. Due to this uniqueness, this Lie groupoid is weakly equivalent to one that comes just from a manifold $P$ (it is 0-truncated)

$$\tilde{P}\stackrel{\simeq }{\to }P.$$\tilde P \stackrel{\simeq}{\to} P \,.

This $P$ is traditionally written as

$$P=(\coprod _i{U}_i\times G)/\sim ,$$P = \left( \coprod_{i} U_i \times G \right)/{\sim} \,,

where the equivalence relation is precisely that exhibited by the morphisms in $\tilde{P}$. This is the traditional way to construct a $G$-principal bundle from cocycle functions $\{g_{ij}\}$. We may think of $\tilde{P}$ as being $P$. It is a particular representative of $P$ in the $(\mathrm{\infty },1)$-topos of Lie groupoids.

While it is easy to see in components that the $P$ obtained this way does indeed have a principal $G$-action on it, for later generalizations it is crucial that we can also recover this in a general abstract way. For notice that there is a canonical action

$$(EG)\times G\to EG$$(\mathbf{E}G) \times G \to \mathbf{E}G

given by the action of $G$ on the space of objects, which are themselves identified with $G$.

Then consider the pasting diagram of pullbacks

$$\begin{array}{ccc}\tilde{P}\times G& \to & EG\times G\\ \downarrow & & \downarrow \\ \tilde{P}& \to & EG\\ \downarrow & & \downarrow \\ C(U)& \stackrel{g}{\to }& BG\\ {\downarrow }^{\simeq }\\ X\end{array}.$$\array{ \tilde P \times G &\to& \mathbf{E}G \times G \\ \downarrow && \downarrow \\ \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

The morphism $\tilde{P}\times G\to \tilde{P}$ exhibits the principal $G$-action of $G$ on $\tilde{P}$.

In summary we find

Observation

For $\{U_i\to X\}$ a good open cover, there is an equivalence of categories

$$\mathrm{SmoothFunc}(C(\{U_i\}),BG)\simeq G\mathrm{Bund}(X)$$SmoothFunc(C(\{U_i\}), \mathbf{B}G) \simeq G Bund(X)

between the functor category of smooth functors and smooth natural transformations, and the groupoid of smooth $G$-principal bundles on $X$.

It is no coincidence that this statement looks akin to the maybe more familiar statement which says that equivalence classes of $G$-principal bundles are classified by homotopy-classes of morphisms of topological spaces

$${\pi }_0\mathrm{Top}(X,BG)\simeq {\pi }_{0}G\mathrm{Bund}(X),$$\pi_0 Top(X, \mathbf{B}G) \simeq \pi_0 G Bund(X) \,,

where $BG\in$ Top is the topological classifying space of $G$. The category Top of topological spaces, regarded as an (∞,1)-category, is the archetypical (∞,1)-topos the way that Set is the archetypical topos. And it is equivalent to ∞Grpd, the $(\mathrm{\infty },1)$-category of bare ∞-groupoids. What we are seeing above is a first indication of how cohomology of bare $\mathrm{\infty }$-groupoids is lifted to a richer $(\mathrm{\infty },1)$-topos to cohomology of $\mathrm{\infty }$-groupoids with extra structure.

In fact, all of the statements that we have considered so far become conceptually simpler in the $(\mathrm{\infty },1)$-topos. We had already remarked that the anafunctor span $X\stackrel{\simeq }{\leftarrow }C(U)\stackrel{g}{\to }BG$ is really a model for what is simply a direct morphism $X\to BG$ in the $(\mathrm{\infty },1)$-topos. But more is true: that pullback of $EG$ which we considered is just a model for the homotopy pullback of just the point

$$\begin{array}{ccc}⋮& & ⋮\\ \tilde{P}\times G& \to & EG\times G\\ \downarrow & & \downarrow \\ \tilde{P}& \to & EG\\ \downarrow & & \downarrow \\ C(U)& \stackrel{g}{\to }& BG\\ {\downarrow }^{\simeq }\\ X\\ \\ \\ & \mathrm{in}\mathrm{the}\mathrm{model}\mathrm{category}& \end{array}\begin{array}{ccc}⋮& & ⋮\\ P\times G& \to & G\\ \downarrow & {⇙}_{\simeq }& \downarrow \\ P& \to & *\\ \downarrow & {⇙}_{\simeq }& \downarrow \\ X& \stackrel{}{\to }& BG\\ .\\ .\\ \\ \\ & \mathrm{in}\mathrm{the}(\mathrm{\infty },1)-\mathrm{topos}\end{array}.$$\array{ \vdots && \vdots \\ \tilde P \times G &\to& \mathbf{E}G \times G \\ \downarrow && \downarrow \\ \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X \\ {} \\ {} \\ & in\;the\;model\;category & } \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \array{ \vdots && \vdots \\ P \times G &\to& G \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G \\ . \\ . \\ \\ \\ & in\;the\;(\infty,1)-topos } \,.

Universal principal bundle

• groupal model for universal principal infinity-bundles

Principal bundles

also

$$EG=G⫽G=(G\times G\stackrel{\stackrel{\cdot }{\to }}{\underset{p_1}{\to }}G).$$\mathbf{E}G = G \sslash G = \left( G\times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}} G \right) \,.

canonical map

$$\begin{array}{c}EG\\ \downarrow \\ BG\end{array}$$\array{ \mathbf{E}G \\ \downarrow \\ \mathbf{B}G }

universal principal bundle

for $X$ a smooth space a $G$-principal bundle over $X$ is a smooth space $P$ with map $P\to X$ such that there is a diagram

$$\begin{array}{ccccc}P& \stackrel{\simeq }{\leftarrow }& \tilde{P}& \to & EG\\ \downarrow & & \downarrow & \mathrm{pb}& \downarrow \\ X& \stackrel{\simeq }{\leftarrow }& \tilde{X}& \underset{g}{\to }& BG\end{array}$$\array{ P &\stackrel{\simeq}{\leftarrow}& \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow &pb& \downarrow \\ X &\stackrel{\simeq}{\leftarrow}& \tilde X &\underset{g}{\to}& \mathbf{B}G }

where the left horizontal morphisms are weak equivalences and the right square is a pullback

Weakly principal simplicial bundles

The principal ∞-bundles that we wish to model are already the main and simplest example of the application of these three items:

Consider an object $BG\in [C^{\mathrm{op}},\mathrm{sSet}]$ which is an $\mathrm{\infty }$-groupoid with a single object, so that we may think of it as the delooping of an ∞-group $G$, let $*$ be the point and $*\to BG$ the unique inclusion map. The good replacement of this inclusion morphism is the $G$-universal principal ∞-bundle $EG\to BG$ given by the pullback diagram

$$\begin{array}{ccc}EG& \to & *\\ \downarrow & & \downarrow \\ B{G}^{\Delta [1]}& \to & BG\\ \downarrow \\ BG\end{array}$$\array{ \mathbf{E}G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G^{\Delta[1]} &\to& \mathbf{B}G \\ \downarrow \\ \mathbf{B}G }

An ∞-anafunctor $X\stackrel{\simeq }{\leftarrow }\hat{X}\to BG$ we call a cocycle on $X$ with coefficients in $G$, and the (∞,1)-pullback $P$ of the point along this cocycle, which by the above discussion is the ordinary limit

$$\begin{array}{ccccc}P& \to & EG& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ & & B{G}^I& \to & BG\\ \downarrow & & \downarrow \\ \hat{X}& \stackrel{g}{\to }& BG\\ {\downarrow }^{\simeq }\\ X\end{array}$$\array{ P &\to& \mathbf{E}G &\to& * \\ \downarrow && \downarrow && \downarrow \\ && \mathbf{B}G^I &\to& \mathbf{B}G \\ \downarrow && \downarrow \\ \hat X &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }

we call the principal ∞-bundle $P\to X$ classified by the cocycle.

It is now evident that our discussion of ordinary smooth principal bundles above is the special case of this for $BG$ the nerve of the one-object groupoid associated with the ordinary Lie group $G$.

So we find the complete generalization of the situation that we already indicated there, which is summarized in the following diagram:

$$\begin{array}{ccc}⋮& & ⋮\\ \tilde{P}\times G& \to & EG\times G\\ \downarrow & & \downarrow \\ \tilde{P}& \to & EG\\ \downarrow & & \downarrow \\ C(U)& \stackrel{g}{\to }& BG\\ {\downarrow }^{\simeq }\\ X\\ \\ \\ & \mathrm{in}\mathrm{the}\mathrm{model}\mathrm{category}& \end{array}\begin{array}{ccc}⋮& & ⋮\\ P\times G& \to & G\\ \downarrow & & \downarrow \\ P& \to & *\\ \downarrow & {⇙}_{\simeq }& \downarrow \\ X& \stackrel{}{\to }& BG\\ .\\ .\\ \\ \\ & \mathrm{in}\mathrm{the}(\mathrm{\infty },1)-\mathrm{topos}\end{array}.$$\array{ \vdots && \vdots \\ \tilde P \times G &\to& \mathbf{E}G \times G \\ \downarrow && \downarrow \\ \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X \\ {} \\ {} \\ & in\;the\;model\;category & } \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \array{ \vdots && \vdots \\ P \times G &\to& G \\ \downarrow && \downarrow \\ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G \\ . \\ . \\ \\ \\ & in\;the\;(\infty,1)-topos } \,.

• simplicial principal bundle

Semantic Layer

Principal $\mathrm{\infty }$-bundles

• homotopy fiber

• homotopy colimit

• principal infinity-bundle

Syntactic Layer

• connected type?

Structure sheaves

Model layer

• structure sheaf

Semantics layer

Let $H$ be a cohesive (∞,1)-topos $(\Pi ⊣♭⊣♯)$ equipped with differential cohesion $(\mathrm{Red}⊣{\Pi }_{\inf }⊣{♭}_{\inf })$.

Definition

For $X\in H$, write

$${\mathrm{Sh}}_H(X)\hookrightarrow H_{/X}$$Sh_{\mathbf{H}}(X) \hookrightarrow \mathbf{H}_{/X}

for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on the formally étale maps into $X$, def. 62.

We call this the petit (∞,1)-topos of $X$.

Proposition

The petit topos ${\mathrm{Sh}}_H(X)$ of def. 74 is indeed an (∞,1)-topos. Moreover the defining inclusion into the slice (∞,1)-topos is both reflective? as well as coreflective.

This is proven at differential cohesion – structure sheaves.

Definition

For $X\in H$ write

$${𝒪}_{(-)}:H\stackrel{X^{*}}{\to }{H}_{/X}\stackrel{R}{\to }{\mathrm{Sh}}_H(X)$$\mathcal{O}_{(-)} \;\colon\; \mathbf{H} \stackrel{X^*}{\to} \mathbf{H}_{/X} \stackrel{R}{\to} Sh_{\mathbf{H}}(X)

for the (∞,1)-functor which is the composite of the base change to $X$ followed by the co-reflection of prop. 41. We call this the structure sheaf of $X$.

Remark

For $X,A\in H_{\mathrm{th}}$ and for $U\to X$ a formally étale morphism in $H_{\mathrm{th}}$, we have that

$${𝒪}_X(A)(U)≔{Sh}_H(X)(U,{𝒪}_X)\simeq H(U,A)=A(U).$$\mathcal{O}_{X}(A)(U) \coloneqq \Sh_{\mathbf{H}}(X)( U , \mathcal{O}_{X} ) \simeq \mathbf{H}(U,A) = A(U) \,.

This means that ${𝒪}_X(A)$ behaves as the sheaf of $A$-valued functions over $X$.

Syntax layer

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Manifolds and Orbifolds

For $n\in \mathbb{N}$ a manifold of dimension $n$ is an object $X$ that locally looks like a Cartesian space ${ℝ}^n$, hence that can be thought of as being glued together from Cartesian spaces by gluing these along diffeomorphisms.

A natural way to make this precisely is to say that a manifold of dimension $X$ is an object such that first of all there is a cover, hence a 1-epimorphism of the form

$$p:\left(\coprod _{i\in i}{ℝ}^n\right)\to X.$$p \;\colon\; \left(\coprod_{i \in i} \mathbb{R}^n\right) \to X \,.

This encodes that $X$ can surjectively covered by Cartesian spaces, but it does not yet ensure that $X$ is locally equivalent to a Cartesian space in the intended sense. That intended sense is that $p$ is a local diffeomorphism.

Hence a manifold is a smooth space which receives a map out of a coproduct of Cartesian spaces that is a 1-epimorphism and a local diffeomorphism.

By the discussion above at Structure sheaves the general way to say local diffeomorphism is to say formally étale morphism. Hence more generally we can consider the notion of a smooth groupoid which received a map out of a coproduct of Cartesian spaces that is a 1-epimorphism and a formally étale morphism. If here the souce-fibers of the groupoid are in addition compact, then this is what is called an orbifold.

Model Layer

1. Smooth manifolds

2. Tangent bundle and frame bundle

Smooth manifolds

A smooth manifold of dimension $n$

a smooth space with an atlas

$$\{{ℝ}^n\underset{\simeq }{\overset{{\varphi }_i^{-1}}{\to }}{U}_i\hookrightarrow X\}$$\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i^{-1}}{\to} U_i \hookrightarrow X\}

of coordinate charts. On each overlap $U_i\cap U_j$ of two charts, the partial derivatives of the corresponding coordinate transformations

$${\varphi }_j\circ {\varphi }_i^{-1}:U_i\cap U_j\subset {ℝ}^n\to {ℝ}^n$$\phi_j\circ \phi_i^{-1} : U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n

form the Jacobian matrix of smooth functions

$$(({\lambda }_{ij}{)}^{\mu }{}_{\mu })≔[\frac{d}{d{x}^{\nu }}{\varphi }_j\circ {\varphi }_i^{-1}(x^{\mu })]:U_i\cap U_j\to {\mathrm{GL}}_n$$((\lambda_{i j})^{\mu}{}_{\mu}) \coloneqq \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu) \right] : U_i \cap U_j \to GL_n

with values in invertible matrices, hence in the general linear group $\mathrm{GL}(n)$. By construction (by the chain rule), these functions satisfy on triple overlaps of coordinate charts the matrix product equations

$$({\lambda }_{ij}{)}^{\mu }{}_{\lambda }({\lambda }_{jk}{)}^{\lambda }{}_{\nu }=({\lambda }_{ik}{)}^{\mu }{}_{\nu },$$(\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu} = (\lambda_{i k})^\mu{}_{\nu} \,,

(here and in the following sums over an index appearing upstairs and downstairs are explicit)

hence the equation

$${\lambda }_{ij}\cdot {\lambda }_{jk}={\lambda }_{ik}$$\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}

in the group $C^{\mathrm{\infty }}(U_i\cap U_j\cap U_k,\mathrm{GL}(n))$ of smooth $\mathrm{GL}(n)$-valued functions on the chart overlaps.

This is the cocycle condition for a smooth Cech cocycle in degree 1 with coefficients in $\mathrm{GL}(n)$ (precisely: with coefficients in the sheaf of smooth functions with values in $\mathrm{GL}(n)$ ). We write

$$[({\lambda }_{ij})]\in H_{\mathrm{smooth}}^1(X,{\mathrm{GL}}_n).$$[(\lambda_{i j})] \in H^1_{smooth}(X, GL_n) \,.

Formulated as smooth groupoids

• $X$ itself is a Lie groupoid $(X\overrightarrow{\to }X)$ with trivial morphism structure;

• from the atlas $\{U_i\to X\}$ we get the corresponding Cech groupoid

  $$C(\{U_i\})=(\coprod _{i,j}{U}_i\cap U_j\overrightarrow{\to }\coprod _i{U}_i)=\{\begin{array}{ccc}& & (x,j)\\ & \nearrow & =& \searrow \\ (x,i)& & \to & & (x,k)\end{array}\mathrm{for}x\in U_i\cap U_j\cap U_k\},$$C(\{U_i\}) = (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i) = \left\{ \array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\; for\, x \in U_i \cap U_j \cap U_k \right\} \,,

  whose objects are the points in the atlas, with morphisms identifying lifts of a point in $X$ to different charts of the atlas;

Tangent bundle

We discuss how the tangent bundle of a manifold $X$ naturally arises in the above perspecive in terms of the map ${\tau }_X:X\to B\mathrm{GL}(n)$ that modulates it.

The above situation is neatly encoded in the existence of a diagram of Lie groupoids of the form

$$\begin{array}{ccc}C(\{U_i\})& \stackrel{{\tau }_X}{\to }& B\mathrm{GL}(n).\\ {}^{\simeq }\downarrow \\ X\end{array},$$\array{ C(\{U_i\}) &\stackrel{\tau_X}{\to}& \mathbf{B} GL(n). \\ {}^{\mathllap{\simeq}}\downarrow \\ X } \,,

where

• the left morphism is stalk-wise (around small enough neighbourhoods of each point) an equivalence of groupoids (we make this more precise in a moment);

• the horizontal functor ${\tau }_X$ has as components the functions ${\lambda }_{ij}$ and its functoriality is the cocycle condition ${\lambda }_{ij}\cdot {\lambda }_{jk}={\lambda }_{ik}$.

A transformation of smooth functors ${\lambda }_1\Rightarrow {\lambda }_2:C(\{U_i\})\to B\mathrm{GL}(n)$ is precisely a coboundary between two such cocycles.

This defines a morphism of smooth groupoids

$${\tau }_X:X\to B\mathrm{GL}(n).$$\tau_X \;\colon\; X \to \mathbf{B}GL(n) \,.

The homotopy fiber of this map is a $\mathrm{GL}(n)$-principal bundle called the frame bundle of $X$, while the canonically associated bundle via the canonical representation of $\mathrm{GL}(n)$ on ${ℝ}^n$ is the tangent bundle

$$TX\to X.$$T X \to X \,.

Semantics Layer

Let $H$ be a cohesive (∞,1)-topos $(\Pi ⊣♭⊣♯)$ equipped with differential cohesion $(\mathrm{Red}⊣{\Pi }_{\inf }⊣{♭}_{\inf })$. Let

$${𝔸}^1\in H$$\mathbb{A}^1 \in \mathbf{H}

be an line object that exhibits the cohesive structure.

Definition

An étale ∞-groupoid of dimension $n$ is an object $X\in H$ such that there exists a map $p:\left({\coprod }_i{𝔸}^n\right)\to X$ such that

1. $p$ is a 1-epimorphism;

2. $p$ is a formally étale morphism, def. 62

If $X$ here is 0-truncated then we call it it manifold. It $X$ is 1-truncated we call it an orbifold.

Definition

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Syntax Layer

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Reduction of structure groups

Model Layer

$G$-Structure

$$BG\to BK$$\mathbf{B}G \to \mathbf{B}K

given a $K$-principal bundle

$$\begin{array}{ccc}\tilde{X}& \to & BK\\ {\downarrow }^{\simeq }\\ X\end{array}$$\array{ \tilde X &\to &\mathbf{B}K \\ \downarrow^{\mathrlap{\simeq}} \\ X }

a reduction of the structure group along $G\to K$ is

$$\begin{array}{ccccc}\tilde{X}& & \to & & BG\\ & \searrow & {⇙}_e& \swarrow \\ & & BK\end{array}$$\array{ \tilde X &&\to&& \mathbf{B}G \\ & \searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}K }

Examples

Vielbein, orthogonal structure, Riemannian geometry

• vielbein, orthogonal structure

reduction of the structure group along

$BO(n)\to B\mathrm{GL}(n)$

$$\begin{array}{ccccc}\tilde{X}& & \to & & BO(n)\\ & {}_{⊢T\Sigma }\searrow & {⇙}_e& \swarrow \\ & & B\mathrm{GL}(n)\end{array}$$\array{ \tilde X &&\to&& \mathbf{B}O(n) \\ & {}_{\mathllap{\vdash T \Sigma}}\searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}GL(n) }

$e$ is vielbein: definition of an orthonormal frame? at each point

Electromagnetism in gravitational background

example: the other 2 Maxwell equations: $d\star F=j_{\mathrm{el}}$.

Einstein-Maxwell theory

Almost complex structure

• almost complex structure

Almost Hermitean structure

• almost Hermitean structure?

Almost symplectic structure

• almost symplectic structure

Metaplectic structure

• metaplectic structure

Metalinear structure

• metalinear structure

Generalized complex geometry

• generalized complex geometry

Type II geometry

• type II geometry

Generalized Calabi-Yau structure

• generalized Calabi-Yau manifold

Exceptional generalized geometry

• exceptional generalized geometry

Spin structure, String structure, Fivebrane structure

• spin structure

• string structure

• fivebrane structure

Semantics Layer

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Syntax Layer

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Representations and Associated bundles

Model Layer

Actions

• action

Spin geometry

• spin group

• spin representation

$$\begin{array}{ccc}V& \to & V⫽\mathrm{Spin}\\ & & \downarrow \\ & & B\mathrm{Spin}\end{array}$$\array{ V &\to& V \sslash Spin \\ && \downarrow \\ && \mathbf{B}Spin }

• spinor bundle

• spinor

$$\begin{array}{ccccc}X& & \stackrel{\psi }{\to }& & V⫽\mathrm{Spin}\\ & \searrow & ⇙& \swarrow \\ & & B\mathrm{Spin}\end{array}$$\array{ X &&\stackrel{\psi}{\to}&& V \sslash Spin \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}Spin }

Associated bundle

• associated bundle

Representations up to coherent homotopy

• infinity-representation

Semantic Layer

$\mathrm{\infty }$-Actions

∞-action

$$\begin{array}{ccc}V& \to & V⫽G\\ & & \downarrow \\ & & BG\end{array}$$\array{ V &\to& V\sslash G \\ && \downarrow \\ && \mathbf{B}G }

Associated $\mathrm{\infty }$-bundles

• associated infinity-bundle

$$\begin{array}{ccc}E& \to & V⫽G\\ \downarrow & \mathrm{pb}& \downarrow \\ \tilde{X}& \to & BG\\ {\downarrow }^{\simeq }\\ X\end{array}$$\array{ E &\to& V\sslash G \\ \downarrow &pb& \downarrow \\ \tilde X &\to& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }

• section

$$\begin{array}{ccccc}X& & \stackrel{\sigma }{\to }& & V⫽G\\ & \searrow & {⇙}_{\simeq }& \swarrow \\ & & BG\end{array}$$\array{ X &&\stackrel{\sigma}{\to}&& V \sslash G \\ & \searrow &\swArrow_{\simeq}& \swarrow \\ && \mathbf{B}G }

Syntactic Layer

The context of a pointed connected type: representation theory

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Dependent product over a pointed connected type: invariants

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Dependent sum over a pointed connected type: quotients

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Modules

for the moment see the sub-entry geometry of physics - modules

Flat connections

Model Layer

Flat 1-connections

• flat connection

$X$ connected, ${\pi }_1(X)\in$Grp its fundamental group for any choice of basepoint, then the holonomy pairing

$$\mathrm{hol}:[S^1,X]\times H_{\mathrm{conn}}^1(X,G)\to G$$hol \colon [S^1,X]\times H^1_{conn}(X,G) \to G

descends to homotopy classes of (based) loops

$$\mathrm{hol}:H_{\mathrm{conn},\mathrm{flat}}^1(X,G)\stackrel{\simeq }{\to }{\mathrm{Hom}}_{\mathrm{Grp}}({\pi }_1(X),G)/G$$hol \colon H^1_{conn,flat}(X,G) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X), G)/G

to a bijection from equivalence classes of flat? $G$-principal connections to the quotient set of group homomorphisms ${\pi }_1(X)\to G$ modulo the adjoint action of $G$ on itself.

Semantic Layer

Definition

For $G\in \mathrm{Grp}(H)$ and $X\in H$ a flat $G$-connection $\nabla$ on $X$ is a morphism

$$\nabla :X\to ♭BG.$$\nabla \colon X \to \flat \mathbf{B}G \,.

We write

$$H_{\mathrm{flat}}(X,BG)≔H(X,♭BG)$$\mathbf{H}_{flat}(X, \mathbf{B}G) \coloneqq \mathbf{H}(X, \flat \mathbf{B}G)

and accordingly

$$H_{\mathrm{flat}}^{1}X,G≔{\pi }_0{H}_{\mathrm{flat}}(X,G)$$H^1_{flat}{X, G} \coloneqq \pi_0 \mathbf{H}_{flat}(X,G)

for the cohomology of $X\in H$ with flat coefficients.

Remark

By adjunction,

$$\frac{X\stackrel{\nabla }{\to }♭BG}{\Pi (X)\stackrel{\mathrm{transport}(\nabla )}{\to }BG}$$\frac{X \stackrel{\nabla}{\to} \flat \mathbf{B}G}{\Pi(X) \stackrel{transport(\nabla)}{\to} \mathbf{B}G}

a flat $G$-connection is equivalently a morphism

$$\mathrm{transport}(\nabla ):\Pi (X)\to BG.$$transport(\nabla) \colon \Pi(X) \to \mathbf{B}G \,.

Since $\Pi (X)$ is the fundamtal infinity-groupoid? of $X$, this manifestly encodes the higher parallel transport of the flat connection.

Definition

Write

$${\mathrm{UnderlyingBundle}}_{BG}:♭BG\to BG$$UnderlyingBundle_{\mathbf{B}G} \colon \flat \mathbf{B}G \to \mathbf{B}G

for the $(\mathrm{Disc}⊢\Gamma )$-counit-

Definition

For $\nabla :X\to ♭BG$ the composite

$$\mathrm{UnderlyingBundle}(\nabla ):X\stackrel{\nabla }{\to }♭BG\stackrel{{\mathrm{UnderlyingBundle}}_{BG}}{\to }BG$$UnderlyingBundle(\nabla) \colon X \stackrel{\nabla}{\to} \flat\mathbf{B}G \stackrel{UnderlyingBundle_{\mathbf{B}G}}{\to} \mathbf{B}G

modulates a $G$-principal ∞-bundle on $X$, by def. \ref{spring}. This we call the underlying $G$-principal bundle of $\nabla$.

$${\mathrm{ConstantPaths}}_X:X\to \Pi (X)$$ConstantPaths_{X} \colon X \to \Pi(X)

Syntactic Layer

$$BG:\mathrm{Type}⊢\mathrm{UnderlyingBundle}:♭BG\to BG$$\mathbf{B}G \colon Type \;\vdash \; UnderlyingBundle \colon \flat \mathbf{B}G \to \mathbf{B}G

de Rham Coefficients

Model Layer

Lie-algebra valued differential 1-forms

Definition

Let $G$ be a Lie group, and write $𝔤$ for its Lie algebra. The set of Lie algebra valued differential 1-forms is the tensor product

$${\Omega }^1(U,𝔤)={\Omega }^1(U){\otimes }_{ℝ}𝔤.$$\Omega^1(U,\mathfrak{g}) = \Omega^1(U) \otimes_{\mathbb{R}} \mathfrak{g} \,.

flat forms:

$${\Omega }_{\mathrm{flat}}^1(U,𝔤)=\{\omega \in {\Omega }^1(U,𝔤)\mid F_{\omega }=d\omega +[\omega ,\omega ]=0\}.$$\Omega^1_{flat}(U, \mathfrak{g}) = \left\{ \omega \in \Omega^1(U,\mathfrak{g}) | F_\omega = \mathbf{d} \omega + [\omega, \omega] = 0 \right\} \,.

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This is a smooth space

$${\Omega }_{\mathrm{flat}}^1(-,𝔤)\in \mathrm{Smooth}0\mathrm{Type}$$\Omega^1_{flat}(-,\mathfrak{g}) \in Smooth 0 Type

For $𝔤=\mathrm{Lie}(ℝ)$ we have

$${\Omega }^1(-,\mathrm{Lie}(ℝ))={\Omega }^1$$\Omega^1(-,Lie(\mathbb{R})) = \Omega^1

and we write

$${\Omega }_{\mathrm{flat}}^1(-,\mathrm{Lie}(ℝ))={\Omega }_{\mathrm{cl}}^1$$\Omega^1_{flat}(-,Lie(\mathbb{R})) = \Omega^1_{cl}

Below we see

$${♭}_{\mathrm{dR}}BG\simeq {\Omega }_{\mathrm{flat}}^1(-,𝔤)$$\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})

The de Rham complex

• de Rham complex

• de Rham cohomology

• de Rham theorem

Below we see that

$${♭}_{\mathrm{dR}}{B}^nℝ\simeq {♭}_{\mathrm{dR}}{B}^{n}U(1)\simeq \mathrm{DK}[{\Omega }^1(-)\stackrel{d}{\to }{\Omega }^2(-)\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }_{\mathrm{cl}}^n(-)].$$\flat_{dR}\mathbf{B}^n \mathbb{R} \simeq \flat_{dR}\mathbf{B}^n U(1) \simeq DK[ \Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to}\cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \,.

Semantic Layer

De Rham coefficient objects

Definition

For $G\in \mathrm{Gpr}(H)$, its de Rham coefficient object is the homotopy pullback

$${♭}_{\mathrm{dR}}BG≔♭BG{\times }_{BG}*$$\flat_{dR} \mathbf{B}G \coloneqq \flat \mathbf{B}G \times_{\mathbf{B}G} *

in

$$\begin{array}{ccc}{♭}_{\mathrm{dR}}BG& \stackrel{\mathrm{UnderlyingConnection}}{\to }& ♭BG\\ \downarrow & \mathrm{pb}& {\downarrow }^{\mathrm{UnderlyingBundle}}\\ *& \to & BG\end{array}.$$\array{ \flat_{dR} \mathbf{B}G &\stackrel{UnderlyingConnection}{\to}& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow^{\mathrlap{UnderlyingBundle}} \\ * &\to& \mathbf{B}G } \,.

Remark

This pullback diagram expresses that elements of ${♭}_{\mathrm{dR}}BG$ are flat $G$-connections $\nabla :X\to ♭BG$, def. 78 equipped with a trivialization of their underlying $G$-principal bundle, def. 80.

Recovering smooth differential forms from cohesive de Rham coefficients

Let $H=$ Smooth∞Grpd. All smooth manifolds and sheaves on smooth manifolds etc. in the following are canonically regarded as objects in this $H={\mathrm{Sh}}_{\mathrm{\infty }}(\mathrm{CartSp})$.

Proposition

For $G$ a Lie group, the de Rham coefficient object ${♭}_{\mathrm{dR}}BG$, def. 82 of its delooping is given by the sheaf of flat Lie algebra valued differential 1-forms ${\Omega }_{\mathrm{flat}}^1(-,𝔤)$, def. 81, for $𝔤$ the Lie algebra of $G$:

$${♭}_{\mathrm{dR}}BG\simeq {\Omega }_{\mathrm{flat}}^1(-,𝔤).$$\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g}) \,.

This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for BG with G a Lie group.

Write $U(1)$ for the circle group regared as a Lie group in the standard way.

Proposition

For $n\in \mathbb{N}$, the de Rham coefficient object ${♭}_{\mathrm{dR}}{B}^{n}U(1)$, def. 82, of the $n$-fold delooping of $U(1)$ is given by the image under the Dold-Kan correspondence

$$\mathrm{DK}::\mathrm{Sh}(\mathrm{CartSp},{\mathrm{Ch}}_{•})\to \mathrm{Sh}(\mathrm{CartSp},\mathrm{sSet})\to L_{\mathrm{lwhe}}\mathrm{Sh}(\mathrm{CartSp},\mathrm{sSet})\simeq H$$DK \colon : Sh(CartSp, Ch_\bullet) \to Sh(CartSp, sSet) \to L_{lwhe} Sh(CartSp, sSet) \simeq \mathbf{H}

of the truncated de Rham complex of sheaves of differential forms,

$$\begin{array}{rl}{♭}_{\mathrm{dR}}{B}^{n}U(1)& \simeq {♭}_{\mathrm{dR}}{B}^nℝ\\ & \simeq \mathrm{DK}[{\Omega }^1(-)\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }_{\mathrm{cl}}^n(-)]\\ & \simeq \mathrm{DK}[{\Omega }_{\mathrm{cl}}^1(-)\to 0\to \cdots \to 0]\end{array}.$$\begin{aligned} \flat_{dR}\mathbf{B}^n U(1) &\simeq \flat_{dR} \mathbf{B}^n \mathbb{R} \\ & \simeq DK[\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \\ & \simeq DK[\Omega^1_{cl}(-) \to 0 \to \cdots \to 0] \end{aligned} \,.

This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for the circle n-groups.

Syntactic Layer

$$\begin{array}{rl}{♭}_{\mathrm{dR}}(BG:\mathrm{Type}):& \mathrm{Type}\\ ≔& \sum _{\nabla :♭BG}(\mathrm{UnderlyingBundle}(\nabla )=*)\end{array}$$\begin{aligned} \flat_{dR}(\mathbf{B}G \colon Type)\; \colon & Type \\ \coloneqq & \;\; \sum_{\nabla \colon \flat \mathbf{B}G} ( UnderlyingBundle(\nabla) = * ) \end{aligned}

Maurer-Cartan forms

Model Layer

Maurer-Cartan form on a Lie group

• Maurer-Cartan form

$${\theta }_G:G\to {\Omega }_{\mathrm{flat}}^1(-,𝔤)$$\theta_G \colon G \to \Omega^1_{flat}(-,\mathfrak{g})

Consider

$${♭}_{\mathrm{dR}}Bℝ={\Omega }_{\mathrm{cl}}^1$$\flat_{dR} \mathbf{B}\mathbb{R} = \Omega^1_{cl}

the Maurer-Cartan form on $ℝ$ is the de Rham differential

$${\theta }_{ℝ}=d:ℝ\to {\Omega }_{\mathrm{cl}}^1\hookrightarrow {\Omega }^1.$$\theta_{\mathbb{R}} = \mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl} \hookrightarrow \Omega^1 \,.

Semantic Layer

Maurer-Cartan form on a cohesive $\mathrm{\infty }$-group

Let $H$ be a cohesive (infinity,1)-topos $(\Pi ⊣♭⊣♯)$. We discuss a general formulation of Maurer-Cartan forms on cohesive infinity-groups

Let $G\in \mathrm{Grp}(H)$ be a group object.

Use the pasting law together with the fact that $♭$ is right adjoint and hence preserves limits to get $\theta$ in

$$\begin{array}{ccc}G& \to & *\\ {\downarrow }^{\theta }& \mathrm{pb}& \downarrow \\ {♭}_{\mathrm{dR}}BG& \to & ♭BG\\ \downarrow & \mathrm{pb}& \downarrow \\ *& \to & BG\end{array}$$\array{ G &\to& * \\ \downarrow^{\mathrlap{\theta}} & pb & \downarrow \\ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow \\ * &\to& \mathbf{B}G }

Definition

This is the Maurer-Cartan form on $G$

$$\theta :G\to {♭}_{\mathrm{dR}}BG.$$\theta \;\colon\; G \to \flat_{dR} \mathbf{B}G \,.

Definition

For $S:X\to G$ a morphism, write

$$S^{-1}dS≔S^{*}{\theta }_G:X\stackrel{S}{\to }G\stackrel{{\theta }_G}{\to }{♭}_{\mathrm{dR}}BG$$S^{-1} \mathbf{d} S \coloneqq S^* \theta_G \;\colon\; X \stackrel{S}{\to} G \stackrel{\theta_G}{\to} \flat_{dR}\mathbf{B}G

for its composite with the map of def. 83, hence the pullback of the Maurer-Cartan form along $S$. We also call this the de Rham differential of $S$.

Maurer-Cartan forms on smooth $\mathrm{\infty }$-groups

Proposition

For $G$ a Lie group canonically regarded in $H=$Smooth∞Grpd the general abstract morphism

$${\theta }_G:G\to {♭}_{\mathrm{dR}}BG$$\theta_G \colon G \to \flat_{dR}\mathbf{B}G

is identified, via the identification ${♭}_{\mathrm{dR}}BG\simeq {\Omega }_{\mathrm{flat}}^1(-,𝔤)$ of prop. 42 and the Yoneda lemma, with the traditional Maurer-Cartan form

$${\theta }_G\in {\Omega }_{\mathrm{flat}}^1(G,𝔤).$$\theta_G \in \Omega^1_{flat}(G, \mathfrak{g}) \,.

Cohesive differentiation

The Maurer-Cartan form on the line object

$${\theta }_{ℝ}:ℝ\to {\Omega }_{\mathrm{cl}}^1(-,ℝ)$$\theta_{\mathbb{R}} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})

is the de Rham differential,

$$d={\theta }_{ℝ}.$$\mathbf{d} = \theta_{\mathbb{R}} \,.

Universal curvature characteristic forms

For $G=B^{n}U(1)$

$$\mathrm{curv}:B^{n}U(1)\to {♭}_{\mathrm{dR}}{B}^{n+1}U(1)$$curv \colon \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)

sends a circle $n$-bundle to the curvature of a pseudo-connection on it.

Syntactic Layer

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Principal connections

Model Layer

Circle-principal connections

Dirac charge quantization says that the electromagnetic field is only locally in general a map

$$\begin{array}{ccc}& & {\Omega }^1(-)\\ & {}^A\nearrow & {\downarrow }^d\\ X& \stackrel{\omega }{\to }& {\Omega }_{\mathrm{cl}}^2\end{array}$$\array{ && \Omega^1(-) \\ & {}^{\mathllap{A}}\nearrow & \downarrow^{\mathrlap{\mathbf{d}}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl} }

globally it is instead a map

$$\begin{array}{ccc}& & BU(1{)}_{\mathrm{conn}}\\ & {}^{\nabla }\nearrow & {\downarrow }^{F_{(-)}}\\ X& \stackrel{\omega }{\to }& {\Omega }_{\mathrm{cl}}^2\end{array}$$\array{ && \mathbf{B}U(1)_{conn} \\ & {}^{\nabla}\nearrow & \downarrow^{F_{(-)}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl} }

where

$$\begin{array}{ccc}BU(1{)}_{\mathrm{conn}}& \stackrel{F_{(-)}}{\to }& {\Omega }_{\mathrm{cl}}^2\\ \downarrow & \mathrm{pb}& \downarrow \\ BU(1{)}_{\mathrm{diff}}& \to & {\Omega }_{\mathrm{cl}}^{1\le •\le 2}\\ {\downarrow }^{\simeq }\\ BU(1)\end{array}$$\array{ \mathbf{B}U(1)_{conn} &\stackrel{F_{(-)}}{\to}& \Omega^2_{cl} \\ \downarrow &pb& \downarrow \\ \mathbf{B}U(1)_{diff} &\to& \Omega^{1 \leq \bullet \leq 2}_{cl} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}U(1) }

circle bundle with connection

the smooth groupoid is

$$BU(1{)}_{\mathrm{conn}}={\Omega }^1(-)⫽U(1)$$\mathbf{B}U(1)_{conn} = \Omega^1(-) \sslash U(1)

quotient of ${\Omega }^1(-)$ by $U(1)$-gauge transformations

for

$$A,A\prime :X\to {\Omega }^1(-)$$A,A' : X \to \Omega^1(-)

a gauge transformation $A\to A\prime$ is $\lambda :X\to U(1)$ with

$$A\prime =A+d\log \lambda$$A' = A + \mathbf{d} log \lambda

Dirac charge quantization and the electromagnetic field

• Dirac charge quantization

• electromagnetic field

Principal 1-connection

There are different equivalent definitions of the classical notion of a connection. One that is useful for our purposes is that a connection $\nabla$ on a $G$-principal bundle $P\to X$ is a rule ${\mathrm{tra}}_{\nabla }$ for parallel transport along paths: a rule that assigns to each path $\gamma :[0,1]\to X$ a morphism ${\mathrm{tra}}_{\nabla }(\gamma ):P_x\to P_y$ between the fibers of the bundle above the endpoints of these paths, in a compatible way:

$$\begin{array}{cccccccc}{P}_x& \stackrel{{\mathrm{tra}}_{\nabla }(\gamma )}{\to }& P_y& \stackrel{{\mathrm{tra}}_{\nabla }(\gamma \prime )}{\to }& P_z& & & P\\ & & & & & & & \downarrow \\ x& \stackrel{\gamma }{\to }& y& \stackrel{\gamma \prime }{\to }& z& & & X\end{array}.$$\array{ P_x &\stackrel{tra_\nabla(\gamma)}{\to}& P_y &\stackrel{tra_\nabla(\gamma')}{\to}& P_z &&& P \\ && && &&& \downarrow \\ x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z &&& X } \,.

In order to formalize this, we introduce a (diffeological) Lie groupoid to be called the path groupoid of $X$. (Constructions and results in this section are from ([SWI]).

Definition

For $X$ a smooth manifold let $[I,X]$ be the set of smooth functions $I=[0,1]\to X$. For $U$ a Cartesian space, we say that a $U$-parameterized smooth family of points in $[I,X]$ is a smooth map $U\times I\to X$. (This makes $[I,X]$ a diffeological space).

Say a path $\gamma \in [I,X]$ has sitting instants if it is constant in a neighbourhood of the boundary $\partial I$. Let $[I,P{]}_{\mathrm{si}}\subset [I,P]$ be the subset of paths with sitting instants.

Let $[I,X{]}_{\mathrm{si}}\to [I,X{]}_{\mathrm{si}}^{\mathrm{th}}$ be the projection to the set of equivalence classes where two paths are regarded as equivalent if they are cobounded by a smooth thin homotopy.

Say a $U$-parameterized smooth family of points in $[I,X{]}_{\mathrm{si}}^{\mathrm{th}}$ is one that comes from a $U$-family of representatives in $[I,X{]}_{\mathrm{si}}$ under this projection. (This makes also $[I,X{]}_{\mathrm{si}}^{\mathrm{th}}$ a diffeological space.)

Remark

The passage to the subset and quotient $[I,X{]}_{\mathrm{si}}^{\mathrm{th}}$ of the set of all smooth paths in the above definition is essentially the minimal adjustment to enforce that the concatenation of smooth paths at their endpoints defines the composition operation in a groupoid.

Definition

The path groupoid $P_1(X)$ is the groupoid

$$P_1(X)=([I,X{]}_{\mathrm{si}}^{\mathrm{th}}\overrightarrow{\to }X)$$\mathbf{P}_1(X) = ([I,X]_{si}^{th} \stackrel{\to}{\to} X)

with source and target maps given by endpoint evaluation and composition given by concatenation of classes $[\gamma ]$ of paths along any orientation preserving diffeomorphism $[0,1]\to [0,2]\simeq [0,1]{\coprod }_{1,0}[0,1]$ of any of their representatives

$$[{\gamma }_2]\circ [{\gamma }_1]:[0,1]\stackrel{\simeq }{\to }[0,1]\coprod _{1,0}[0,1]\stackrel{({\gamma }_2,{\gamma }_1)}{\to }X.$$[\gamma_2] \circ [\gamma_1] : [0,1] \stackrel{\simeq}{\to} [0,1] \coprod_{1,0} [0,1] \stackrel{(\gamma_2 , \gamma_1)}{\to} X \,.

This becomes an internal groupoid in diffeological spaces with the above $U$-families of smooth paths. We regard it as a groupoid-valued presheaf, an object in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Grpd}]$:

$$P_1(X):U\mapsto (\mathrm{Diff}(U\times I,X{)}_{\mathrm{si}}^{\mathrm{th}}\overrightarrow{\to }\mathrm{Diff}(U,X)).$$\mathbf{P}_1(X) : U \mapsto (Diff(U \times I, X)_{si}^{th} \stackrel{\to}{\to} Diff(U,X) ) \,.

Observe now that for $G$ a Lie group and $BG$ its delooping Lie groupoid discussed above, a smooth functor $\mathrm{tra}:P_1(X)\to BG$ sends each (thin-homotopy class of a) path to an element of the group $G$

$$\mathrm{tra}:(x\stackrel{[\gamma ]}{\to }y)\mapsto (•\stackrel{\mathrm{tra}(\gamma )\in G}{\to }•)$$tra : (x \stackrel{[\gamma]}{\to} y) \mapsto ( \bullet \stackrel{tra(\gamma) \in G}{\to} \bullet )

such that composite paths map to products of group elements

$$\mathrm{tra}:\left(\begin{array}{ccc}& & y\\ & {}^{[\gamma ]}\nearrow & =& {\searrow }^{[\gamma \prime ]}\\ x& & \stackrel{[\gamma \prime ]\circ [\gamma ]}{\to }& & z\end{array}\right)\mapsto \left(\begin{array}{ccc}& & •\\ & {}^{\mathrm{tra}(\gamma )}\nearrow & =& {\searrow }^{\mathrm{tra}(\gamma \prime )}\\ •& & \stackrel{\mathrm{tra}(\gamma )\mathrm{tra}(\gamma \prime )}{\to }& & •\end{array}\right)$$tra : \left( \array{ && y \\ & {}^{\mathllap{[\gamma]}}\nearrow &=& \searrow^{\mathrlap{[\gamma']}} \\ x &&\stackrel{[\gamma']\circ [\gamma]}{\to}&& z } \right) \mapsto \left( \array{ && \bullet \\ & {}^{\mathllap{tra(\gamma)}}\nearrow &=& \searrow^{\mathrlap{tra(\gamma')}} \\ \bullet &&\stackrel{tra(\gamma)tra(\gamma')}{\to}&& \bullet } \right)

and such that $U$-families of smooth paths induce smooth maps $U\to G$ of elements.

There is a classical construction that yields such an assignment: the parallel transport of a Lie-algebra valued 1-form.

Definition

Suppose $A\in {\Omega }^1(X,𝔤)$ is a degree-1 differential form on $X$ with values in the Lie algebra $𝔤$ of $G$. Then its parallel transport is the smooth functor

$${\mathrm{tra}}_A:P_1(X)\to BG$$tra_A : \mathbf{P}_1(X) \to \mathbf{B}G

given by

$$[\gamma ]\mapsto P\exp ({\int }_{[0,1]}{\gamma }^{*}A)\in G,$$[\gamma] \mapsto P \exp(\int_{[0,1]} \gamma^* A) \; \in G \,,

where the group element on the right is defined to be the value at 1 of the unique solution $f:[0,1]\to G$ of the differential equation

$$d_{\mathrm{dR}}f+{\gamma }^{*}A\wedge f=0$$d_{dR} f + \gamma^*A \wedge f = 0

for the boundary condition $f(0)=e$.

Theorem

This construction $A\mapsto {\mathrm{tra}}_A$ induces an equivalence of categories

$$[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Grpd}](P_1(X),BG)\simeq B{G}_{\mathrm{conn}}(X),$$[CartSp^{op},Grpd](\mathbf{P}_1(X), \mathbf{B}G) \simeq \mathbf{B}G_{conn}(X) \,,

where on the left we have the hom-groupoid of groupoid-valued presheaves and where on the right we have the groupoid of Lie-algebra valued 1-forms whose

• objects are 1-forms $A\in {\Omega }^1(X,𝔤)$,

• morphisms $g:A_1\to A_2$ are labeled by smooth functions $g\in C^{\mathrm{\infty }}(X,G)$ such that $A_2=g^{-1}Ag+g^{-1}dg$.

This equivalence is natural in $X$, so that we obtain another smooth groupoid.

Definition

Define $B{G}_{\mathrm{conn}}:{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Grpd}$ to be the (generalized) Lie groupoid

$$B{G}_{\mathrm{conn}}:U\mapsto [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Grpd}](P_1(-),BG)$$\mathbf{B}G_{conn} : U \mapsto [CartSp^{op}, Grpd](\mathbf{P}_1(-), \mathbf{B}G)

whose $U$-parameterized smooth families of groupoids form the groupoid of Lie-algebra valued 1-forms on $U$.

Remark

This equivalence in particular subsumes the classical facts that parallel transport $\gamma \mapsto P\exp ({\int }_{[0,1]}{\gamma }^{*}A)$

• is invariant under orientation preserving reparameterizations of paths;

• sends reversed paths to inverses of group elements.

Observation

There is an evident natural smooth functor $X\to P_1(X)$ that includes points in $X$ as constant paths. This induces a natural morphism $B{G}_{\mathrm{conn}}\to BG$ that forgets the 1-forms.

Defintion

Let $P\to X$ be a $G$-principal bundle that corresponds to a cocycle $g:C(U)\to BG$ under the construction discussed above. Then a connection $\nabla$ on $P$ is a lift $\nabla$ of the cocycle through $B{G}_{\mathrm{conn}}\to BG$.

$$\begin{array}{ccc}& & B{G}_{\mathrm{conn}}\\ & {}^{\nabla }\nearrow & \downarrow \\ C(U)& \stackrel{g}{\to }& BG\end{array}.$$\array{ && \mathbf{B}G_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G } \,.

Observation

This is equivalent to the traditional definitions.

A morphism $\nabla :C(U)\to B{G}_{\mathrm{conn}}$ is

• on each $U_i$ a 1-form $A_i\in {\Omega }^1(U_i,𝔤)$;

• on each $U_i\cap U_j$ a function $g_{ij}\in C^{\mathrm{\infty }}(U_i\cap U_j,G)$;

such that

• on each $U_i\cap U_j$ we have $A_j=g_{ij}^{-1}(A+d_{\mathrm{dR}})g_{ij}$;

• on each $U_i\cap U_j\cap U_k$ we have $g_{ij}\cdot g_{jk}=g_{ik}$.

Definition

Let $[I,X{]}_{\mathrm{si}}^{\mathrm{th}}\to [I,X{]}^h$ the projection onto the full quotient by smooth homotopy classes of paths.

Write ${\Pi }_1(X)=([I,X{]}^h\overrightarrow{\to }X)$ for the smooth groupoid defined as $P_1(X)$, but where instead of thin homotopies, all homotopies are divided out.

Proposition

The above restricts to a natural equivalence

$$[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Grpd}]({\Pi }_1(X),BG)\simeq ♭BG,$$[CartSp^{op}, Grpd](\mathbf{\Pi}_1(X), \mathbf{B}G) \simeq \mathbf{\flat}\mathbf{B}G \,,

where on the left we have the hom-groupoid of groupoid-valued presheaves, and on the right we have the full sub-groupoid $♭BG\subset B{G}_{\mathrm{conn}}$ on those $𝔤$-valued differential forms whose curvature 2-form $F_A=d_{\mathrm{dR}}A+[A\wedge A]$ vanishes.

A connection $\nabla$ is flat precisely if it factors through the inclusion $♭BG\to B{G}_{\mathrm{conn}}$.

For the purposes of Chern-Weil theory we want a good way to extract the curvature 2-form in a general abstract way from a cocycle $\nabla :X\stackrel{\simeq }{\leftarrow }C(U)\to B{G}_{\mathrm{conn}}$. In order to do that, we first need to discuss connections on 2-bundles.

Covariant derivatives

• covariant derivative

$$\begin{array}{c}(V⫽G{)}_{\mathrm{conn}}\\ \downarrow \\ B{G}_{\mathrm{conn}}\end{array}$$\array{ (V\sslash G)_{conn} \\ \downarrow \\ \mathbf{B}G_{conn} }

$$\begin{array}{ccccc}\tilde{X}& & \stackrel{(\sigma ,\nabla \sigma )}{\to }& & (V⫽G{)}_{\mathrm{conn}}\\ & \searrow & ⇙& \swarrow \\ & & B{G}_{\mathrm{conn}}\end{array}$$\array{ \tilde X &&\stackrel{(\sigma, \nabla \sigma)}{\to}&& (V \sslash G)_{conn} \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G_{conn} }

Deligne cohomology and Cheeger-Simons differential characters

• Deligne cohomology

• Cheeger-Simons differential character

• Beilinson-Deligne cup product

• fiber integration in ordinary differential cohomology

Circle-principal 2-connection

• circle 2-group principal 2-bundle

• circle 2-bundle with connection

Magnetic charge and the B-field

• magnetic charge, B-field

Circle-principal 3-connection

• circle 3-group principal 3-bundle

• circle 3-bundle with connection

The 3d Chern-Simons action functional and the C-field

• supergravity C-field

Circle-principal $n$-connection

• ordinary differential cohomology

• circle n-bundle with connection

Definition

Write $U(1)\in \mathrm{Smooth}0\mathrm{Type}$ for the smooth space of the circle group. Write

$$d\log :U(1)\to {\Omega }^1$$\mathbf{d}log \colon U(1) \to \Omega^1

for the homomorphism of smooth spaces which over an abstract coordinate system $U\in$ CartSp is given by the function

$$d{\log }_u:C^{\mathrm{\infty }}(U,U(1))\to {\Omega }^1(U)$$\mathbf{d} log_u \colon C^\infty(U,U(1)) \to \Omega^1(U)

which sends a $U(1)$-valued $f:U\to U(1)$ – for which one can always find an $ℝ$-valued function $\hat{f}:U\to ℝ$ for whuch $f=\hat{f}\mathrm{mod}\mathbb{Z}$ – to the differential 1-form $d\hat{f}$.

Definition

For $n\in \mathbb{N}$, the chain complex of smooth spaces (of sheaves on CartSp)

$$(U(1)\stackrel{d\log }{\to }{\Omega }^1\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }^n)$$(U(1) \stackrel{\mathbf{d} log}{\to} \Omega^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n)

(regarded as a chain complex of abelian groups and as sitting in degrees $n$ through 0) is called the (smooth) Deligne complex in these degrees.

Definition

For $n\in \mathbb{N}$, write

$$B^{n}U(1{)}_{\mathrm{conn}}=\mathrm{DK}(U(1)\stackrel{d\log }{\to }{\Omega }^1\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }^n)\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$$\mathbf{B}^n U(1)_{conn} = DK(U(1) \stackrel{\mathbf{d} log}{\to} \Omega^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n) \in Smooth\infty Grpd

for the smooth ∞-groupoid presented by the Deligne complex under the Dold-Kan correspondence map, def. 73.

For $X\in$ Smooth∞Grpd we say that

• a morphism $\nabla :X\to B^{n}U(1{)}_{\mathrm{conn}}$ modulates a circle n-bundle with connection on $X$.

We also call $B^{n}U(1{)}_{\mathrm{conn}}$ the universal moduli ∞-stack of circle $n$-bundles with connection.

Definition

Write

$$\begin{array}{ccccccccccc}{B}^{n}U(1{)}_{\mathrm{conn}}& ≔& \mathrm{DK}(& U(1)& \stackrel{d\log }{\to }& {\Omega }^1& \stackrel{d}{\to }& \cdots & \stackrel{d}{\to }& {\Omega }^n& )\\ {\downarrow }^{U_{B^{n}U(1{)}_{\mathrm{conn}}}}& ≔& \mathrm{DK}(& {\downarrow }^{\mathrm{id}}& & \downarrow & & & & \downarrow )\\ B^{n}U(1)& ≔& \mathrm{DK}(& U(1)& \stackrel{}{\to }& 0& \stackrel{}{\to }& \cdots & \stackrel{d}{\to }& 0& )\end{array}$$\array{ \mathbf{B}^n U(1)_{conn} & \coloneqq & DK( & U(1) &\stackrel{\mathbf{d} log}{\to}& \Omega^1 &\stackrel{\mathbf{d}}{\to}& \cdots & \stackrel{\mathbf{d}}{\to} & \Omega^n & ) \\ \downarrow^{\mathrlap{U_{\mathbf{B}^n U(1)_{conn}}}} & \coloneqq& DK( & \downarrow^{\mathrlap{id}} && \downarrow && && \downarrow ) \\ \mathbf{B}^n U(1) & \coloneqq & DK( & U(1) &\stackrel{}{\to}& 0 &\stackrel{}{\to}& \cdots & \stackrel{\mathbf{d}}{\to} & 0 & ) }

for the canonical forgetful morphism from moduli of $n$-connections to those of the underlying principal bundles.

Definition

More generally, we may consider the intermediate stages