Higher toposes of laws of motion


In (Lawvere 67, Lawvere 86, Lawvere 97) there was proposed a notion of toposes of laws of motion meant to formalize classical continuum mechanics in synthetic differential geometry/in topos theory. This page here gives an introductory survey of the refinements possible when lifting this from topos theory to higher topos theory and of the applications of the resulting formalism to quantum field theory.

This text originates in a talk at the Eighth Scottish Category Theory Seminar. Accordingly, these notes amplify aspects of category theory and topos theory and generally stick to a Lawverian perspective.



physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory



Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?



Introduction and motivation

Urban legend has it that there was a time when only three people understood Einstein‘s theory of classical gravity – “general relativity”.

Whether true or not, one of the three was David Hilbert. He made sure every beginning student today can understand general relativity, he did so by giving it a clear and precise (= rigorous) formalization in mathematics:

classical Einstein gravity is simply the study of the critical points of the integral of the scalar curvature density functional on the moduli space of pseudo-Riemannian metrics on spacetime.

By trusting that a fundamental theory of physics should have a fundamental formulation in mathematics, Hilbert was able to essentially scoop Einstein (see here for the history), that’s why this functional is now called the Einstein-Hilbert action functional.

Hilbert had promoted this general idea before as part of the famous eponymous Hilbert's problems in mathematics, from 1900. Here Hilbert's 6th problem asks mathematicians generally to find axioms for theories in physics.

Since then a list of such axiomatizations has been found, for instance

mechanicssymplectic geometry
gravityRiemannian geometry
gauge theoryChern-Weil theory
quantum mechanicsoperator algebra
topological local quantum field theorymonoidal (∞,n)-category theory

Two aspects of this list are noteworthy: on the one hand, it contains crown jewels of mathematics, on the other the items appear unrelated and piecemeal.

As a student, William Lawvere was exposed to the proposal to axiomatize thermodynamics as what was called “rational thermodynamics”. He realized that a fundamental foundation of such continuum physics first of all requires a good foundation of differential geometry itself. Looking over his life publication record (see here) one sees that he pursued the following grand plan.


  1. lay the foundations of mathematics in topos theory (“ETCS”)

  2. lay the foundations of geometry in topos theory (synthetic differential geometry, cohesion)

  3. lay the foundations of classical continuum physics in synthetic differential geometry (toposes of laws of motion).

Lawvere became famous for his groundbreaking contributions to the first two items (categorical logic, elementary topos theory, algebraic theories, SDG). For some reason the motivation of all this by the third item is not as widely recognized, even thought Lawvere continuously emphasized this third point, see the list of quotations here.

Grandiose as this plan is, we have to note that in the above form it falls short in each item, by modern standards:

  1. modern mathematics is naturally founded not in topos theory/type theory, but in higher topos theory/homotopy type theory.

  2. modern geometry is not just about “variable sets” (sheaves) but is higher geometry about “variable homotopy types”, “geometric homotopy types”, “higher stacks”;

  3. modern physics goes beyond classical continuum physics; at high energy (small distance) classical physics is refined by quantum physics, specifically by quantum field theory.

Therefore what is needed is a foundation of high energy physics in higher differential geometry formulated in higher topos theory.

In the following we illustrate three aspects of such a refined theory, following (dcct, sythQFT). We close by indicating how this theory serves to solve subtle open problems in modern quantum field theory and string theory.

I) Mapping spaces in gauge theory and General covariance

One basic point emphasized in (Lawvere 67) is that central to the formulation of physics is the existence of mapping spaces which satisfy the exponential law.

(notice: mapping space … space of trajectoriespath integral)

Indeed, generically a physical system is specified by

such that a trajectory or history of field configurations is a map

ΣX. \Sigma \longrightarrow X \,.

For instance for Σ=\Sigma = \mathbb{R} the abstract worldline and XX spacetime, then ΣX\Sigma \to X may be taken to be the trajectory of a particle in spacetime.

(Notice that after second quantization the roles change. First the domain Σ\Sigma is worldvolume and the codomain XX is spacetime (“sigma-model)”, then after second quantization spacetime XX becomes the domain (hence becomes Σ\Sigma in the above).)

Hence the mapping space [Σ,X][\Sigma,X] is the space of all trajectories (the path space, famous as the domain of the infamous path integral).

Lawvere observed that this not only needs to exist as a decent “space”, it also needs to satisfy the axiom of an cartesian internal hom. Because if we consider a split

Σ×Σ d1 \Sigma \coloneqq \mathbb{R} \times \Sigma_{d-1}

into time and space, then we want that spacetime field configurations

×Σ d1X \mathbb{R} \times \Sigma_{d-1} \longrightarrow X

are equivalently trajectories of fields on space

[Σ d1,X] \mathbb{R} \longrightarrow [\Sigma_{d-1}, X]

and also equivalently a collection of field trajectories for each point of space

Σ d1[,X]. \Sigma_{d-1} \longrightarrow [\mathbb{R}, X] \,.

This led Lawvere to recognize that physics (prequantum physics, to be precise) is to be formulated in a cartesian closed category, such as a topos.

The category SmthMfdSmthMfd of smooth manifolds is too small to accomplish this. But the category of sheaves Sh(SmthMdf)Sh(SmthMdf) on the site of smooth manifolds is the canonical improvement. Objects in here include smooth manifolds, also diffeological spaces and general smooth spaces.

Better still, there is the category of sheaves Sh(FSmthMfd)Sh(FSmthMfd) on the site of formal smooth manifolds – known as the Cahiers topos . This also contains infinitesimal objects and indeed interprets the axioms of synthetic differential geometry.

But actually in modern physics one needs a bit more than this. Physics is fundamentally governed by gauge equivalence, which means that there is no sense in asking if field configurations are equal, we must ask if they are equivalent.

A fundamental example of this is Einstein‘s notion of general covariance. This says that if

s:UΣ s \;\colon\; U \hookrightarrow \Sigma

is a region in spacetime, and ϕ:ΣΣ\phi \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma is a diffeomorphism acting on spacetime, then the “translated region”

ϕ *s:UΣΣ \phi^\ast s \;\colon\; U \hookrightarrow \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma

is “the same”, for all physical purposes. Stated this way in ordinary topos theory this is confusing, and historically it was confusing: this confusion is essentially what is known as the “hole paradox”.

This apparent paradox is resolved in higher topos theory. Here a space SS has internal symmetries, it is a groupoid, a homotopy type. This means that it is not sensible to ask if two maps into it are equal, but between any two maps there is a space of equivalences between them.

For general covariance this means by the above that spacetime is not actually the spacetime manifold Σ\Sigma, but is the action groupoid/quotient stack

Σ//Diff(Σ) \Sigma//Diff(\Sigma)

of Σ\Sigma by its diffeomorphism group (regarded as a diffeological group). By the very definition of “stack”, this Σ//Diff(Σ)\Sigma//Diff(\Sigma) is the thing which is such that maps UΣ//Diff(Σ)U \longrightarrow \Sigma//Diff(\Sigma) into it behave just as they should as demanded by general covariance.

This is the formalization of “general covariance” for regions inside spacetime. The other thing now is general covariance for fields on spacetime. It turns out that the formalism automatically handles these now:

For notice that the above means now that we also need to consider the mapping spaces refined to higher topos theory. A fundamental fact is that for GGrp(H)G \in Grp(\mathbf{H}) group object then the higher slice topos over its delooping is equivalently the collection of G-actions

H /BGGAct(H). \mathbf{H}_{/\mathbf{B}G} \simeq G Act(\mathbf{H}) \,.

Under this equivalence a higher action is identified with the universal associated bundle which it induces

(Σ Σ//Diff(Σ) BDiff(Σ))H /BDiff(Σ)Diff(Σ)Act(H). \left( \array{ \Sigma &\longrightarrow& \Sigma//Diff(\Sigma) \\ && \downarrow \\ && \mathbf{B}Diff(\Sigma) } \right) \;\; \in \;\; \mathbf{H}_{/\mathbf{B}Diff(\Sigma)} \simeq Diff(\Sigma)Act(\mathbf{H}) \,.

Lawvere also introduced categorical logic and understood dependent product \prod and dependent sum \sum as base change. In higher topos theory this becomes representation theory as follows:

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

Using this we have the mapping space of covariant fields formed in the context of Diff(Σ)Diff(\Sigma)-covariance/equivariance

BDiff(Σ)[Σ//Diff(Σ),X] \mathbf{B}Diff(\Sigma) \;\; \vdash \;\; \left[ \Sigma//Diff(\Sigma) \,,\; X \right]

Theorem. Down in the absolute context, under dependent sum, this is

BDiff(Σ)[Σ//Diff(Σ),X][Σ,X]//Diff(Σ). \vdash \;\; \underset{\mathbf{B}Diff(\Sigma)}{\sum} \left[ \Sigma//Diff(\Sigma) \,,\; X \right] \;\; \simeq \;\; [\Sigma, X]//Diff(\Sigma) \,.

Here [Σ,X][\Sigma, X] is the space of fields on spacetime as one would find it in ordinary topos theory, and [Σ,X]//Diff(Σ)[\Sigma,X]//Diff(\Sigma) is its homotopy quotient by the action of the diffeomorphism group by pullback of fields. This is precisely the group of gauge equivalences on fields in general relativity.

So we obtain a formalization of the famous insight of Einstein, derived by lifting Lawvere‘s argument about mapping spaces in physics from topos theory to higher topos theory.

In a slogan we may conclude, in view of the above table, that in the formalization of physics in higher topos theory we have:

in mapping spaces.

II) Toposes of laws of motion and Hamilton-Jacobi-Lagrange mechanics

In (Lawvere 97) it was observed that equations of motion in physics can (almost, see below) be formalized in synthetic differential geometry as follows.

Let H\mathbf{H} be an ambient synthetic differential topos (such as the Cahiers topos of smooth spaces and formal smooth manifolds).

The canonical line object 𝔸 1=\mathbb{A}^1 = \mathbb{R} of this models the continuum line, the abstract worldline. Let

D D \hookrightarrow \mathbb{R}

be the inclusion of the first order infinitesimal neighbourhood of the origin of \mathbb{R} – in the internal logic this is D={x|x 2=0}D = \{x \in \mathbb{R}| x^2 = 0\}, externally it is the spectrum of the ring of dual numbers over \mathbb{R}.

Then consider XHX \in \mathbf{H} any object which we are going to think of as a configuration space of a physical system. For instance if the system is a particle propagating on a spacetime, then XX is that spacetime. Or XX may be the phase space of the system.

Accordingly the mapping space [,X]H[\mathbb{R}, X] \in \mathbf{H} is the smooth path space of XX. This is the space of potential trajectories of the physical system.

If XX is thought of as phase space, then every point in there determines a unique trajectory starting at that point. This means that time evolution is then an action of \mathbb{R} on XX. As XX here might be any space, we have the collection

Act(H)Topos \mathbb{R}Act(\mathbf{H}) \in Topos

of all \mathbb{R}-actions on objects in H\mathbf{H}. This is again a topos, and hence this is a first version of what one might call a topos of laws of motion.

On the other hand, if we think of XX as configuration space, then it is (in the simplest but common case of physical systems) a tangent vector in XX that determines a trajectory, hence a point in [D,X][D,X]. There is the canonical projection [,X][D,X][\mathbb{R},X] \longrightarrow [D,X] from the smooth path space to the tangent bundle, which sends each path to its tangent vector/derivative at 00 \in \mathbb{R}. A section of this map is hence an assignment that sends each tangent vector to a trajectory which starts out with this tangent. Specifying such a section is hence part of what it means to have equations of motion in physics. Accordingly in Toposes of laws of motion Lawvere called the collection of such data a Galilean topos of laws of motion.

Of course this is not quite yet what is actually used and needed in physics. On p. 9 of (Lawvere 97) this problem is briefly mentioned:

But what about actual dynamical systems in the spirit of Galileo, for example, second-order ODE’s? (Of course, the symplectic or Hamiltonian systems that are also much studied do address this question of states of Becoming versus locations of Being, but in a special way which it may not be possible to construe as a topos;

We observe now (following Classical field theory via Cohesive homotopy types) that it does exist as a “higher topos”.

First notice that in physics a phase space is not any space XX, but is a space XX equipped with a closed differential 2-form, a “presymplectic form”. Speaking in terms of mapping spaces as above let Ω cl 2\mathbf{\Omega}^2_{cl} be the moduli space of closed 2-forms, then this means that a phase space is really a map

X Ω cl 2 \array{ X \\ & \searrow \\ && \mathbf{\Omega}^2_{cl} }

In fact this is still not quite the accurate statement. Rather a phase space is a “prequantization” of such data. This means the following.

The circle group S 1S^1 naturally acts on the space of differential 1-forms Ω 1\mathbf{\Omega}^1 by

Ω 1×S 1Ω 1 \mathbf{\Omega^1} \times S^1 \longrightarrow \mathbf{\Omega}^1
(A,g)A+dlog(g), (A, g) \mapsto A + \mathbf{d}log(g) \,,

where d\mathbf{d} is the de Rham differential. The resulting quotient stack we write

BS conn 1=Ω 1//S 1. \mathbf{B}S^1_{conn} = \mathbf{\Omega}^1//S^1 \,.

The de Rham differential d:Ω 1Ω cl 2\mathbf{d} \;\colon\; \mathbf{\Omega}^1 \longrightarrow \mathbf{\Omega}^2_{cl} descends to this quotient to yield a map

F ():BS conn 1Ω cl 2. F_{(-)} \;\colon\; \mathbf{B}S^1_{conn} \longrightarrow \mathbf{\Omega}^2_{cl} \,.

A prequantization of a presymplectic form is a lift \nabla through this map

X BU(1) conn ω F () Ω 2. \array{ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}U(1)_{conn} \\ & {}_{\mathllap{\omega}}\searrow & \downarrow^{\mathrlap{F_{(-)}}} \\ && \mathbf{\Omega}^2 } \,.

Now suppose ω\omega is actually a symplectic form. Then:

Theorem. Concrete actions of \mathbb{R} on (X,)H /BS conn 1(X, \nabla) \in \mathbf{H}_{/\mathbf{B}S^1_{conn}} are equivalent to “HamiltoniansH[X,]H \in [X,\mathbb{R}], where under the equivalence an element tt \in \mathbb{R} is sent to the slice automorphism

X exp(t{H,}) X θ exp(iS t) θ BU(1) conn, \array{ X && \stackrel{\exp(t \{H,-\})}{\longrightarrow} && X \\ & {}_{\mathllap{\theta}} \searrow & \swArrow_{\exp( i S_t )} & \swarrow_{\mathrlap{\theta}} \\ && \mathbf{B}U(1)_{conn} } \,,

where exp(t{H,})\exp(t \{H,-\}) denotes the flow of Hamilton's equations of motion induced by HH and where S t= 0 tLdtS_t = \int_0^t L \, d t is the Hamilton-Jacobi action given by the integral of the Lagrangian LL (the Legendre transform of HH).

This statement subsumes the core ingredients of classical mechanics. See at prequantized Lagrangian correspondence for details.

In conclusion we find that \mathbb{R}-actions in the higher slice topos H /BS conn 1\mathbf{H}_{/\mathbf{B}S^1_{conn}} over the moduli stack of circle group-principal connections are equivalent to actual laws of motion in classical mechanics

LawsOfMotion(H)Act(H /BS conn 1). LawsOfMotion(\mathbf{H}) \simeq \mathbb{R}Act\left( \mathbf{H}_{/\mathbf{B}S^1_{conn}} \right) \,.

More precisely, this applies to laws of motion in mechanics. One obtains more generally the Hamilton-de Donder-Weyl equations of motion? of nn-dimensional local classical field theory by replacing BS conn 1\mathbf{B} S^1_{conn} here with B nS conn 1\mathbf{B}^n S^1_{conn} (Schreiber 13, Schreiber 13b).

III) Extensive/intensive duality and Cohomological quantization

In physics and especially in continuum mechanics and thermodynamics, a physical quantity associated with a physical system extended in space is called

For instance for a solid body its temperature is intensive, but its mass is extensive: there is a temperature assigned to every point of the body (in the idealization of classical continuum mechanics anyway) but a mass is assigned only to every little “extended” piece of the body, not to a single point.

This terminology in physics apparently originates with Richard Tolman in 1917.

In (Lawvere 86) it is amplified that this duality is generally a fundamental one also in mathematics: given a topos H\mathbf{H} with a commutative ring object RCRing(H)R \in CRing(\mathbf{H}), then

  • the space of intensive quantities on an object XHX \in \mathbf{H} is the mapping space [X,R] HCRing(H)[X,R]_{\mathbf{H}} \in CRing(\mathbf{H}) formed in H\mathbf{H};

  • the space of extensive quantities on XX is the RR-linear dual, namely the mapping space [X,R] *[[X,R],R] RMod[X,R]^\ast \coloneqq [[X,R], R]_{R Mod} formed in RR-modules in H\mathbf{H}.

  • the integration map is the canonical evaluation pairing

    X:[X,R]×[X,R] *R. \int_X \;\colon\; [X,R] \times [X,R]^\ast \longrightarrow R \,.

Viewed this way, this naturally generalizes to the case where H\mathbf{H} is in fact an (∞,1)-topos and RCRing(H)R \in CRing(\mathbf{H}) an E-∞ ring. In this case [X,R][X,R] is called the RR-cohomology spectrum of XX and [X,R] *[X,R]^\ast is the corresponding generalized homology spectrum. In this form intensive and extensive properties appear in physics in the context of motivic quantization of local prequantum field theory.

More generally, for χ\chi an RR-(∞,1)-line bundle over XX then the corresponding extensive object is the χ\chi-twisted Thom spectrum R +χ(X)R_{\bullet + \chi}(X) and the intensive object is the χ\chi-twisted cohomology spectrum R +χ(X)=[R +χ(X),R] RModR^{\bullet + \chi}(X) = [R_{\bullet+ \chi}(X),R]_{R Mod}. See at motivic quantization for how this appears in physics.

In particular ordinary quantum mechanics is recovered by settin R=R = KU, the complex K-theory spectrum (Nuiten 13).

The monoidal (infinity,1)-category KUModKU Mod is the refined ambient home for Hilb=ModHilb = \mathbb{C} Mod (used for finite quantum mechanics in terms of dagger-compact categories).

Application: Cohesion and Superstring anomaly cancellation

On first sight, formalization of physics in (higher) topos theory might seem like a fruitless exercise. But on the contrary, it is hardly possible to understand the deep structure of quantum field theory without such (geometric) homotopy theory.

We close here by briefly indicating one example problem of recent interest, concerned with the fine-structure of quantum anomaly cancellation in 2d QFT.

One reason why the need for geometric homotopy theory in QFT is not mentioned in the bulk of the QFT literature is that traditionally the bulk of the discussion of quantum field theory is in perturbation theory (perturbative both in Planck's constant and in terms of the coupling constant). This perspective tends to hide the rich nature of what QFT fundamentally is, as non-perturbative quantum field theory.

Phenomena that arise from the global structure of a moduli of field configurations in physics are alien to perturbation theory, and hence are anomalies. Such an anomalous action functional is something that ought to be a function [Σ,X]S 1[\Sigma,X] \longrightarrow S^1 on configuration space, but possibly comes out just as a section of a bundle over configuration space (examples include gravitational anomalies, the conformal anomaly, the Freed-Witten-Kapustin anomaly, the Green-Schwarz anomaly, the Diaconescu-Moore-Witten anomaly.)

The anomaly bundles on [Σ,X][\Sigma,X] typically arise as the transgression of higher bundles on the moduli space of fields XX itself (see at twisted smooth cohomology in string theory for more on this). So these are phenomena which are intrinsically phenomena in geometric homotopy theory/(infinity,1)-topos theory.

We consider now specifically a general aspect of what is called the Freed-Witten-Kapustin anomaly. It is usually read out as follows:

Just as above we saw that the basic example of a quantum field theory on Σ=\Sigma = \mathbb{R} descibes the dynamics of a particle, so the basic example of a quantum field theory on a 2-dimensional Σ\Sigma describes the dynamics of a string. This naturally feels forces excerted in particular by two background gauge fields called the B-field and the RR-field.

The global nature of these fields is more subtle than for, say, the electromagnetic field, since they are higher gauge fields. To a first approximation one finds that the RR-field is a cocycle in twisted K-theory, where the twist is the B-field which in turn is a cocycle in ordinary cohomology.

But this is not the full story, in the full story these fields are cocycles in differential cohomology. The RR-field is a cocycle in twisted differential K-theory twisted by the B-field which is a cocycle in ordinary differential cohomology.

In (DFM 09) is indicated the rich subtleties in the quantum anomaly consistency conditions on these background fields, assuming that twisted differential K-theory exists with some properties, but without having constructed it.

The infamous “landscape of string theory vacua” is essentially the moduli space of certain 2d field theories satisfying consistency conditions like this.

The central problem in showing the existence of a differential cohomology theory is to show that this cohomology theory sits inside a double square diagram called the “differential cohomology diagram”.

Now a miracle happens. After developing synthetic differential geometry, Lawvere explored a more fundamental axiomatization of differential geometry, which he called cohesion (Lawvere 94, Lawvere 07 ) (Earlier: “being and becoming” (Lawvere 91)). Slightly paraphrased, cohesion means that the ambient type theory is equipped with an adjoint triple of (co-)modalities

\int \;\dashv\; \flat \;\dashv\; \sharp

called: shape modality \dashv flat modality \dashv sharp modality.

This has an immediate extension to homotopy type theory (cohesive homotopy type theory). But there it has more dramatic consequences. In (Bunke-Nikolaus-Völkl 13) it was observed that on stable homotopy types AA cohesion implies that the canonical diagram formed from modality units and counits

dRΩA dRΣA θ A dRΩA A dRΣA θ A A A, \array{ && \int_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \flat \int_{dR} \Omega A && && A && && \int \flat_{dR}\Sigma A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{\int \theta_A}} \\ && \flat A && \longrightarrow && \int A } \,,

is guaranteed to consist of homotopy pullback squares, by the nature of adjoint triples of modalities (see at tangent cohesion for more on this).

In (Bunke-Nikolaus-Völkl 13) it is shown that this is universally the “differential cohomology diagram” which hence exhibits every stable homotopy type AA in cohesive homotopy type theory as a differential cohomology theory, hence as the moduli stack for abelian higher gauge fields in quantum field theory. Hence cohesive homotopy type theory is a universal ambient context for differential cohomology and hence for higher gauge fields appearing in quantum field theory – whence the title “differential cohomology in a cohesive topos”.

Using this and the twisted cohomology available in tangent cohesion (Bunke-Nikolaus) show the existence of twisted differential K-theory, the way it needs to exist for 2d QFT to be consistent.


Last revised on April 22, 2020 at 18:39:19. See the history of this page for a list of all contributions to it.