nLab
pro-manifold

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A pro-manifold is a pro-object in a category of manifolds, i.e. a formal projective limit of manifolds.

Details depend on what exactly is understood by “manifold”, i.e. whether topological manifolds or smooth manifold, etc.

Typically one wants to mean pro-objects in manifolds of finite dimensions, the point being then that a pro-manifold is like an infinite-dimensional manifold but with “mild” infinite dimensionality, expressed by the very fact that it may be presented as a formal projective limit of finite dimensional manifolds.

To amplify this specification, one should properly speak of “pro-(finite dimensional smooth manifolds)”, but beware that people often abbreviate to “pro-manifold” regardless. Also “pro-finite manifold” is in use, which however, strictly speaking, is a misnomer since a “finite manifold” is one with a finite number of points.

An important example of pro-objects in finite-dimensional smooth manifolds are infinite jet bundles. These are the formal projective limits of the underlying finite-order jet bundles.

Pro-Cartesian spaces

Embedding into smooth loci

Definition

Write CartSp for the full subcategory of that of smooth manifolds on the Cartesian spaces, i.e. on those of the form n\mathbb{R}^n, for nn \in \mathbb{N}. Write

ProCartSpPro(CartSp) ProCartSp \coloneqq Pro(CartSp)

for its category of pro-objects, the pro-Cartesian spaces.

Proposition

The functor which sends a formal cofiltered limit of Cartesian spaces to its actual cofiltered limit of smooth loci is a fully faithful functor, hence constitutes a full subcategory inclusion of pro-Cartesian spaces (def. ) into smooth loci:

Pro(CartSp)SmthLoc. Pro(CartSp) \hookrightarrow SmthLoc \,.
Proof

Since Pro(𝒞)(Ind(𝒞 op)) opPro(\mathcal{C}) \simeq (Ind(\mathcal{C}^{op}))^{op} (remark) it is sufficient to show that the functor in question is on opposite categories a fully faithful functor of the form

Ind(CartSp op)SmthLoc op=SmthAlg , Ind(CartSp^{op}) \hookrightarrow SmthLoc^{op} = SmthAlg_{\mathbb{R}} \,,

where SmothAlg SmothAlg_{\mathbb{R}} is the category of smooth algebras.

Now, there is the fully faithful functor

i:CartSpSmthLoc i \;\colon\; CartSp \hookrightarrow SmthLoc

(prop.) hence a fully faithful functor

i op:CartSp opSmthAlg . i^{op} \colon CartSp^{op} \hookrightarrow SmthAlg_{\mathbb{R}} \,.

Moreover, the image of the latter is in compact objects i op:CartSp op(SmthAlg ) cptSmthAlgi^{op} \colon CartSp^{op} \hookrightarrow (SmthAlg_{\mathbb{R}})_{cpt} \hookrightarrow SmthAlg, because

C ( n)y( n)SmthAlg Func ×(CartSp,Set) C^\infty(\mathbb{R}^n) \simeq y(\mathbb{R}^n) \in SmthAlg_{\mathbb{R}} \simeq Func_\times(CartSp,Set)

is co-representable, hence compact (by the Yoneda lemma and since colimits are computed objectwise prop.).

This implies that the composite

Ind(CartSp op)Ind(i op)Ind(SmthAlg )LSmthAlg Ind(CartSp^{op}) \overset{Ind(i^{op})}{\hookrightarrow} Ind(SmthAlg_{\mathbb{R}}) \overset{L}{\longrightarrow} SmthAlg_{\mathbb{R}}

is also fully faithful (prop.).

Here Ind(i op)Ind(i^{op}) takes formal filtered colimits in CartSp opCartSp^{op} to the corresponding formal colimits in SmthAlg SmthAlg_{\mathbb{R}} (prop.), while LL takes formal filtered colimits to actual filtered colimits (prop.). Hence this is indeed the functor in question.

The site of towers of Cartesian spaces and pro-morphisms

under construction

Definition

Write

TowCartSpProCartSp TowCartSp \hookrightarrow ProCartSp

for the full subcategory of the category of pro-Cartesian spaces (def. ) on those pro-objects in CartSp which are presented as formal sequential limits of tower diagrams, i.e. where the indexing category 𝒦= \mathcal{K} = \mathbb{N}_{\geq}.

Definition

For UTowCartSpU \in TowCartSp a tower of Cartesian spaces (def. ), say that a tower of good open covers of UU is a sequence of morphisms {U iϕ iU}\{U_i \overset{\phi_i}{\to} U\} in TowCartSpTowCartSp such that these are the formal sequential limit of a cofiltered diagram of good open covers {U i kϕ i kU k}\{U_i^k \overset{\phi_i^k}{\to} U^k\}.

U i k lim f U i ϕ i k ϕ i U k lim f U \array{ U_i^{k} &\overset{\underset{\longleftarrow}{\lim}^f}{\mapsto}& U_i \\ {}^{\mathllap{\phi_i^k}}\downarrow && \downarrow^{\mathrlap{\phi_i}} \\ U^k &\overset{\underset{\longleftarrow}{\lim}^f}{\mapsto}& U }
Definition

The collection of towers of good open covers on TowCartSpTowCartSp, according to def. , constitutes a coverage.

Proof

By the definition of coverage (def.) we need to check that for every tower of good open covers {U iϕ iU}\{U_i \overset{\phi_i}{\to} U\} and for every morphism VgUV \overset{g}{\longrightarrow} U in TowCartSpTowCartSp, there exists a tower of good open covers {V jψ jV}\{V_j \overset{\psi_j}{\longrightarrow} V\} of VV such that for each index jj we may find an index ii and a morphism V jU iV_j \overset{}{\to} U_i such as to make a commuting diagram of the form

V j U i ϕ i V g U. \array{ V_j &\overset{}{\longrightarrow}& U_i \\ \downarrow && \downarrow^{\mathrlap{\phi_i}} \\ V &\underset{g}{\longrightarrow}& U } \,.

Now by this prop. the bottom morphism is represented by a sequence of component morphisms

V n(k)U k. V^{n(k)} \overset{}{\longrightarrow} U^k \,.

Since ordinary good open covers do form a coverage on CartSp (prop.) each of these component diagrams may be completed

V˜ j n(k) U i k ϕ i k V n(k) g k U k \array{ \tilde V^{n(k)}_j &\overset{}{\longrightarrow}& U^k_i \\ \downarrow && \downarrow^{\mathrlap{\phi^k_i}} \\ V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k }

by first forming the pullback open cover (g k) *U i kV n(k)(g^k)^\ast U^k_i \to V^{n(k)} and then refining this to a good open cover V˜ j n(k)V n(k)\tilde V^{n(k)}_j \to V^{n(k)}. By the universal property of the pullback, there are morphisms

V˜ n(k+1)(g k) *U i k \tilde V^{n(k+1)} \longrightarrow (g^k)^\ast U^k_i

that make the evident cube commute

V˜ j n(k+1) U i k+1 ϕ i k+1 V n(k+1) g k+1 U k+1(g k) *U i k U i k ϕ i k V n(k) g k U k \array{ \tilde V^{n(k+1)}_j &\overset{}{\longrightarrow}& U^{k+1}_i \\ \downarrow && \downarrow^{\mathrlap{\phi^{k+1}_i}} \\ V^{n(k+1)} &\underset{g^{k+1}}{\longrightarrow}& U^{k+1} } \;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ (g^{k})^\ast U_i^k &\overset{}{\longrightarrow}& U^k_i \\ \downarrow && \downarrow^{\mathrlap{\phi^k_i}} \\ V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k }

Take

V j n(0)V˜ j n(0) V^{n(0)}_j \coloneqq \tilde V^{n(0)}_j

and then inductively define

V j n(k+1) V^{n(k+1)}_j

to be a refinement by a good open cover of the joint refinement of {V˜ j n(k+1)}\{\tilde V^{n(k+1)}_j\} with the pullback of {V j n(k)}\{V^{n(k)}_j\} to V n(k+1)V^{n(k+1)}.

This refines the above commuting cubes to

V j n(k+1) U i k+1 ϕ i k+1 V n(k+1) g k+1 U k+1V j n(k) U i k ϕ i k V n(k) g k U k \array{ V_j^{n(k+1)} &\overset{}{\longrightarrow}& U^{k+1}_i \\ \downarrow && \downarrow^{\mathrlap{\phi^{k+1}_i}} \\ V^{n(k+1)} &\underset{g^{k+1}}{\longrightarrow}& U^{k+1} } \;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ V_j^{n(k)} &\overset{}{\longrightarrow}& U^k_i \\ \downarrow && \downarrow^{\mathrlap{\phi^k_i}} \\ V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k }

and hence provides components for the required diagram in TowCartSpTowCartSp.

Last revised on September 20, 2017 at 06:31:03. See the history of this page for a list of all contributions to it.