# nLab pro-manifold

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

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 $\mathbb{R}^n$, for $n \in \mathbb{N}$. Write

$ProCartSp \coloneqq Pro(CartSp)$

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:

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

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

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

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

Now, there is the fully faithful functor

$i \;\colon\; CartSp \hookrightarrow SmthLoc$

(prop.) hence a fully faithful functor

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

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

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

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

This implies that the composite

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

is also fully faithful (prop.).

Here $Ind(i^{op})$ takes formal filtered colimits in $CartSp^{op}$ to the corresponding formal colimits in $SmthAlg_{\mathbb{R}}$ (prop.), while $L$ 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

$TowCartSp \hookrightarrow ProCartSp$

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 $\mathcal{K} = \mathbb{N}_{\geq}$.

###### Definition

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

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

The collection of towers of good open covers on $TowCartSp$, 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 $\{U_i \overset{\phi_i}{\to} U\}$ and for every morphism $V \overset{g}{\longrightarrow} U$ in $TowCartSp$, there exists a tower of good open covers $\{V_j \overset{\psi_j}{\longrightarrow} V\}$ of $V$ such that for each index $j$ we may find an index $i$ and a morphism $V_j \overset{}{\to} U_i$ such as to make a commuting diagram of the form

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

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

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

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

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

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

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

that make the evident cube commute

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

Take

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

and then inductively define

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

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

This refines the above commuting cubes to

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

and hence provides components for the required diagram in $TowCartSp$.

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