nLab pro-manifold

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 $TowCartSp$.