nLab pro-manifold



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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


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.


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 \,.

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



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}.


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 }

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


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 }


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 10:31:03. See the history of this page for a list of all contributions to it.