nLab
smooth manifold

Context

Higher geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Manifolds and cobordisms

Contents

Idea

A smooth manifold is a space that is locally isomorphic to a Cartesian space n\mathbb{R}^n equipped with its canonical smooth structure.

Definition

Traditional definition

Traditionally, a smooth manifold is defined as follows.

As special topological manifolds

Definition

A manifold is a smooth manifold if its transition functions are smooth functions n n\mathbb{R}^n \to \mathbb{R}^n, or in other words a GG-manifold over the pseudogroup GG of C C^\infty diffeomorphisms between open sets of a Euclidean space.

So a smooth manifold is a C kC^k-differentiable manifold for all kk.

A homomorphism of smooth manifolds is a smooth functions. Smooth manifolds and smooth functions form the category Diff.

As special locally ringed spaces

Proposition

A smooth manifold is equivalently a locally ringed space (X,𝒪 X)(X,\mathcal{O}_X) which is locally isomorphic to the ringed space ( n,C ())(\mathbb{R}^n, C^\infty(-) ).

(…)

General abstract geometric definition

There is a more fundamental and general abstract way to think of smooth manifolds, which realizes their theory as a special case of general constructions in higher geometry.

In this context one specifies for instance 𝒢\mathcal{G} a geometry (for structured (∞,1)-toposes) and then plenty of geometric notions are defined canonically in terms of 𝒢\mathcal{G}. The theory of smooth manifolds appears if one takes 𝒢=\mathcal{G} = CartSp.

Alternatively one can specify differential cohesion and proceed as discussed at differential cohesion – structures - Cohesive manifolds (separated).

This is discussed in The geometry CartSp below.

The geometry CartSpCartSp

Let CartSp be the category of Cartesian spaces and smooth functions between them. This has finite products and is in fact (the syntactic category) of a Lawvere theory: the theory of smooth algebras.

Moreover, CartSpCartSp is naturally equipped with the good open cover coverage that makes it a site.

Both properties together make it a pregeometry (for structured (∞,1)-toposes) (if the notion of Grothendieck topology is relaxed to that of coverage in StrSp).

For 𝒳\mathcal{X} a topos, a product-preserving functor

𝒪:𝒢𝒳 \mathcal{O} : \mathcal{G} \to \mathcal{X}

is a 𝒢\mathcal{G}-algebra in 𝒳\mathcal{X}. This makes 𝒳\mathcal{X} is 𝒢\mathcal{G}-ringed topos. For 𝒢=\mathcal{G} = CartSp this algebra is a smooth algebra in 𝒳\mathcal{X}. If 𝒳\mathcal{X} has a site of definition XX, then this is a [sheaf] of smooth algebras on XX.

If 𝒪\mathcal{O} sends covering families {U iU}\{U_i \to U\} in 𝒢\mathcal{G} to effective epimorphism i𝒪(U i)𝒪(U)\coprod_i \mathcal{O}(U_i) \to \mathcal{O}(U) we say that it is a local 𝒢\mathcal{G}-algebra in 𝒳\mathcal{X}, making 𝒳\mathcal{X} a 𝒢\mathcal{G}-locally ringed topos.

The big topos Sh(𝒢)Sh(\mathcal{G}) itself is canonically equipped with such a local 𝒢\mathcal{G}-algebra, given by the Yoneda embedding jj followed by sheafification LL

𝒪:𝒢jPSh(𝒢)LSh(𝒢). \mathcal{O} : \mathcal{G} \stackrel{j}{\to} PSh(\mathcal{G}) \stackrel{L}{\to} Sh(\mathcal{G}) \,.

It is important in the context of locally representable locally ringed toposes that we regard Sh(𝒢)Sh(\mathcal{G}) as equipped with this local 𝒢\mathcal{G}-algebra. This is what remembers the site and gives a notion of local representability in the first place.

The big topos Sh(CartSp)Sh(CartSp) is a cohesive topos of generalized smooth spaces. Its concrete sheaves are precisely the diffeological spaces. See there for more details. We now discuss how with Sh(CartSp)Sh(CartSp) regarded as a CartSpCartSp-structured topos, smooth manifolds are precisely its locally representable objects.

Cartesian spaces as representable objects of Sh(CartSp)Sh(CartSp)

The representables themselves should evidently be locally representable and canonically have the structure of CartSpCartSp-structured toposes.

Indeed, every object UCartSpU \in \mathrm{CartSp} is canonically a CartSp-ringed space, meaning a topological space equipped with a local sheaf of smooth algebras. More generally: every object UCartSpU \in CartSp is canonically incarnated as the CartSpCartSp-structured (∞,1)-topos

(𝒳,𝒪 𝒳):=(Sh (,1)(CartSp)/U,𝒪 U:CartSpjSh (,1)(CartSp)U *Sh (,1)(CartSp)/U) (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) := (Sh_{(\infty,1)}(CartSp)/U , \;\;\; \mathcal{O}_U : CartSp \stackrel{j}{\to} Sh_{(\infty,1)}(CartSp) \stackrel{U^*}{\to} Sh_{(\infty,1)}(CartSp)/U)

given by the over-(∞,1)-topos of the big (∞,1)-sheaf (∞,1)-topos over CartSpCartSp and the structure sheaf given by the composite of the (∞,1)-Yoneda embedding and the inverse image of the etale geometric morphism induced by UU.

Smooth manifolds as locally representable objects of Sh(CartSp)Sh(CartSp)

Definition

Say a concrete object XX in the sheaf topos Sh(CartSp)Sh(CartSp) – a diffeological space – is locally representable if there exists a family of open embeddings {U iX} iX\{U_i \hookrightarrow X\}_{i \in X} with U iCartSpjSh(CartSp)U_i \in CartSp \stackrel{j}{\hookrightarrow} Sh(CartSp) such that the canonical morphism out of the coproduct

iU iX \coprod_i U_i \to X

is an effective epimorphism in Sh(CartSp)Sh(CartSp).

Let LocRep(CartSp)Sh(CartSp)LocRep(CartSp) \hookrightarrow Sh(CartSp) be the full subcategory on locally representable sheaves.

Proposition

There is an equivalence of categories

DiffLocRep(CartSp) Diff \simeq LocRep(CartSp)

of the category Diff of smooth manifolds with that of locally representable sheaves for the pre-geometry CartSpCartSp.

Proof

Define a functor DiffLocRep(CartSp)Diff \to LocRep(CartSp) by sending each smooth manifold to the sheaf over CartSpCartSp that it naturally represents. By definition of manifold there is an open cover {U iX}\{U_i \hookrightarrow X\}. We claim that iU iX\coprod_i U_i \to X is an effective epimorphism, so that this functor indeed lands in LocRep(CartSp)LocRep(CartSp). (This is a standard argument of sheaf theory in Diff, we really only need to observe that it goes through over CartSp, too.)

For that we need to show that

i,jU i× XU j iU iX \coprod_{i, j} U_i \times_X U_j \stackrel{\to}{\to} \coprod_i U_i \to X

is a coequalizer diagram in Sh(CartSp)Sh(CartSp) (that the Cech groupoid of the cover is equivalent to XX.). Notice that the fiber product here is just the intersection in XX U i× XU jU iU jU_i \times_X U_j \simeq U_i \cap U_j. By the fact that the sheaf topos Sh(CartSp)Sh(CartSp) is by definition a reflective subcategory of the presheaf topos PSh(CartSp)PSh(CartSp) we have that colimits in Sh(CartSp)Sh(CartSp) are computed as the sheafification of the corresponding colimit in PSh(CartSp)PSh(CartSp). The colimit in PSh(CartSp)PSh(CartSp) in turn is computed objectwise. Using this, we see that that we have a coequalizer diagram

i,jU i× XU j iU iS({U i}) \coprod_{i, j} U_i \times_X U_j \stackrel{\to}{\to} \coprod_i U_i \to S(\{U_i\})

in PSh(CartSp)PSh(CartSp), where S({U i})S(\{U_i\}) is the sieve corresponding to the cover: the subfunctor S({U i})XS(\{U_i\}) \hookrightarrow X of the functor X:CartSp opSetX : CartSp^{op} \to Set which assigns to VCartSpV \in CartSp the set of smooth functions VXV \to X that have the property that they factor through any one of the U iU_i.

Essentially by the definition of the coverage on CartSpCartSp, it follows that sheafification takes this subfunctor inclusion to an isomorphism. This shows that XX is indeed the tip of the coequalizer in Sh(CartSp)Sh(CartSp) as above, and hence that it is a locally representable sheaf.

Conversely, suppose that for XConc(Sh(CartSp))Sh(CartSp)X \in Conc(Sh(CartSp)) \hookrightarrow Sh(CartSp) there is a family of open embeddings {U iX}\{U_i \hookrightarrow X\} such that we have a coequalizer diagram

i,jU i× XU j iU iX \coprod_{i, j} U_i \times_X U_j \stackrel{\to}{\to} \coprod_{i} U_i \to X

in Sh(CartSp)Sh(CartSp), which is the sheafification of the corresponding coequalizer in PSh(CartSp)PSh(CartSp). By evaluating this on the point, we find that the underlying set of XX is the coequalizer of the underlying set of the U iU_i in SetSet. Since every plot of XX factors locally through one of the U iU_i it follows that XX is a diffeological space.

It follows that in the pullback diagrams

U i× XU j U j U i X \array{ U_i \times_X U_j &\to& U_j \\ \downarrow && \downarrow \\ U_i &\to& X }

the object U iU jU_i \cap U_j is the diffeological space whose underlying topological space is the intersection of U iU_i and U jU_j in the topological space underlying XX. In particular the inclusions U i× XU jU iU_i \times_X U_j \hookrightarrow U_i are open embeddings.

As locally representable CartSpCartSp-structured (,1)(\infty,1)-toposes

We may switch from regarding smooth manifolds as objects in the big topos XSh(CartSp)X \in Sh(CartSp) to regrading them as toposes themselves, by passing to the over-topos Sh(CartSp)/XSh(CartSp)/X. This remembers the extra (smooth) structure on the topological space XX by being canonically a locally ringed topos with the structure sheaf of smooth functions on XX: a CartSp-structured (∞,1)-toposes

For every choice of geometry (for structured (∞,1)-toposes) there is a notion of 𝒢\mathcal{G}-locally representable structured (∞,1)-topos (StrSp).

Claim

Smooth manifolds are equivalently the 0-localic CartSp-generalized schemes of locally finite presentation.

Sketch of proof

The statement says that a smooth manifold XX may be identified with an ∞-stack on CartSp (an ∞-Lie groupoid) which is represented by a CartSp-structured (∞,1)-topos (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) such that

  1. 𝒳\mathcal{X} is a 0-localic (∞,1)-topos;

  2. There exists a family of objects {U i𝒳}\{U_i \in \mathcal{X}\} such that the canonical morphism iU i* 𝒳\coprod_i U_i \to *_{\mathcal{X}} to the terminal object in 𝒳\mathcal{X} is a regular epimorphism;

  3. For every iIi \in I there is an equivalence

(𝒳/U i,𝒪 𝒳|U i)t i(Sh (,1)( n),𝒪( n)). (\mathcal{X}/U_i, \mathcal{O}_{\mathcal{X}|U_i}) \underoverset{\simeq}{t_i}{\to} (Sh_{(\infty,1)}(\mathbb{R}^n), \mathcal{O}(\mathbb{R}^n)) \,.

The second and third condition say in words that (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) is locally equivalent to the ordinary cannonically CartSp-locally ringed space n\mathbb{R}^n (for nn \in \mathbb{N} the dimension. The first condition then says that these local identifications cover 𝒳\mathcal{X}.

(…)

(…)

References

A textbook reference is

Discussion of smooth manifolds as colimits of the Cech nerves of their good open covers is also at

The general abstract framework of higher geometry referred to above is discussed in

Revised on November 12, 2013 10:10:45 by David Corfield (87.114.15.30)