nLab
smooth space

Idea

A smooth space or smooth set as discussed here, is a joint generalization of smooth manifolds and diffeological spaces: it is a space that can be probed by smooth Cartesian spaces (in the sense discussed in the exposition at motivation for sheaves, cohomology and higher stacks).

Following the logic of space and quantity, a smooth space is, in full generality, a space that may be probed by standard smooth test spaces. See generalized smooth space for more on the general idea and for examples and variations.

Here standard smooth test spaces may be taken to be smooth manifolds. But since manifolds themselves are built from gluing together smooth open balls $D_{\mathrm{int}}^n\subset {ℝ}^n$ or equivalently Cartesian spaces ${ℝ}^n$, one may just as well consider Cartesian spaces test spaces. Finally, since $D^n$ is diffeomorphic to ${ℝ}^n$, one can just as well take just the cartesian smooth spaces ${ℝ}^n$ as test objects.

Definition

The category of smooth spaces is the sheaf topos

$$\mathrm{SmoothSp}:=\mathrm{Sh}(\mathrm{Diff})$$SmoothSp := Sh(Diff)

of sheaves on the site Diff of smooth manifolds equipped with its standard coverage (Grothendieck topology) given by open covers of manifolds.

Since $\mathrm{Diff}$ is equivalent to the category of manifolds embedded into ${ℝ}^{\mathrm{\infty }}$, $\mathrm{Diff}$ is an essentially small category, so there are no size issues involved in this definition.

But since manifolds themselves are defined in terms of gluing conditons, the Grothendieck topos $\mathrm{SmoothSp}$ depends on much less than all of $\mathrm{Diff}$.

Let

$$\mathrm{Ball}:=\{(D_{\mathrm{int}}^n\to D_{\mathrm{int}}^m)\in \mathrm{Diff}\mid n,m\in \mathbb{N}\}$$Ball := \{ (D^n_{int} \to D^m_{int}) \in Diff | n,m \in \mathbb{N}\}

and

$$\mathrm{CartSp}:=\{({ℝ}^n\to {ℝ}^m)\in \mathrm{Diff}\mid n,m\in \mathbb{N}\}$$CartSp := \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff | n,m \in \mathbb{N}\}

be the full subcategories $\mathrm{Ball}$ and CartSp of $\mathrm{Diff}$ on open balls and on cartesian spaces, respectively. Then the corresponding sheaf toposes are still those of smooth spaces:

$$\begin{array}{rl}\mathrm{SmoothSp}& \simeq \mathrm{Sh}(\mathrm{Ball})\\ & \simeq \mathrm{Sh}(\mathrm{CartSp})\end{array}.$$\begin{aligned} SmoothSp &\simeq Sh(Ball) \\ & \simeq Sh(CartSp) \end{aligned} \,.

Examples

• The category of ordinary manifolds is a full subcategory of smooth spaces:

  $$\mathrm{Diff}\hookrightarrow \mathrm{SmoothSp}.$$Diff \hookrightarrow SmoothSp \,.

  When one regards smooth spaces concretely as sheaves on $\mathrm{Diff}$, then this inclusion is of course just the Yoneda embedding.

• The full subcategory

  $$\mathrm{DiffSp}\subset \mathrm{SmoothSp}$$DiffSp \subset SmoothSp

  on concrete sheaves is called the category of diffeological spaces.

  • The standard class of examples of smooth spaces that motivate their use even in cases where one starts out being intersted just in smooth manifolds are mapping spaces: for $X$ and $\Sigma$ two smooth spaces (possibly just ordinary smooth manifolds), by the closed monoidal structure on presheaves the mapping space $[\Sigma ,X]$, i.e. the space of smooth maps $\Sigma \to X$ exists again naturally as a smooth. By the general formula it is given as a sheaf by the assignment

    $$[\Sigma ,X]:U\mapsto \mathrm{SmoothSp}(\Sigma \times U,X).$$[\Sigma,X] : U \mapsto SmoothSp(\Sigma \times U, X) \,.

    If $X$ and $\Sigma$ are ordinary manifolds, then the hom-set on the right sits inside that of the underlying sets $\mathrm{SmoothSp}(\Sigma \times U,X)\subset \mathrm{Set}(\mid \Sigma \mid \times \mid U\mid ,\mid X\mid )$ so that $[\Sigma ,X]$ is a diffeological space.

    The above formula says that a $U$-parameterized family of maps $\Sigma \to X$ is smooth as a map into the smooth space $[\Sigma ,X]$ precisely if the corresponding map of sets $U\times \Sigma \to X$ is an ordinary morphism of smooth manifolds.

• The canonical examples of smooth spaces that are not diffeological spaces are the sheaves of (closed) differential forms:

  $$K^n:U\mapsto {\Omega }_{\mathrm{closed}}^n(U).$$K^n : U \mapsto \Omega^n_{closed}(U) \,.

• The category

  $$\mathrm{SimpSmoothSp}:={\mathrm{SmoothSp}}^{{\Delta }^{\mathrm{op}}}$$SimpSmoothSp := SmoothSp^{\Delta^{op}}

  equivalently that of sheaves on $\mathrm{Diff}$ with values in simplicial sets

  $$\cdots \simeq \mathrm{Sh}(\mathrm{Diff},\mathrm{SSet})$$\cdots \simeq Sh(Diff, SSet)

  of simplicial objects in smooth spaces naturally carries the structure of a homotopical category (for instance the model structure on simplicial sheaves or that of a Brown category of fibrant objects (if one restricts to locally Kan simplicial sheaves)) and as such is a presentation for the (∞,1)-topos of smooth ∞-stacks.

Properties

Connectedness, locality, cohesion

The topos of smooth space is

• a locally connected and connected topos (discussed here);

• a local topos (discussed here);

• hence a cohesive topos (discussed here).

Topos points and stalks

Lemma

For every $n\in N$ there is a topos point

$$D^n:\mathrm{Set}\stackrel{\stackrel{(D^n{)}^{*}}{\leftarrow }}{\stackrel{D_{*}^n}{\to }}\mathrm{SmoothSp}$$D^n : Set \stackrel{\stackrel{(D^n)^*}{\leftarrow}} {\stackrel{D^n_*}{\to}} SmoothSp

where the inverse image morphism – the stalk – is given on $A\in \mathrm{SmoothSp}$ by

$$(D^n{)}^{*}A:={colim}_{{ℝ}^n\supset U\ni 0}A(U),$$(D^n)^* A := \colim_{\mathbb{R}^n \supset U \ni 0} A(U) \,,

where the colimit is over all open neighbourhoods of the origin in ${ℝ}^n$.

Lemma

SmoothSp has enough points: they are given by the $D^n$ for $n\in \mathbb{N}$.

Variants and generalizations

Synthetic differential geometry

The site CartSp${}_{\mathrm{smooth}}$ may be replaced by the site CartSp${}_{\mathrm{th}}$ (see there) whose objects are products of smooth Cartesian spaces with infinitesimally thickened points. The corresponding sheaf topos $\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$ is called the Cahiers topos. It contains smooth spaces with possibly infinitesimal extension and is a model for synthetic differential geometry (a “smooth topos”), which $\mathrm{Sh}(\mathrm{CartSp})$ is not.

The two toposes are related by an adjoint quadruple of functors that witness the fact that the objects of $\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{th}})$ are possiby infinitesimal extensions of objects in $\mathrm{Sh}(\mathrm{CartSp})$. For more discussion of this see synthetic differential ∞-groupoid

Higher smooth geometry

The topos of smooth spaces has an evident generalization from geometry to higher geometry, hence from differential geometry to higher differential geometry: to an (∞,1)-topos of smooth ∞-groupoids. See there for more details.

References

Lecture notes are at

• geometry of physics – smooth spaces

The concrete smooth spaces are known as diffeological spaces. See there for more references.

Aspects of the category of smooth spaces is discussed with an eye towards its generalization to smooth ∞-groupoids and their homotopy localization in section 3.4, from page 29 on in

• Daniel Dugger, Sheaves and Homotopy Theory (web, pdf)

The topos points of $\mathrm{Sh}(\mathrm{Diff})$ are discussed there in example 4.1.2 on p. 36. (they are mentioned before on p. 31).

As a cohesive topos smooth spaces are discuss in sections 1.2, 1.3 and 3.3 in

• Urs Schreiber, differential cohomology in a cohesive topos