nLab smooth set

Idea

The concept of a smooth set or smooth space, in the sense discussed here, is a generalization of that of smooth manifolds beyond that of diffeological spaces: A smooth set is a generalized smooth space that may be probed by smooth Cartesian spaces.

For expository details see at geometry of physics – smooth sets.

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

Definition

The category of smooth spaces is the sheaf topos

$$SmoothSp := Sh(Diff)$$

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

Since $Diff$ is equivalent to the category of manifolds embedded into $\mathbb{R}^\infty$, $Diff$ 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 $SmoothSp$ depends on much less than all of $Diff$.

Let

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

and

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

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

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

Examples

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

  $$Diff \hookrightarrow SmoothSp \,.$$

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

• The full subcategory

  $$DiffSp \subset SmoothSp$$

  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

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

    If $X$ and $\Sigma$ are ordinary manifolds, then the hom-set on the right sits inside that of the underlying sets $SmoothSp(\Sigma \times U , X) \subset Set(|\Sigma| \times |U|, |X| )$ 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:

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

• The category

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

  equivalently that of sheaves on $Diff$ with values in simplicial sets

  $$\cdots \simeq Sh(Diff, SSet)$$

  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

Cohesion

Proposition

(smooth sets form a cohesive topos)

The category $SmoothSet$ of smooth sets is a cohesive topos

(1)$$SmoothSet \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } Set$$

Proof

First of all (by this Prop) smooth sets indeed form a sheaf topos, over the site CartSp of Cartesian spaces $\mathbb{R}^n$ with smooth functions between them, and equipped with the coverage of differentiably-good open covers (this def.)

$$SmoothSet \simeq Sh(CartSp) \,.$$

Hence, by Prop. , it is now sufficient to see that CartSp is a cohesive site (Def. ).

It clearly has finite products: The terminal object is the point, given by the 0-dimensional Cartesian space

$$\ast = \mathbb{R}^0$$

and the Cartesian product of two Cartesian spaces is the Cartesian space whose dimension is the sum of the two separate dimensions:

$$\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,.$$

This establishes the first clause in Def. .

For the second clause, consider a differentiably-good open cover $\{U_i \overset{}{\to} \mathbb{R}^n\}$ (this def.). This being a good cover implies that its Cech groupoid is, as an internal groupoid (via this remark), of the form

(2)$$C(\{U_i\}_i) \;\simeq\; \left( \array{ \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} y(U_i) } \right) \,.$$

where we used the defining property of good open covers to identify $y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )$.

The colimit of (2), regarded just as a presheaf of reflexive directed graphs (hence ignoring composition for the moment), is readily seen to be the graph of the colimit of the components (the universal property follows immediately from that of the component colimits):

(3)$$\begin{aligned} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \ast \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \ast } \right) \end{aligned} \,.$$

Here we first used that colimits commute with colimits, hence in particular with coproducts (this prop.) and then that the colimit of a representable presheaf is the singleton set (this Lemma).

This colimiting graph carries a unique composition structure making it a groupoid, since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point.

Precisely this operation on Cech groupoids of good open covers of topological spaces is what Borsuk's nerve theorem is about, a classical result in topology/homotopy theory. This theorem implies directly that the set of connected components of the groupoid (4) is in bijection with the set of connected components of the Cartesian space $\mathbb{R}^n$, regarded as a topological space. But this is evidently a connected topological space, which finally shows that, indeed

$$\pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,.$$

The second item of the second clause in Def. follows similarly, but more easily: The limit of the Cech groupoid is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since $CartSp$ has a terminal object $\ast = \mathbb{R}^0$, which is hence an initial object in the opposite category $CartSp^{op}$, limits over $CartSp^{op}$ yield simply the evaluation on that object:

(4)$$\begin{aligned} \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \phantom{A} \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} Hom_{CartSp}\left( \ast, U_i \underset{\mathbb{R}^n}{\cap} U_j \right) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} Hom_{CartSp}( \ast, U_i ) } \right) \end{aligned} \,.$$

Here we used that colimits (here coproducts) of presheaves are computed objectwise, and then the definition of the Yoneda embedding $y$.

But the equivalence relation induced by this graph on its set of objects $\underset{i}{\coprod} Hom_{CartSp}( \ast, U_i )$ precisely identifies pairs of points, one in $U_i$ the other in $U_j$, that are actually the same point of the $\mathbb{R}^n$ being covered. Hence the set of equivalence classes is the set of points of $\mathbb{R}^n$, which is just what remained to be shown:

$$\pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,.$$

Topos points and stalks

Lemma

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

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

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

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

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

Lemma

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

Distribution theory

Since a space of smooth functions on a smooth manifold is canonically a smooth set, it is natural to consider the smooth linear functionals on such mapping spaces. These turn out to be equivalent to the continuous linear functionals, hence to distributional densities. See at distributions are the smooth linear functionals for details.

Variants and generalizations

Synthetic differential geometry

The site CartSp${}_{smooth}$ may be replaced by the site CartSp${}_{th}$ (see there) whose objects are products of smooth Cartesian spaces with infinitesimally thickened points. The corresponding sheaf topos $Sh(CartSp_{th})$ 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 $Sh(CartSp)$ is not.

The two toposes are related by an adjoint quadruple of functors that witness the fact that the objects of $Sh(CartSp_{th})$ are possiby infinitesimal extensions of objects in $Sh(CartSp)$. 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.

$\,$

Related concepts

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

$\,$

References

The category of sheaves of the site of smooth manifolds appears as a model for homotopy types in

• Denis-Charles Cisinski, Ch. 6 in: Faisceaux localement asphériques (2003) [pdf, pdf]

and in the context of simplicial homotopy theory in:

• Daniel Dugger, section 3.4, from page 29 on in: Sheaves and Homotopy Theory [web, pdf]

  (the topos points of $Sh(Diff)$ are discussed there in example 4.1.2 on p. 36, mentioned before on p. 31)

The equivalent incarnation over the smaller site CartSp and discussion as a cohesive topos appears in

• Urs Schreiber, Def. 1.2.16, Def. 1.3.58 of: differential cohomology in a cohesive topos [arXiv:1310.7930]

The terminology “smooth set” is due to

• nLab: smooth set — from revision 10 on (Feb 2013)

• Schreiber 2013, §1.2.2, p. 48

  (which otherwise speaks of “smooth 0-types”)

• Urs Schreiber: geometry of physics – smooth sets — from revision 3 on (Nov 2014)

• Igor Khavkine, Urs Schreiber, Def. 2.1 in: Synthetic geometry of differential equations: I. Jets and comonad structure [arXiv:1701.06238]

Discussion of smooth sets as a convenient category for variational calculus of Lagrangian classical field theory:

• Grigorios Giotopoulos, Hisham Sati, Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]

Exposition:

• Grigorios Giotopoulos: Classical field theory in the topos of smooth sets, talk at CQTS (Oct 2023) [pdf, video:YT]

• Grigorios Giotopoulos: Towards Non-Perturbative Lagrangian Field Theory via the Topos of Smooth Sets, talk at M-Theory and Mathematics 2024 (Jan 2024) [video: kt]

• Grigorios Giotopoulos: Sheaf Topos Theory as a setting for Physics, talk at Workshop on Noncommutative and Generalized Geometry in String Theory, Corfu Summer Institute (2024) [pdf]

  (also on super smooth sets)