nLab
smooth set

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Cohesive toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Smooth spaces

Idea

The concept of a smooth set or smoth 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 int n nD^n_{int} \subset \mathbb{R}^n or equivalently Cartesian spaces n\mathbb{R}^n, one may just as well consider Cartesian spaces test spaces. Finally, since D nD^n is diffeomorphic to n\mathbb{R}^n, one can just as well take just the cartesian smooth spaces n\mathbb{R}^n as test objects.

Definition

The category of smooth spaces is the sheaf topos

SmoothSp:=Sh(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 DiffDiff is equivalent to the category of manifolds embedded into \mathbb{R}^\infty, DiffDiff 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 SmoothSpSmoothSp depends on much less than all of DiffDiff.

Let

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

and

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

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

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

Examples

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

    DiffSmoothSp. Diff \hookrightarrow SmoothSp \,.

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

  • The full subcategory

    DiffSpSmoothSp 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 XX and Σ\Sigma two smooth spaces (possibly just ordinary smooth manifolds), by the closed monoidal structure on presheaves the mapping space [Σ,X][\Sigma,X], i.e. the space of smooth maps ΣX\Sigma \to X exists again naturally as a smooth. By the general formula it is given as a sheaf by the assignment

      [Σ,X]:USmoothSp(Σ×U,X). [\Sigma,X] : U \mapsto SmoothSp(\Sigma \times U, X) \,.

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

      The above formula says that a UU-parameterized family of maps ΣX\Sigma \to X is smooth as a map into the smooth space [Σ,X][\Sigma,X ] precisely if the corresponding map of sets U×ΣXU \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:UΩ closed n(U). K^n : U \mapsto \Omega^n_{closed}(U) \,.
  • The category

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

    equivalently that of sheaves on DiffDiff with values in simplicial sets

    Sh(Diff,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

Cohesion

Proposition

(smooth sets form a cohesive topos)

The category SmoothSetSmoothSet of smooth sets is a cohesive topos

(1)SmoothSetAAAΠ 0AAA AADiscAA AAAΓAAA AAcoDiscAASet 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 n\mathbb{R}^n with smooth functions between them, and equipped with the coverage of differentiably-good open covers (this def.)

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

Hence, by Prop. \ref{CategoriesOfSheavesOnCohesiveSiteIsCohesive}, it is now sufficient to see that CartSp is a cohesive site (Def. \ref{OneCohesiveSite}).

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

*= 0 \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:

n 1× n 2 n 1+n 2. \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,.

This establishes the first clause in Def. \ref{OneCohesiveSite}.

For the second clause, consider a differentiably-good open cover {U i n}\{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)(i,jy(U i nU j) iy(U i)). 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)× Xy(U j)y(U i XU j)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)limCartSp opC({U i} i) (limCartSp opi,jy(U i nU j) limCartSp opiy(U i)) (i,jlimCartSp opy(U i nU j) ilimCartSp opy(U i)) (i,j* i*). \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 n\mathbb{R}^n, regarded as a topological space. But this is evidently a connected topological space, which finally shows that, indeed

π 0limCartSp opC({U i} i)*. \pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,.

The second item of the second clause in Def. \ref{OneCohesiveSite} 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 CartSpCartSp has a terminal object *= 0\ast = \mathbb{R}^0, which is hence an initial object in the opposite category CartSp opCartSp^{op}, limits over CartSp opCartSp^{op} yield simply the evaluation on that object:

(4)limCartSp opC({U i} i) (limCartSp opi,jy(U i nU j) limCartSp opiy(U i)A) (i,jHom CartSp(*,U i nU j) iHom CartSp(*,U i)). \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 yy.

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

π 0limCartSp opC({U i} i)Hom CartSp(*, n). \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 nNn \in N there is a topos point

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

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

(D n) *A:=colim nU0A(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 n\mathbb{R}^n.

Lemma

SmoothSp has enough points: they are given by the D nD^n for nn \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{}_{smooth} may be replaced by the site CartSp th{}_{th} (see there) whose objects are products of smooth Cartesian spaces with infinitesimally thickened points. The corresponding sheaf topos Sh(CartSp th)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)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)Sh(CartSp_{th}) are possiby infinitesimal extensions of objects in Sh(CartSp)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.

References

Lecture notes are at

Aspects of the category of smooth sets is discussed, with an eye towards its generalization to smooth ∞-groupoids and their homotopy localization in

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

As a cohesive topos, smooth sets are discussed in

Last revised on June 16, 2018 at 10:19:58. See the history of this page for a list of all contributions to it.