nLab smooth set

Idea

The concept of a smooth sets (or smooth spaces) [Schreiber 2013, 2014], is a generalization of that of smooth manifolds beyond that of diffeological spaces: A smooth set is a generalized smooth space that is characterized by how it may be probed by smooth Cartesian spaces.

For example, moduli spaces of differential forms exist as smooth sets but not as diffeological spaces (much less as smooth manifolds). Thereby, smooth sets constitute a natural foundation for the geometry of physics, specifically of variational calculus and Lagrangian field theory [Schreiber 2013b, 2017, Giotopoulos & Sati 2023].

Hence the category of smooth sets is a convenient categories of spaces for differential topology, in a precise technical sense: It is a topos and in fact a cohesive topos [Schreiber 2013 §4.4].

From a broader perspective, smooth sets are equivalently the 0-truncated smooth ∞-groupoids, the latter generalizing smooth sets from geometry to higher geometry, specifically from differential geometry to higher differential geometry, forming a cohesive $\infty$-topos (exposition in Schreiber 2025). Thereby, smooth sets are part of a hierarchy of ever more convenient categories of spaces for differential topology:

From Sati-Schreiber 2025

Definition

One way to define category of smooth sets is as the sheaf topos

$$SmthSet \coloneqq Sh(SmthMfd)$$

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

Since $SmthMfd$ is equivalent to the category of manifolds embedded into $\mathbb{R}^\infty$, $SmthMfd$ 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 conditions, the Grothendieck topos $SmthSet$ depends on much less than all of $SmthMfd$.

Let

$$Ball \coloneqq \big\{ (D^n_{int} \to D^m_{int}) \in SmthMfd \vert n,m \in \mathbb{N} \big\}$$

and

$$CartSp \coloneqq \big\{ (\mathbb{R}^n \to \mathbb{R}^m) \in SmthMfd \vert| n,m \in \mathbb{N} \big\}$$

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

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

Examples

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

  $$SmthMfd \hookrightarrow SmoothSet \,.$$

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

• The full subcategory

  $$DiffSp \subset SmoothSet$$

  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

    $$[\Sigma,X] \colon U \mapsto SmthSet(\Sigma \times U, X) \,.$$

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

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

• The category

  $$SimpSmthSet \coloneqq SmthSet^{\Delta^{op}}$$

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

  $$\cdots \simeq Sh(SmthMfd, sSet)$$

  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 ofsmooth ∞-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

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

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

$SmthSet$ 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.

Shape

The shape (or smooth singular complex) functor sends a smooth set $X$ to the simplicial set $\Pi X$ whose set of $n$-simplices is $X(\Delta^n)$, i.e., smooth maps from a smooth $n$-simplex to $X$.

Various choices of cosimplicial objects $\Delta$ yield weakly equivalent shapes. Some of the more popular choices include the following.

• The cosimplicial object $n\mapsto\{x\in\mathbf{R}^{n+1}\mid \sum_i x_i=1\}$ of extended smooth simplices, which is degreewise representable because it is isomorphic to $\mathbf{R}^n$.

• The subobject of the above object given by taking $x_i\ge0$, which yields the more traditional notion of a singular complex. This subobject is no longer a representable presheaf, so working with it may be more involved.

The left adjoint to the shape functor is known as the smooth geometric realization functor. It is computed much like the usual geometric realization, but working in smooth sets instead of topological spaces.

The relative category of smooth sets

The category of smooth sets can be turned into a relative category by declaring a morphism of smooth sets to be a weak equivalence if its shape is a weak equivalence of simplicial sets.

The realization-shape adjunction described above has degreewise weak equivalences as its unit and counit and therefore is an equivalence of relative categories. Thus, smooth sets form yet another model for homotopy types, with the previous statement forming another variant of the homotopy hypothesis.

Model structures on smooth sets

The relative category of smooth sets described above admits a variety of model structures, some of which include the following (Cisinski 2002, Théorème 3.9).

• The injective model structure, in which cofibrations are defined as monomorphisms.

• The projective model structure, transferred from the Kan–Quillen model structure on simplicial sets along the shape functor.

References on the projective model structure also include Clough 2023 (Proposition 7.1.5), Pavlov 2022 (Theorem 7.4).

By Pavlov 2022 (Proposition 8.9), the projective model structure is a cartesian model structure.

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.

Related concepts

• smooth groupoid, smooth $\infty$-groupoid

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 is considered as a model for homotopy types in

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

• Denis-Charles Cisinski, Théories homotopiques dans les topos, Journal of Pure and Applied Algebra 174:1 (2002), 43–82. doi:10.1016/s0022-4049(01)00176-1.

• Adrian Clough, The homotopy theory of differentiable sheaves, arXiv:2309.01757.

• Dmitri Pavlov, Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle, Homology, Homotopy, and Applications 26:2 (2024), 375–408. arXiv:2210.12845.

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)

and as a model for generalized smooth spaces in

• Ib Madsen, Michael Weiss, appendix A.1 of: The stable moduli space of Riemann surfaces: Mumford’s conjecture, Annals of Mathematics 165 3 (2007) 843–941 [arXiv:math/0212321, doi:10.4007/annals.2007.165.843]

  (where a special case of the smooth Oka principle is proven)

The equivalent incarnation over the dense subsite CartSp and the understanding as a cohesive topos is due to:

• 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]

  (generalization to formal smooth sets)

further discussed in the context of (higher, singular) cohesive toposes in:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Ex. 1.26 in: The Character Map in Non-Abelian Cohomology, World Scientific (2023) [doi:10.1142/13422]

• Hisham Sati, Urs Schreiber, Ntn. 4.3.15 in: Equivariant Principal $\infty$-Bundles, Cambridge University Press (2025) [arXiv:2112.13654]

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

• Urs Schreiber: Classical field theory via Cohesive homotopy types [arXiv:1311.1172, pdf]

• Urs Schreiber: Geometry of Physics – Perturbative Quantum Field Theory, lecture notes for a course Mathematical Quantum Field Theory, Hamburg University (2017) [pdf, PhysicsForums version]

• Grigorios Giotopoulos, Hisham Sati: Field Theory via Higher Geometry I: Smooth Sets of Fields, Journal of Geometry and Physics 213 (2025) 105462 [arXiv:2312.16301, doi:10.1016/j.geomphys.2025.105462]

Exposition:

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

• Urs Schreiber: Higher Topos Theory in Physics, Encyclopedia of Mathematical Physics 2nd ed$\;$4 (2025) 62-76 [doi:10.1016/B978-0-323-95703-8.00210-X, ISBN:9780323957038, arXiv:2311.11026]

• 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)

• Grigorios Giotopoulos: Sheaf Topos Theory: A powerful setting for Lagrangian Field Theory [arXiv:2504.08095]

• Alberto Ibort, Arnau Mas: Smooth sets of fields: A pedagogical introduction, Geometric Mechanics (2025) [arXiv:2510.20422, doi:10.1142/S2972458925400052, ResearchGate:394210704]

In analogy with smooth sets one may consider “D-topological sets” (among D-topological infinity-groupoids), forming the sheaf topos over the site $Cart_{top}$ of Cartesian spaces with continuous maps between them.

On $Sh(Cart_{top})$ as a classifying topos:

• Jonas Frey: $Sh(Cart)$ as a classifying topos (2023) [github pdf]