Topos of Smooth Sets

• Topos of Smooth Sets is a categorical framework that models smooth spaces as sheaves over probe domains, capturing both infinite-dimensional and singular structures.
• It extends classical smooth manifolds by incorporating diffeological spaces, C∞-rings, and generalized smooth functions through robust algebraic and topological methods.
• The framework underpins advanced constructions in differential geometry and field theory, integrating homotopy theory, variational calculus, and supergeometric extensions.

The topos of smooth sets is a categorical framework in which smooth spaces, including infinite-dimensional and singular objects, are rigorously encoded as sheaves (or, in higher settings, as ∞-sheaves) over a site of probe spaces such as Euclidean spaces, Cartesian spaces, or generalized test domains. This approach subsumes classical smooth manifolds and their function spaces, extends to generalizations such as diffeological spaces, C^∞-rings, Colombeau generalized functions, and supports constructions central to modern differential geometry and field theory via methods from algebraic topology, homotopy theory, and category theory.

1. Categorical and Sheaf-Theoretic Foundations

A smooth set is defined as a sheaf on a site of open subsets of Euclidean spaces, or more generally, on Cartesian spaces. Each object $X$ assigns to every open set $U$ a set $X(U)$ of "plots," and to every smooth map $\varphi: U \to V$ a pullback map $X(\varphi): X(V) \to X(U)$, satisfying contravariance and the sheaf gluing condition:

• Functoriality: $X(\psi \circ \varphi) = X(\varphi) \circ X(\psi)$.
• Gluing: If $\{U_i\}$ covers $U$ and $f_i \in X(U_i)$ agree on overlaps, a unique $f \in X(U)$ exists with $f|_{U_i} = f_i$.

The category of smooth sets ($\mathrm{SmoothSet}$) is a Grothendieck topos, thus complete, cocomplete, and Cartesian closed. Classical manifolds fully faithfully embed via the Yoneda functor $y(M)(U) = C^\infty(U, M)$, with $$\mathrm{Hom}_{\mathrm{SmoothSet}}(y(M), y(N)) \cong C^\infty(M,N)$$ (Ibort et al., 23 Oct 2025).

In higher settings, smooth spaces are modeled by simplicial presheaves on cartesian spaces, $H = [\mathsf{CartSp}^{\mathrm{op}},\mathsf{sSet}]$, with the category $H$ forming an (∞-)topos after localization. This synthesizes homotopical and smooth information (Bunk, 2020).

2. Algebraic Characterization: C^∞-Rings and Topological Structures

In synthetic differential geometry and algebraic approaches, smooth sets are modeled via $C^\infty$-rings—algebras whose operations include all smooth functions. The paper (Borisov, 2011) introduces three natural topologies on rings of smooth functions:

• Germ topology: Ideals are "germ determined" if membership depends on local behavior at all points.
• Jet topology (near-point determined): Membership is determined by finite jets at all points.
• Point topology: Membership depends solely on pointwise values.

Closed ideals in these topologies correspond to germ determined, near-point determined, or point determined ideals. The chain of reflective subcategories

$$\text{Point determined} \subset \text{Near-point determined} \subset \text{Germ determined}$$

mirrors increasing strength of the topological constraint.

Morphisms between near-point determined C^∞-rings are automatically smooth ($R$-algebra morphisms are already $\mathsf{Co}$-morphisms), ensuring categorical robustness: algebraic maps cannot violate smoothness (Borisov, 2011). This rigidity is foundational for encoding smooth sets via algebraic data.

3. Generalized and Singular Structures

Colombeau theory and its extensions utilize "internal sets" and "strongly internal sets," defined via stable nets in test domains. Generalized smooth functions (GSF) are set-theoretical maps on generalized points generated by nets $(u_\varepsilon)$ of smooth functions, subject to moderateness conditions on all derivatives. These morphisms compose unrestrictedly, forming a subcategory of topological spaces where categorical operations remain valid under composition (Giordano et al., 2014).

A further extension constructs a Grothendieck topos of generalized smooth functions (Giordano et al., 2021), utilizing a non-Archimedean ring of scalars $\widetilde{\mathbb{R}}_\rho$ containing infinitesimals and infinities. Function spaces are encoded via nets, sharp/Fermat topologies, and glued as sheaves over a site of glueable families of strongly internal sets with dynamic compatibility conditions. This framework accommodates singularities and robust nonlinear operations, extends Schwartz distributions, and enables differential calculus on both classical and nonstandard domains. The resultant topos contains diffeological/convenient spaces as full subcategories, supporting rigorous analysis of singular ODEs and PDEs.

4. Homotopical and Higher Structures

Smooth sets support rich homotopy theories when organized as simplicial presheaves. The model category $H$ localizes with respect to morphisms that become weak equivalences under the smooth singular complex functor

$S_e = \delta^* \circ \Delta_e^*$

where $$\Delta_e^n = \{(t^0, \dots, t^n) \in \mathbb{R}^{n+1} \mid \sum t^i = 1\}$$ are extended affine simplices. After localization ($\mathbb{R}$-localization), the smooth space functors become homotopy-invariant (Bunk, 2020).

Critical features:

• Quillen Equivalences: The localized model category of smooth spaces is Quillen equivalent to standard homotopy categories, confirming that smoothification refines geometric data without altering homotopy types.
• Fibrant Replacement and Concordance Sheaf: Mapping spaces are computed via "concordance sheaves," formalized as

$$^{p/i}F = R^{p/i}\Bigl(\delta^*(F^{\#\Delta_e})\Bigr)$$

with mapping spaces

$$\mathrm{Map}_{H^{pI}}(F,G) \cong \underline{H}\left(Q^pF, R(\delta^*(G^{\#\Delta_e}))\right)$$

providing explicit control and linking sheaf-theoretic and homotopy-theoretic constructions (Bunk, 2020).

5. Variational Calculus and Gauge Structures in Field Theory

In field theory, the topos of smooth sets provides a rigorous setting for spaces of sections, connections, and field configurations (Giotopoulos et al., 2023, Giotopoulos, 10 Apr 2025). Field spaces are encoded as sheaves: $$\mathcal{F}(R^k) = \{\varphi^k: \mathbb{R}^k \times M \to F \mid \pi \circ \varphi^k = \mathrm{pr}_2\}$$ and Lagrangians are smooth maps $L: J^\infty_M F \to \Omega^d(M)$, with actions $$S = \int_M L \circ j^\infty : \mathcal{F} \to \mathbb{R}$$ defined as morphisms in the topos. The infinite jet bundle is realized as a projective limit in the topos.

The tangent bundle of a smooth set $X$ is generally described as $T X = [D, X]$, internal hom with the infinitesimal disk as a probe. Jet geometry extends via local coordinates, and the bicomplex of differential forms splits horizontally/vertically: $$d = d_H + d_V, \quad d_H^2 = d_V^2 = 0, \quad d_H d_V + d_V d_H = 0$$ enabling rigorous definitions of Euler-Lagrange equations and conserved currents (Giotopoulos et al., 2023, Ibort et al., 23 Oct 2025).

Higher gauge fields and non-perturbative structures require encoding spaces as ∞-groupoids valued in Kan complexes or in sheaves of such, affording substantial categorical flexibility.

6. Analytical Testing: Arc-Smooth Maps and Function Space Properties

Arc-smoothness, tested via smoothness of composites with smooth curves, is categorically adequate for characterizing smooth, analytic, and ultradifferentiable functions on various closed sets, including those with Hölder boundaries or fat subanalytic sets (Rainer, 10 Mar 2025). These sets fulfill set-theoretic identities: $$\mathrm{AC}^\infty(X) = C^\infty(X), \quad \mathrm{AC}^\omega(X) = C^\omega(X)$$ which are bornological isomorphisms in their convenient locally convex topologies. Such identifications extend to vector-valued function spaces, with exponential laws

$$\mathrm{AC}^\infty(X_1, \mathrm{AC}^\infty(X_2, E)) \cong \mathrm{AC}^\infty(X_1 \times X_2, E)$$

holding in scalar, analytic, and ultradifferentiable Braun–Meise–Taylor classes under robust weight conditions. These exponential laws guarantee function space behavior suitable for internal manipulation in smooth topoi.

7. Fermionic and Supergeometric Extensions

The topos framework generalizes to sheaves on sites of super-Cartesian spaces $(\mathbb{R}^{k|q})$ for encoding fermionic fields, with each object awaiting parametrizations by even and odd coordinates. Fermionic field spaces, such as $$C_{\mathrm{ferm}}(R^{0|q}) = \{\psi^{0|q}: R^{0|q} \times M \to V_{\mathrm{odd}}\}$$, encode anticommutative properties by assigning higher "odd plots." The category of super smooth sets, $\mathrm{Sup} = \mathrm{Sh}(\mathrm{SupCartSp})$, is defined analogously to the bosonic case, and further generalization to thickened Cartesian spaces supports synthetic differential geometry and the fine structure of infinitesimal neighborhoods.

Non-perturbative gauge fields, higher moduli stacks, and smooth groupoids are captured as sheaves valued in Kan complexes, fully integrating homotopical and categorical data required for genuine gauge-theoretic field spaces (Giotopoulos, 10 Apr 2025).

Summary Table: Core Themes and Correspondences

Theme	Model/Key Idea	Reference
Smooth sets (basic)	Sheaves on Euclidean spaces	(Ibort et al., 23 Oct 2025, Bunk, 2020)
Algebraic models	C^∞-rings, topologies	(Borisov, 2011)
Generalized/singular	Internal sets, GSF, nets	(Giordano et al., 2014, Giordano et al., 2021)
Homotopy theory	Simplicial presheaves, Quillen	(Bunk, 2020)
Field theory	Jet bundles, bicomplex, tangent	(Giotopoulos et al., 2023, Giotopoulos, 10 Apr 2025)
Analytical testing	Arc-smooth maps, exponential laws	(Rainer, 10 Mar 2025)
Fermionic/supergeometry	Super-Cartesian sheaves	(Giotopoulos, 10 Apr 2025)

Concluding Perspective

The topos of smooth sets synthesizes differential geometry, algebraic analysis, and categorical topology, enabling precise and robust treatment of a variety of smooth spaces, function spaces, generalized and singular objects, infinite-dimensional geometries, and field-theoretic structures. Category-theoretic techniques such as sheaves, Kan extensions, limits, exponentials, and topological localizations underpin the entire framework, ensuring compatibility, composability, and the ability to carry out advanced analytical and geometric constructions. This paradigm not only clarifies foundational questions in modern geometry and mathematical physics but also equips practitioners with a unifying setting for the paper of smooth phenomena across domains.