Cahiers Topos of Formal Smooth Sets

• Cahiers Topos is a Grothendieck topos that formalizes smooth geometry by using formal smooth sets and infinitesimal objects.
• Its foundation is built on finitely presented C∞-rings and Weil algebras, extending classical smooth manifolds via fully faithful embeddings.
• The topos internalizes differential tools like tangent and jet bundles, providing a categorical approach to synthetic differential geometry and variational principles.

The Cahiers topos, also referred to as the topos of formal smooth sets, is a Grothendieck topos designed to serve as a synthetic foundation for smooth geometry, incorporating infinitesimal objects and supporting models of synthetic differential geometry (SDG). It extends the category of smooth manifolds by embedding them fully faithfully as reduced objects, while its structure underlies the development of jet bundles, tangent bundles, and differential equations in a categorically internal setting. The formalism of the Cahiers topos makes possible rigorous manipulations of infinitesimals and microlinearity axioms, recasting classical constructions through internal hom and sheaf-theoretic mechanisms (Berni et al., 2018, Giotopoulos et al., 28 Dec 2025, Giotopoulos et al., 22 Jan 2026).

1. Algebraic and Site-Theoretic Foundations

The construction of the Cahiers topos begins with the algebraic formalism of $\mathcal{C}^{\infty}$-rings, generalizing the concept of smooth functions $C^\infty(\mathbb{R}^n)$ to arbitrary sets of generators. The core category $C^\infty \mathrm{Rng}_{fp}$ consists of finitely presented $\mathcal{C}^{\infty}$-rings, i.e., rings of the form $A \simeq C^\infty(\mathbb{R}^n)/I$ where $I$ is a finitely generated ideal. The dual algebraic perspective introduces Weil algebras, finite-dimensional, local Artinian $R$-algebras $W \cong R \oplus V$ with nilpotent ideal $V$. These induce formal spectra $D = \mathrm{Spec}(W)$ representing infinitesimal disk objects.

A central construct is the category of formal Cartesian spaces, whose objects are tensor products $C^\infty(\mathbb{R}^k) \otimes W$ for $W$ a Weil algebra, and whose morphisms are algebra maps consistent with the $\mathcal{C}^{\infty}$-structure. The Grothendieck site $\mathrm{FrmCrtSp}$ is equipped with a coverage by families $\{ U_i \times D \hookrightarrow U \times D \}$ for differentiably good open covers $\{U_i \to U\}$.

2. The Definition and Structure of the Cahiers Topos

The Cahiers topos is defined as the category of sheaves on the site $\mathrm{FrmCrtSp}$ (or equivalently $\mathrm{ThCartSp}$), written $$\mathrm{FrmSmthSets} = \mathrm{Sh}(\mathrm{FrmCrtSp})$$ or $\FormalSmooth = \Sh(\ThCartSp)$. Objects of this topos, called formal smooth sets or infinitesimally thickened smooth sets, satisfy the descent condition relative to the specified coverage.

Sheaves are subcanonical: every representable $y(\mathbb{R}^k \times D)$ is a sheaf. There is a criterion ("Petit criterion" [Editor’s term]) that a presheaf $F$ is a sheaf if, for every fixed infinitesimal point $D$, the assignment $\mathbb{R}^k \mapsto F(\mathbb{R}^k \times D)$ is a sheaf on $\mathbb{R}^k$.

Fundamental categorical properties include the existence of all small limits and colimits, exponentials, a subobject classifier $\Omega$, and power objects $\mathcal{P}X$. The Yoneda embedding is fully faithful, embedding smooth manifolds (with corners) through $C^\infty(-): \mathrm{Man}^{\mathrm{cor}} \to \CAlg_R^{op}$ and $y(M)(\mathbb{R}^k \times D) \simeq \mathrm{Hom}_{\CAlg_R}(C^\infty(M), C^\infty(\mathbb{R}^k)\otimes W)$.

3. Internal Geometry: Tangent and Jet Bundles

The Cahiers topos internalizes classical constructions of differential geometry. The tangent bundle of any object $X$ is defined as the exponential $T X = X^D$, recovering the classical tangent bundle for manifolds, $T M \simeq [D, M] \simeq y(TM)$, where $D = \mathrm{Spec}(R[\epsilon]/\epsilon^2)$. The bundle projection and fiberwise $R$-module structure are induced internally by the map $\iota_0: * \to D$ and scaling.

Key to SDG is the Kock–Lawvere axiom: internally, $\forall x:D. x^2=0$ and $$\forall f:R \to R, \forall x:D, f(x) = f(0) + x f'(0)$$. Synthetic microlinearity is ensured, as certain infinitesimal cocone diagrams are mapped to pullback diagrams under $\Hom(-,X)$, giving the tangent bundle its vector bundle structure.

The apparatus of jet bundles is realized as projective limits in the topos. Infinite jet bundles $J^\infty_M F$ are recovered as limits of finite jets inside $\FormalSmooth$: $J^\infty_MF \simeq \lim_n y(J^n_MF)$, recognizing sections over infinitesimal neighborhoods. The transfer to the Cahiers topos preserves projective limits defining infinite jet bundles, as proven in (Giotopoulos et al., 22 Jan 2026).

4. Classifying Toposes for Smooth Algebraic Theories

Beyond the core Cahiers topos, refinements encompass classifying toposes for local $\mathcal{C}^{\infty}$-rings and von Neumann regular $\mathcal{C}^{\infty}$-rings. The smooth Zariski topos $$Z^\infty = \mathrm{Sh}(C^\infty\mathrm{Rng}_{fp}^{op}, J_{\mathrm{Zar}})$$ classifies local objects, where covers correspond to families $\{A \to A[a_i^{-1}]\}$ with $\langle a_1,\dots,a_n \rangle = A$. Sheaf conditions enforce geometric-logic locality.

For von Neumann regular objects, one defines the category $vN\,\mathrm{Rng}_{fp}$ and its presheaf topos $\mathrm{Set}^{vN\,\mathrm{Rng}_{fp}^{op}}$, classifying such ring objects via the universal property: $$\mathrm{Geom}(\mathcal{E}, \,\mathrm{Set}^{vN\,\mathrm{Rng}_{fp}^{op}}) \simeq \{$$vN–regular $\mathcal{C}^{\infty}$-rings in $\mathcal{E}\}$ (Berni et al., 2018).

5. Embedding and Reduction: Ordinary Geometry as a Subcategory

The embedding $$i_1: \mathrm{SmthSets} = \mathrm{Sh}(\mathrm{CartSp}) \hookrightarrow \mathrm{Sh}(\mathrm{FrmCrtSp})$$ demonstrates that ordinary smooth sets (manifolds and their sheaves) are a reflective and coreflective subcategory of the Cahiers topos, with reduced objects defined via the reduction-reflection unit $\varepsilon: X \to \mathrm{reduced}(X)$. This inclusion is fully faithful and interacts compatibly with the limit- and colimit-preserving structure of the topos. Fréchet-smooth manifolds embed through this mechanism (Giotopoulos et al., 22 Jan 2026).

6. Synthetic Differential Geometry and Field-Theoretic Applications

The Cahiers topos is a model for SDG, with infinitesimal thickening and microlinearity required for rigorous field-theoretic formulations. In the context of local Lagrangian field theory, infinitesimal spaces exist and interact appropriately with infinite jet bundles and spaces of fields. The variational principle for Lagrangian theories is realized as intersections within the topos, and perturbative field theory corresponds to restriction to infinitesimal neighborhoods (Giotopoulos et al., 28 Dec 2025).

The comonadic construction of jet bundles utilizes the adjunctions from infinitesimal projection $\pi: \Sigma^D \to \Sigma$, giving rise to a cofree coalgebra structure on $J^\infty_\Sigma(E) = \pi_*\pi^* E$, with counit and comultiplication internal to the topos. This framework recovers Vinogradov's category of "diffieties," substantiating SDG approaches to PDEs within the categorical environment.

7. Universal Properties and Significance

The Cahiers topos exhibits universal properties as a classifying topos for $\mathcal{C}^{\infty}$-rings (and refinements), with equivalences of categories reflecting the classification of ring-objects in any Grothendieck topos. Its subcanonical and microlinear structure ensures the compatibility of internal and external differential geometry, rigorously anchoring infinitesimal reasoning.

A plausible implication is that the Cahiers topos provides a unifying setting for advancements in synthetic approaches to geometric analysis, field theory, and the formalization of variational principles, embedding classical and modern smooth geometry within a single topos-theoretic architecture (Berni et al., 2018, Giotopoulos et al., 28 Dec 2025, Giotopoulos et al., 22 Jan 2026).