Formalization of Synthetic Differential Geometry

• Formalization of SDG is a rigorous framework that introduces nilpotent infinitesimals via foundational systems where differential constructions are internal.
• Various approaches—including sheaf-topos, homotopy type theory, and categorical tangent structures—embed classical smooth geometry fully and faithfully.
• Machine-checked proofs in Lean and type-theoretic methods highlight SDG’s practical implications for synthetic tangent bundles, jets, and variational calculus.

Searching arXiv for the cited SDG formalization papers to ground the article in current and relevant research. Formalization of Synthetic Differential Geometry (SDG) is the effort to state infinitesimal differential geometry in rigorous foundational systems in which nilpotent infinitesimals exist, ordinary smooth geometry embeds faithfully, and differential-geometric constructions may be carried out internally. In current work this formalization appears in several forms: sheaf-topos models such as the Cahiers topos of formal smooth sets and the topos of thickened smooth sets, axiomatic developments in homotopy type theory, categorical reconstructions via tangent structures and Weil algebras, and machine-checked proofs in Lean. Across these settings, the recurrent primitives are a line object $R$, infinitesimal objects such as $D=\{\,d:R\mid d^2=0\,\}$ or more general spectra of Weil algebras, the Kock–Lawvere axiom, and microlinearity; these support synthetic tangent bundles, jets, differential operators, and variational constructions (Giotopoulos et al., 22 Jan 2026, Nishimura, 2016, Leung, 2016, Brasca et al., 18 Mar 2026).

1. Topos-theoretic models and site presentations

A central formalization strategy places SDG inside a Grothendieck topos built from smooth or thickened probes. In one such presentation, $\mathrm{CartSp}$ is the category of Euclidean spaces $\mathbb R^n$ and ordinary smooth maps, $\mathrm{FrmPnt}$ is the full subcategory of the opposite of commutative $\mathbb R$-algebras consisting of Weil algebras, and a formal Cartesian space is an object $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$. Equipping $\mathrm{CartSp}$ with the usual open-cover Grothendieck topology and $\mathrm{FrmCrtSp}$ with the topology generated by families $\{(U_i\times D\to U\times D)\}$, one obtains $D=\{\,d:R\mid d^2=0\,\}$0 and $D=\{\,d:R\mid d^2=0\,\}$1. The embedding $D=\{\,d:R\mid d^2=0\,\}$2 is fully faithful and preserves representables, and its essential image is the class of reduced objects characterized by the counit $D=\{\,d:R\mid d^2=0\,\}$3 being an isomorphism (Giotopoulos et al., 22 Jan 2026).

A closely related $D=\{\,d:R\mid d^2=0\,\}$4-algebraic presentation starts from thickened Cartesian probes. Here $D=\{\,d:R\mid d^2=0\,\}$5 has objects $D=\{\,d:R\mid d^2=0\,\}$6 for $D=\{\,d:R\mid d^2=0\,\}$7 any finite-dimensional nilpotent Weil algebra, and the resulting sheaf topos is $D=\{\,d:R\mid d^2=0\,\}$8, equivalently $D=\{\,d:R\mid d^2=0\,\}$9. The internal hom makes $\mathrm{CartSp}$0 Cartesian closed, and this setting is used as a fully $\mathrm{CartSp}$1-algebraic model of the Cahiers topos (Giotopoulos et al., 28 Dec 2025).

Menni’s construction modifies the usual well-adapted models by restricting the site to the full subcategory $\mathrm{CartSp}$2 of finitely generated $\mathrm{CartSp}$3-rings that are connected and pointed, thereby producing a pre-cohesive geometric morphism

$\mathrm{CartSp}$4

with four adjoints $\mathrm{CartSp}$5, where $\mathrm{CartSp}$6 computes connected components. A refinement to the subcategory $\mathrm{CartSp}$7 of $\mathrm{CartSp}$8-determined rings yields another pre-cohesive setting (Menni, 2024).

Formal setting	Site or probes	Structural feature
$\mathrm{CartSp}$9	formal Cartesian spaces $\mathbb R^n$0	$\mathbb R^n$1 is fully faithful; reduced objects are its essential image
$\mathbb R^n$2	thickened probes $\mathbb R^n$3	Cartesian closed; equivalent to Dubuc’s $\mathbb R^n$4-algebraic Cahiers topos
$\mathbb R^n$5, $\mathbb R^n$6	connected pointed or $\mathbb R^n$7-determined $\mathbb R^n$8-rings	pre-cohesive morphism to $\mathbb R^n$9; connected components via $\mathrm{FrmPnt}$0

These site descriptions are not merely model-building devices. They fix the exact ambient categories in which infinitesimal objects, exponentials, sheaf conditions, and embeddings of classical spaces are defined, and they determine what it means for a synthetic construction to recover a classical one.

2. Infinitesimals, Kock–Lawvere axioms, and microlinearity

The axiomatic core of SDG is the existence of infinitesimal objects and a principle that functions on them are polynomial of bounded degree. In Nishimura’s homotopy-type-theoretic formulation, $\mathrm{FrmPnt}$1 is postulated to be a set which is a $\mathrm{FrmPnt}$2-algebra, Weil algebras are finitely presented $\mathrm{FrmPnt}$3-algebras of the form $\mathrm{FrmPnt}$4 with nilpotent generators, and one sets $\mathrm{FrmPnt}$5 to be the type of $\mathrm{FrmPnt}$6-algebra maps $\mathrm{FrmPnt}$7. The homotopical generalized Kock–Lawvere axiom states that for every Weil algebra $\mathrm{FrmPnt}$8, the canonical $\mathrm{FrmPnt}$9-algebra map $\mathbb R$0 is an equivalence; in particular, the canonical map

$\mathbb R$1

is an equivalence of sets (Nishimura, 2016).

Microlinearity formalizes the idea that infinitesimal limit diagrams are seen correctly by a space. In the same HoTT framework, a type $\mathbb R$2 is microlinear if every finite diagram of infinitesimal types obtained as $\mathbb R$3 of a finite limit diagram of Weil algebras becomes a limit diagram after exponentiation by $\mathbb R$4. One expression of this is

$\mathbb R$5

This condition yields synthetic tangent fibers

$\mathbb R$6

an $\mathbb R$7-module structure on each $\mathbb R$8, differentials $\mathbb R$9, and the equivalence of three notions of vector field: a section $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$0, an infinitesimal flow $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$1, and a transformation $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$2 (Nishimura, 2016).

A different axiomatization, described by Menni as “radically synthetic,” starts from a space $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$3 with a unique point $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$4, forms the endomorphism monoid $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$5, and extracts

$$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$6

The Kock–Lawvere or line-type axiom then forces the additive structure on $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$7. If $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$8 is the subobject of invertible elements, bi-directionality is the condition $$U\times D\in \mathrm{FrmCrtSp}:=(\mathrm{CartSp}\times \mathrm{FrmPnt})^{op}$$9, and it induces a preorder $\mathrm{CartSp}$0 compatible with the rig structure (Menni, 2024).

In the Cahiers-style field-theoretic setting, the infinitesimal interval is

$\mathrm{CartSp}$1

and the synthetic tangent bundle is the internal mapping object

$\mathrm{CartSp}$2

This realizes tangent vectors as infinitesimal curves and is one of the standard synthetic consequences of the Kock–Lawvere principle (Giotopoulos et al., 28 Dec 2025).

3. Categorical reconstructions beyond the topos-internal viewpoint

Formalization of SDG is not restricted to internal topos logic. One influential categorical reconstruction starts from the notion of a tangent structure. A tangent structure on a category $\mathrm{CartSp}$3 consists of an endofunctor $\mathrm{CartSp}$4 together with natural transformations

$\mathrm{CartSp}$5

subject to additive-bundle, lift, flip, coherence, and equalizer axioms. Leung shows that giving such a tangent structure is equivalent to giving a strong monoidal functor

$\mathrm{CartSp}$6

that preserves foundational pullbacks and a distinguished equalizer. In this sense $\mathrm{CartSp}$7 is the free or initial tangent-structure object, and the passage from Weil algebras to endofunctors formalizes the relation between SDG and abstract tangent-structure theory (Leung, 2016).

Another externalization is Nishimura’s program for Frölicher spaces. A Frölicher space $\mathrm{CartSp}$8 is equipped with structure curves and structure functions, and the category $\mathrm{CartSp}$9 of Frölicher spaces is Cartesian closed. Each Weil algebra $\mathrm{FrmCrtSp}$0 defines a Weil prolongation $\mathrm{FrmCrtSp}$1, a space is microlinear if it preserves finite limit diagrams in the category of Weil algebras after applying $\mathrm{FrmCrtSp}$2, and a space is Weil-exponentiable if prolongation interacts correctly with exponentials. The full subcategory of microlinear and Weil-exponentiable Frölicher spaces is Cartesian closed, the tangent bundle is $\mathrm{FrmCrtSp}$3, and the vector fields $\mathrm{FrmCrtSp}$4 form a Lie algebra (Nishimura, 2010).

Burke extends the formalization further into internal category theory. In a well-adapted model of SDG with infinitesimals $\mathrm{FrmCrtSp}$5 and Kock–Lawvere isomorphisms

$\mathrm{FrmCrtSp}$6

he defines enriched mono-coreflective subcategories of internal categories, identifies one-object jet-categories with formal group laws, and proves a synthetic form of Lie’s second theorem. Under enriched path-connectedness and simply-connectedness hypotheses, the functor

$\mathrm{FrmCrtSp}$7

is fully faithful (Burke, 2016).

These reconstructions show that formalization of SDG can be phrased equivalently as topos-internal infinitesimal geometry, as categorical algebra of tangent structures, or as externalized Weil-functor geometry.

4. Jets, comonads, and field-theoretic formalization

A major modern strand of SDG formalization concerns infinite jets and the formal theory of PDEs. Khavkine and Schreiber work in a topos $\mathrm{FrmCrtSp}$8 equipped with a differential cohesion structure

$\mathrm{FrmCrtSp}$9

where $\{(U_i\times D\to U\times D)\}$0 is the full subcategory of reduced objects and the induced idempotent (co-)monads are reduction $\{(U_i\times D\to U\times D)\}$1, infinitesimal-shape $\{(U_i\times D\to U\times D)\}$2, and étale $\{(U_i\times D\to U\times D)\}$3. The formal disk bundle $\{(U_i\times D\to U\times D)\}$4 is defined as a pullback over the unit $\{(U_i\times D\to U\times D)\}$5, and the infinite jet functor over $\{(U_i\times D\to U\times D)\}$6 is the right base-change

$\{(U_i\times D\to U\times D)\}$7

This endofunctor on the slice topos $\{(U_i\times D\to U\times D)\}$8 carries a comonad structure $\{(U_i\times D\to U\times D)\}$9, its co-Kleisli category is the category of differential operators, and formally integrable PDEs are identified with the Eilenberg–Moore coalgebras for $D=\{\,d:R\mid d^2=0\,\}$00 (Khavkine et al., 2017).

A technical gap in this program concerned the classical definition of infinite jets as a projective limit of finite jets. For a surjective submersion $D=\{\,d:R\mid d^2=0\,\}$01 of finite-dimensional manifolds, the classical infinite jet bundle is

$D=\{\,d:R\mid d^2=0\,\}$02

in Fréchet manifolds, while its synthetic counterpart is the limit $D=\{\,d:R\mid d^2=0\,\}$03 in $D=\{\,d:R\mid d^2=0\,\}$04. “Synthetic Differential Jet Bundles are Reduced” proves that these coincide canonically: $D=\{\,d:R\mid d^2=0\,\}$05 The proof reduces to Cartesian spaces, shows directly that $D=\{\,d:R\mid d^2=0\,\}$06 is reduced by a plot-factorization argument using Hadamard’s lemma and a finite-dimensional embedding trick, and then deduces the general case by local triviality and preservation of fibered products (Giotopoulos et al., 22 Jan 2026).

This limit-preservation theorem stabilizes a broader field-theoretic formalization. In thickened smooth sets, the tower $D=\{\,d:R\mid d^2=0\,\}$07 gives

$D=\{\,d:R\mid d^2=0\,\}$08

and there is also a synthetic definition via infinitesimal tubular neighborhoods and sections over $D=\{\,d:R\mid d^2=0\,\}$09-infinitesimal neighborhoods. For bundles of manifolds, these two notions coincide. The same setting carries the horizontal/vertical splitting

$D=\{\,d:R\mid d^2=0\,\}$10

the variational bicomplex, local Lagrangians $D=\{\,d:R\mid d^2=0\,\}$11, a critical locus $D=\{\,d:R\mid d^2=0\,\}$12 defined as a pullback, and the identification $D=\{\,d:R\mid d^2=0\,\}$13 with classical Jacobi fields (Giotopoulos et al., 28 Dec 2025).

The significance of these results is foundational. Earlier work had observed that infinite jets and the PDE comonad become natural in formal smooth sets but tacitly assumed that forming infinite jets commuted with the embedding $D=\{\,d:R\mid d^2=0\,\}$14. The reducedness theorem closes that gap, and this supports a completely internal, coordinate-free, comonadic framework for variational calculus, Poisson structures on diffieties, and the geometry of field theories (Giotopoulos et al., 22 Jan 2026).

5. Type-theoretic and machine-checked formalization

Synthetic differential geometry has also been formalized directly in type theory. Nishimura’s first paper on SDG within homotopy type theory develops the subject synthetically from axioms rather than from a chosen topos model. The framework assumes $D=\{\,d:R\mid d^2=0\,\}$15 is a $D=\{\,d:R\mid d^2=0\,\}$16-algebra, defines infinitesimal types such as

$D=\{\,d:R\mid d^2=0\,\}$17

postulates the generalized Kock–Lawvere axiom for all Weil algebras, defines microlinearity, and then constructs tangent bundles, differentials, vector fields, and Lie brackets internally. Theorems include the $D=\{\,d:R\mid d^2=0\,\}$18-module structure on each tangent fiber $D=\{\,d:R\mid d^2=0\,\}$19, the chain rule $D=\{\,d:R\mid d^2=0\,\}$20, the equivalence of three presentations of vector fields, and the Jacobi identity proved via $D=\{\,d:R\mid d^2=0\,\}$21-diagrams (Nishimura, 2016).

The transition from axiomatic synthetic mathematics to proof assistants is made explicit in “Synthetic Differential Geometry in Lean.” That development works in Lean 4 and mathlib4, introduces only unique choice rather than classical choice, and formalizes infinitesimals, the Kock–Lawvere axiom, derivatives, and Taylor expansion. It defines the $D=\{\,d:R\mid d^2=0\,\}$22-th infinitesimal object $D=\{\,d:R\mid d^2=0\,\}$23, formalizes one-variable and higher-order Kock–Lawvere classes, defines the synthetic derivative by unique choice, and proves a multivariable Taylor theorem for $D=\{\,d:R\mid d^2=0\,\}$24 over infinitesimal neighborhoods $D=\{\,d:R\mid d^2=0\,\}$25: $D=\{\,d:R\mid d^2=0\,\}$26 The project reports that the only extra logical axiom is unique choice, that a custom fork of Lean core, Batteries, and mathlib4 was used to avoid transitive uses of Classical.choice, and that the entire project compiles in about $D=\{\,d:R\mid d^2=0\,\}$27 s on a standard machine with no Classical.choice or excluded-middle used anywhere in SDG/… (Brasca et al., 18 Mar 2026).

This line of work demonstrates that formalization of SDG is not limited to semantic models. It can also be expressed as an axiomatic type theory and then implemented constructively in a proof assistant.

6. Foundational significance, misconceptions, and current directions

A common misconception is that SDG formalization is a single foundational recipe. The literature instead exhibits several non-equivalent but overlapping programs: well-adapted smooth topoi, pre-cohesive variants, homotopy type theory, tangent structures, Frölicher-space externalizations, and mechanized formalization in Lean. Each fixes a different balance between semantic modeling, internal logic, categorical abstraction, and proof automation.

A second misconception is that synthetic methods displace classical differential geometry rather than recover it. Multiple formalizations emphasize full and faithful embeddings of classical objects. Every finite or countable Fréchet manifold defines a sheaf on $D=\{\,d:R\mid d^2=0\,\}$28, and the resulting functor $D=\{\,d:R\mid d^2=0\,\}$29 is fully faithful; composing with $D=\{\,d:R\mid d^2=0\,\}$30 embeds Fréchet manifolds into the Cahiers topos. In thickened smooth sets, manifolds with boundary and corners embed fully faithfully into $D=\{\,d:R\mid d^2=0\,\}$31, and their Weil bundles are recovered synthetically as $D=\{\,d:R\mid d^2=0\,\}$32 (Giotopoulos et al., 22 Jan 2026, Giotopoulos et al., 28 Dec 2025).

A third issue concerns logic and order. Several SDG settings explicitly rely on intuitionistic logic, and Menni’s work shows that even order structure can be derived internally from the connected components of the unit group of the rig $D=\{\,d:R\mid d^2=0\,\}$33. In the more spatial topos $D=\{\,d:R\mid d^2=0\,\}$34, the radically derived sub-rig $D=\{\,d:R\mid d^2=0\,\}$35 and the preorder $D=\{\,d:R\mid d^2=0\,\}$36 coincide exactly with the spatial preorder defined by factorization through $D=\{\,d:R\mid d^2=0\,\}$37 (Menni, 2024).

Current directions extend the formalization rather than merely apply it. The reducedness result for synthetic jet bundles is presented as completing the formal foundation for infinite jet bundles in SDG and, in higher SDG, as pointing toward extensions to $D=\{\,d:R\mid d^2=0\,\}$38-sheaves, higher jet bundles, homotopy-comonads, and a synthetic treatment of PDEs on higher stacks (Giotopoulos et al., 22 Jan 2026). Thickened smooth sets are proposed as synthetic foundations for local Lagrangian field theory in which infinitesimal spaces interact correctly with off-shell and on-shell spaces of fields, and in which the variational principle is recognized as the intersection of thickened smooth sets; a plausible implication is that synthetic infinitesimal neighborhoods may serve as a rigorous framework for perturbative restrictions around a field configuration (Giotopoulos et al., 28 Dec 2025).

Broader developments indicate how these foundations are being used. SDG has been employed to formulate a geometric theory of integration based on equivariant maps from infinitesimal $D=\{\,d:R\mid d^2=0\,\}$39-cuboids, producing Cartan, Leibniz–Fubini, and Nieuwentijdt forms together with a new symmetric differential operator $D=\{\,d:R\mid d^2=0\,\}$40 (Bár, 2024). Other papers use well-adapted topoi to study infinitesimal curvature, monads around a point, and singularity models in general relativity (Heller et al., 2016, Heller et al., 2017, Heller et al., 2016). This suggests that formalization of SDG now functions both as a foundational discipline and as an infrastructure for synthetic treatments of jets, integration, Lie theory, and field theory.