Synthetic differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Cohesive toposes



A site of formal Cartesian spaces.



Let FormalCartSpFormalCartSp (or FormalCartSpFormalCartSp) be the full subcategory of the category of smooth loci on those of the form

n×W, \mathbb{R}^n \times \ell W \,,

consisting of a product of a Cartesian space with an infinitesimally thickened point, i.e. a formal dual of a Weil algebra .

Dually, the opposite category is the full subcategory FormalCartSp opSmoothAlgFormalCartSp^{op} \hookrightarrow SmoothAlg of smooth algebras on those of the form

C ( k×W)=C ( k)W. C^\infty( \mathbb{R}^k \times \ell W) = C^\infty(\mathbb{R}^k) \otimes W \,.

This appears for instance in Kock Reyes (1).


Define a structure of a site on FormalCartSp by declaring a covering family to be a family of the form

{U i×Wp i×IdU×W} \{ U_i \times \ell W \stackrel{p_i \times Id}{\to} U \times \ell W \}

where {U ip iU}\{U_i \stackrel{p_i}{\to} U\} is an open cover of the Cartesian space UU by Cartesian spaces U iU_i.

This appears as Kock (5.1).


The Cahiers topos 𝒞𝒯\mathcal{CT} is the category of sheaves on this site:

𝒞𝒯:=Sh(FormalCartSp). \mathcal{CT} := Sh(FormalCartSp) \,.

This site of definition appears in Kock, Reyes. The original definition is due to Dubuc


The (∞,1)-topos of (∞,1)-sheaves over FormalCartSpFormalCartSp is that of formal smooth ∞-groupoids

FormSmoothGrpdSh (FormalCartSp). FormSmooth\infty Grpd \coloneqq Sh_\infty(FormalCartSp) \,.


geometries of physics

A\phantom{A}(higher) geometryA\phantom{A}A\phantom{A}siteA\phantom{A}A\phantom{A}sheaf toposA\phantom{A}A\phantom{A}∞-sheaf ∞-toposA\phantom{A}
A\phantom{A}discrete geometryA\phantom{A}A\phantom{A}PointA\phantom{A}A\phantom{A}SetA\phantom{A}A\phantom{A}Discrete∞GrpdA\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}CartSpA\phantom{A}A\phantom{A}SmoothSetA\phantom{A}A\phantom{A}Smooth∞GrpdA\phantom{A}
A\phantom{A}formal geometryA\phantom{A}A\phantom{A}FormalCartSpA\phantom{A}A\phantom{A}FormalSmoothSetA\phantom{A}A\phantom{A}FormalSmooth∞GrpdA\phantom{A}



The Cahiers topos was introduced in

  • Eduardo Dubuc, Sur les modèles de la géométrie différentielle synthétique Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no. 3 (1979), p. 231-279 (numdam).

and got its name from this journal publication. The definition appears in theorem 4.10 there, which asserts that it is a well-adapted model for synthetic differential geometry.

A review discussion is in section 5 of

  • Anders Kock, Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 (numdam)

and with a corrected definition of the site of definition in

  • Anders Kock, Gonzalo Reyes, Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 (numdam)

Last revised on June 25, 2018 at 09:05:52. See the history of this page for a list of all contributions to it.