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




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\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

          cohesive topos

          cohesive (∞,1)-topos

          cohesive homotopy type theory



          Presentation over a site

          Structures in a cohesive (,1)(\infty,1)-topos

          structures in a cohesive (∞,1)-topos

          Structures with infinitesimal cohesion

          infinitesimal cohesion?




          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.