nLab
Cahiers topos

Contents

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (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 }

          </semantics></math></div>

          Models

          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

          Backround

          Definition

          Presentation over a site

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

          structures in a cohesive (∞,1)-topos

          Structures with infinitesimal cohesion

          infinitesimal cohesion?

          Models

          Contents

          Idea

          The Cahier topos is a cohesive topos that constitutes a well-adapted model for synthetic differential geometry (a “smooth topos”).

          It is the sheaf topos on the site FormalCartSp of infinitesimally thickened Cartesian spaces.

          Definition

          Definition

          Let FormalCartSp 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).

          Definition

          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).

          Definition

          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 79

          Properties

          Synthetic differential geometry

          Proposition

          The Cahiers topos is a well-adapted model for synthetic differential geometry.

          This is due to Dubuc 79.

          Connectedness, locality and cohesion

          Proposition

          The Cahiers topos is a cohesive topos. See synthetic differential infinity-groupoid for details.

          Convenient vector spaces

          Proposition

          The category of convenient vector spaces with smooth functions between them embeds as a full subcategory into the Cahiers topos.

          The embedding is given by sending a convenient vector space VV to the sheaf given by

          V: k×WC ( k,V)W. V : \mathbb{R}^k \times \ell W \mapsto C^\infty(\mathbb{R}^k, V) \otimes W \,.

          This result was announced in Kock. See the corrected proof in (KockReyes).

          Remark

          Together with prop. this means that the differential geometry on convenient vector spaces may be treated synthetically in the Cahiers topos.

          Synthetic tangent bundles of smooth spaces

          Synthetic tangent spaces

          We discuss here induced synthetic tangent spaces of smooth spaces in the sense of diffeological spaces and more general sheaves on the site of smooth manifolds after their canonical embedding into the Cahiers topos.

          Definition

          Write SmoothLocSmoothLoc for the category of smooth loci. Write

          CartSpSmoothLoc CartSp \hookrightarrow SmoothLoc

          for the full subcategory on the Cartesian spaces n\mathbb{R}^n (nn \in \mathbb{N}). Write

          InfThPointSmoothLoc InfThPoint \hookrightarrow SmoothLoc

          for the full subcategory on the infinitesimally thickened points, and write

          CartSp synthdiffSmoothLoc CartSp_{synthdiff} \hookrightarrow SmoothLoc

          for the full subcategory on those smooth loci which are the cartesian product of a Cartesian space n\mathbb{R}^n (nn \in \mathbb{N}) and an infinitesimally thickened point.

          We regard CartSp as a site by equipping it with the good open cover coverage. We regard InfThPointInfThPoint as equipped with the trivial coverage and CartSp synthdiffCartSp_{synthdiff} as equipped with the induced product coverage.

          The sheaf topos Sh(CartSp)Sh(CartSp) is that of smooth spaces. The sheaf topos Sh(CartSp synthdiff)Sh(CartSp_{synthdiff}) is the Cahier topos.

          Example

          We write

          D([ϵ]/(ϵ 2))InfThPtCartSp synthdiff D \coloneqq \ell(\mathbb{R}[\epsilon]/(\epsilon^2)) \in InfThPt \hookrightarrow CartSp_{synthdiff}

          for the infinitesimal interval, the smooth locus dual to the smooth algebraof dual numbers”.

          Definition

          For XSh(CartSp synthdiff)X \in Sh(CartSp_{synthdiff}) any object in the Cahier topos, its synthetic tangent bundle in the sense of synthetic differential geometry is the internal hom space X DX^D, equipped with the projection map

          X(*D):X DX. X(\ast \to D) \colon X^D \to X \,.
          Proposition

          The canonical inclusion functor i:CartSpCartSp synthdiffi \colon CartSp \to CartSp_{synthdiff} induces an adjoint pair

          Sh(CartSp)i *i !Sh(CartSp synthdiff) Sh(CartSp) \stackrel{\overset{i_!}{\to}}{\underset{i^\ast}{\leftarrow}} Sh(CartSp_{synthdiff})

          where i *i^\ast is given by precomposing a presheaf on CartSp synthdiffCartSp_{synthdiff} with ii. The left adjoint i !i_! has the interpretation of the inclusion of smooth spaces as reduced objects in the Cahiers topos.

          This is discussed in more detail at synthetic differential infinity-groupoid.

          Proposition

          For XSh(CartSp)X \in Sh(CartSp) a smooth space, and for (W)InfThPoint\ell(W) \in InfThPoint an infinitesimally thickened point, the morphisms

          (W)i !X \ell(W) \to i_! X

          in Sh(CartSp synthdiff)Sh(CartSp_{synthdiff}) are in natural bijection to equivalence classes of pairs of morphisms

          (W) nX \ell(W) \to \mathbb{R}^n \to X

          consisting of a morphism in CartSp synthCartSp_{synth} on the left and a morphism in Sh(CartSp)Sh(CartSp) on the right (which live in different categories and hence are not composable, but usefully written in juxtaposition anyway). The equivalence relation relates two such pairs if there is a smooth function ϕ: n n\phi \colon \mathbb{R}^n \to \mathbb{R}^{n'} such that in the diagram

          n (W) ϕ X n \array{ && \mathbb{R}^n \\ & \nearrow & & \searrow \\ \ell(W) && \downarrow^{\mathrlap{\phi}} && X \\ & \searrow && \nearrow \\ && \mathbb{R}^{n'} }

          the left triangle commutes in CartSp synthdiffCartSp_{synthdiff} and the right one in Sh(CartSp)Sh(CartSp).

          Proof

          By general properties of left adjoints of functors of presheaves, i !Xi_! X is the left Kan extension of the presheaf XX along ii. By the Yoneda lemma and the coend formula for these (as discussed there), we have that the set of maps (W)i !X\ell(W) \to i_! X is naturally identified with

          (i !X)((W))=(Lan iX)((W))= nCartSpHom CartSp synthdiff((W), n)×X( n). (i_! X)(\ell(W)) = (Lan_i X)(\ell(W)) = \int^{\mathbb{R}^n \in CartSp} Hom_{CartSp_{synthdiff}}(\ell(W), \mathbb{R}^n) \times X(\mathbb{R}^n) \,.

          Unwinding the definition of this coend as a coequalizer yields the above description of equivalence classes.

          Variants and generalizations

          References

          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)

          It appears briefly mentioned in example 2) on p. 191 of the standard textbook

          With an eye towards Frölicher spaces the site is also considered in section 5 of

          • Hirokazu Nishimura, Beyond the Regnant Philosophy of Manifolds (arXiv:0912.0827)

          The (∞,1)-topos analog of the Cahiers topos (synthetic differential ∞-groupoids) is discussed in section 3.4 of

          Last revised on March 13, 2018 at 05:35:54. See the history of this page for a list of all contributions to it.