nLab
FormalCartSp

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

tangent cohesion

differential cohesion

graded differential 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& 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 }

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

A site of formal Cartesian spaces.

Definition

Definition

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).

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

Definition

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

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

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)
Created on November 4, 2014 07:59:24 by Urs Schreiber (185.26.182.34)