nLab FormalCartSp

Idea

A site of formal Cartesian spaces.

Definition

Definition

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

$$\mathbb{R}^n \times \ell W \,,$$

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 $FormalCartSp^{op} \hookrightarrow SmoothAlg$ of smooth algebras on those of the form

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

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

$$\{ U_i \times \ell W \stackrel{p_i \times Id}{\to} U \times \ell W \}$$

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

This appears as Kock (5.1).

Definition

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

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

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

Definition

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

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

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Related concepts

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

$\,$

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)