nLab Cahiers topos

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

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

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 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 $V$ to the sheaf given by

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

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 $SmoothLoc$ for the category of smooth loci. Write

$$CartSp \hookrightarrow SmoothLoc$$

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

$$InfThPoint \hookrightarrow SmoothLoc$$

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

$$CartSp_{synthdiff} \hookrightarrow SmoothLoc$$

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

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

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

Example

We write

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

for the infinitesimal interval, the smooth locus dual to the smooth algebra “of dual numbers”.

Definition

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

$$X(\ast \to D) \colon X^D \to X \,.$$

Proposition

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

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

where $i^\ast$ is given by precomposing a presheaf on $CartSp_{synthdiff}$ with $i$. The left adjoint $i_!$ 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 $X \in Sh(CartSp)$ a smooth space, and for $\ell(W) \in InfThPoint$ an infinitesimally thickened point, the morphisms

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

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

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

consisting of a morphism in $CartSp_{synth}$ on the left and a morphism in $Sh(CartSp)$ 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 $\phi \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ such that in the diagram

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

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

Proof

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

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

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

Variants and generalizations

• When restricting the site of infinitesimally thickened Cartesian spaces to that of plain Cartesian spaces one obtaines the topos discussed at smooth space. This is still a cohesive topos, but no longer a model for synthetic differential geometry.

• The (∞,1)-sheaf (∞,1)-topos over $CartSp_{th}$ is disucssed at synthetic differential ∞-groupoid. It contains that Cahiers topos as the sub-(1,1)-topos of 0-truncated objects.

• Dubuc topos

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 3 (1979) 231-279 [numdam:CTGDC_1979__20_3_231_0]

and got its name from this journal publication. The definition appears in §4.1 & Thm. 4.10 there, the latter asserting that this topos is a “well-adapted model” for synthetic differential geometry.

A review discussion is in of

• Anders Kock, Section 5 of Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 1 (1986) 3-17 [numdam:CTGDC_1986__27_1_3_0]

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, 28 2 (1987) 99-110 [numdam:CTGDC_1987__28_2_99_0]

where it is first observed (but apparently without proof) that ThCartSp is still a site of definition for the Cahiers topos (cf. formal smooth set).

With its original site of all formal smooth manifolds, the Cahiers topos also appears briefly mentioned in:

• Anders Kock, ex. 2 on p. 191 of: Synthetic Differential Geometry, Cambridge University Press (1981, 2006) [pdf, doi:10.1017/CBO9780511550812]

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

• Urs Schreiber, differential cohomology in a cohesive topos