# nLab Cahiers topos

Contents

### Context

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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

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

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 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 algebraof 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.

## References

The Cahiers topos was introduced in

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 of

and with a corrected definition of the site of definition in

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:

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 October 22, 2023 at 04:16:31. See the history of this page for a list of all contributions to it.