Synthetic differential geometry
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.
This appears for instance in Kock Reyes (1).
Define a structure of a site on FormalCartSp by declaring a covering family to be a family of the form
where is an open cover of the Cartesian space by Cartesian spaces .
This appears as Kock (5.1).
The Cahiers topos is the category of sheaves on this site:
This site of definition appears in Kock, Reyes. The original definition is due to Dubuc
Synthetic differential geometry
This is due to Dubuc.
Connectedness, locality and cohesion
Convenient vector spaces
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 to the sheaf given by
This result was announced in Kock. See the corrected proof in (KockReyes).
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.
The canonical inclusion functor induces an adjoint pair
where is given by precomposing a presheaf on with . The left adjoint 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.
For a smooth space, and for an infinitesimally thickened point, the morphisms
in are in natural bijection to equivalence classes of pairs of morphisms
consisting of a morphism in on the left and a morphism in 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 such that in the diagram
the left triangle commutes in and the right one in .
By general properties of left adjoints of functors of presheaves, is the left Kan extension of the presheaf along . By the Yoneda lemma and the coend formula for these (as discussed there), we have that the set of maps is naturally identified with
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 is disucssed at synthetic differential ∞-groupoid. It contains that Cahiers topos as the sub-(1,1)-topos of 0-truncated objects.
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)
Discussion of manifolds with boundary (and of their gluing along infinitesimal collars) in the Cahiers topos is in
The (∞,1)-topos analog of the Cahiers topos (synthetic differential ∞-groupoids) is discussed in section 3.4 of