derived smooth geometry
Write for the category whose
For definiteness we write
In all these cases, the inclusion is an equivalence of categories: choosing an isomorphism from any of these objects to a Cartesian space amounts to choosing a basis of a vector space, a coordinate system.
For the second condition notice that since an ∞-cohesive site is in particular an ∞-local site we have that is a local (∞,1)-topos. As discussed there, this implies that it is a hypercomplete (∞,1)-topos. By the discussion at model structure on simplicial presheaves this means that it is presented by the Joyal-Jardine-model structure on simplicial sheaves . The claim then follows with the first two statements.
There is a canonical structure of a category with open maps on (…)
(Except that the pullback stability of the open maps holds only in the weaker sense of coverages).
In secton 2 of
is discussed as an example of a “cartesian differential category”.
There are various slight variations of the category (many of them equivalent) that one can consider without changing its basic properties as a category of test spaces for generalized smooth spaces. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see diffeological space.
in detal in section 5 of
and briefly mentioned in example 2) on p. 191 of
following the original article
With an eye towards Frölicher spaces the site is also considered in section 5 of