Examples of cartesian-closed model categories

Question

One of the main settings of my research are Cartesian-closed model categories. I would like to know as many interesting and/or important examples of such categories as possible. "Interesting" and "important" at your discretion.

Classic examples that I know of are certain localizations of categories of simplicial presheaves with an injective model structure (see Charles Rezk, A cartesian presentation of weak n-categories), in particular, sSet.

It also seems to me that some convenient categories of topological spaces are such (especially delta-generated spaces).

Answer 1

There are so many examples. Indeed, the injective model structure on simplicial presheaves, that you mention, is one. But, so are the various model structures of Bergner and Rezk on simplicial presheaves, capturing complete Segal spaces and $\Theta_n$-spaces. The cartesian property here is essential, because you want to be able to enrich in these categories to bootstrap up from $(\infty,n-1)$-categories to $(\infty,n)$-categories. Similarly, Dmitri Ara's category of $n$-quasi-categories is cartesian, for the same reason.

Topological spaces is indeed another example, with any of the following definitions:

1. compactly generated spaces
2. compactly generated weak Hausdorff spaces
3. Delta-generated spaces

Another example is Cat with the folk model structure. And, similarly, groupoids with the folk model structure.

You should also stipulate that the model structure and the Cartesian-closed structure are compatible, i.e., that the model structure satisfies the pushout product axiom. There are examples of Cartesian closed categories with a model strucutre that doesn't match.