Canonical topology on the set of functors between topological categories?

Question

Let $C, D$ be topological categroies. Ist there a canonical topology on the set of functors $Fun(C,D)$? Thank you

Answer 1

I assume that, by topological category, you mean a category enriched in topological spaces. Generically, what you expect is that the category of topological functors, or continuous functors, is itself naturally a topological category. This works with a complete and cocomplete symmetric monoidal closed category of spaces, of which there are various popular examples such as the weakly Hausdorff compactly generated spaces. So there is a space of natural transformations between two topological functors, but the functors themselves do not form a space, other than trivially. (You certainly can't do this with arbitrary functors, since these ignore the topological information.)

What you need for a space of functors is a space of objects in your "topological categories," where now you must switch to talking about categories internal to the category of spaces; in this case, internal category theory becomes a generalization of enriched category theory. The question of exponentials among internal categories is somewhat more subtle-one approach is in the paper Ronnie linked.