Topological Functors II: the Cartesian-closed category of C-maps

Some time ago, I gave an introduction to topological functors. They form a pretty brilliant categorical generalization of topological spaces. The point of today’s post is to give one particular example of the fact that you can somehow generalize some results on topological spaces to topological functors. I will concentrate on showing that a (pretty amazing) construction of certain Cartesian-closed full subcategories of Top, due to Martín Escardó, Jimmie Lawson, and Alex Simpson, generalizes pretty smoothly to a pretty large class of topological functors—the so-called well-fibered topological constructs. More precisely, I will concentrate on the first part of this construction, which builds a Cartesian-closed category Map_C out of a so-called strongly productive class C of objects of a category C that forms the domain of a topological construct with discrete terminal objects. Read the full post.