What is it like to be a lax double functor?

Abstract

I will attempt to impart the vibe that the models of double theories that are the basis of CatColab are certain kinds of families of categories, even though the definition looks like a generalization of the families of sets we know and love as presheaves. Furthermore, CatColab does (on a sufficiently bleeding-edge branch) contain families of sets that live over these families of categories in an appropriate way, which we call instances of models of double theories. (Deep breath.) Probably only two people know what those are, yet, so let me try to tell you, because they’re going to be important; it’s not all that bad, just new.