Title: Discrete Double Fibrations
Abstract: Discrete fibrations over a small category correspond to presheaves on that small category by a category of elements construction. R. Paré proposes that certain lax, span-valued double functors serve as the double categorical analogue of ordinary presheaves and gives an associated category of elements construction. The question thus arises as to whether there is a corresponding notion of “discrete double fibration” and what kind of equivalence between these and lax, span-valued double functors can be obtained. In this talk, we shall review some versions of known elements constructions, see how the double category of elements fits into this pattern, study its fibration properties and see how these lead to a definition.