Presheaves on lax double functors; or, Instances of models of double theories

Abstract

We introduce a notion of (co)presheaf on a lax double functor $X$, which we generally call an instance. In the terminology of double-categorical logic, a lax double functor valued in sets, possibly preserving finite products, is called a model of a double (Lawvere) theory. By varying the double theory, we uniformly define a well-behaved notion of instances of categories, profunctors, monads, monoidal categories, multicategories, and more, and we recover for instance the multifunctors into the category of sets in the last example. We show that instances of $X$ can be described either in terms of modules from the terminal model $I$ to $X,$ satisfying an additional condition on triviality of the left action, or as loose natural transformations from $I$ to $X.$ We propose a notion of discrete opfibration between models of a double theory, establish a comprehensive factorization system, and prove an elements correspondence giving an equivalence between the category of instances of and the category of discrete opfibrations over a model $X.$ We describe properties of the resulting categories of instances, relying on a "collage" construction which we characterize as a lax colimit of a model of a double theory. An appendix gives a detailed treatment of certain morphisms of lax functors relevant also for bicategory theory: (loose) transformations versus modules and modifications versus modulations.