Homodular pseudofunctors and bicategories of modules

Abstract

The universal property for the B\'enabou bicategory of distributors (although we call them "modules") presented here is somewhat implicitly spread over a series of papers and yet, to my knowledge, does not appear in print. The inclusion of a bicategory $\mathscr{W}$ into the bicategory $\mathscr{W}\text{-}\mathrm{Mod}$ of $\mathscr{W}$-enriched categories and modules between them does have a completion property with respect to freely adjoining lax colimits (collages). Here we are interested in the universal property of the construction of $\mathscr{W}\text{-}\mathrm{Mod}$ from $\mathscr{W}\text{-}\mathrm{Cat}$. What we have in mind is an objective version of the notion of {\em homological functor} used by Andr\'e Joyal in 1985.