Homodular pseudofunctors and bicategories of modules
Ross Street
TL;DR
The paper analyzes the universal property of the Bénabou bicategory of distributors (modules) in the enriched bicategory setting, by developing the notion of homodular pseudofunctors and studying coslices, collages, and cofibrations as core tools. It identifies the inclusion $(-)_*:{\mathscr V}\text{-Cat}\to{\mathscr V}\text{-Mod}$ as a universal homodular pseudofunctor and extends this universality to iterative cosmoi, showing that the subbicategory of maps ${\mathscr M}_*$ inherits similar universal properties. It then constructs the Int framework ${\mathscr V}\text{iMod}$, giving an autonomous monoidal bicategory of modules and establishing strong monoidal behavior of the inclusion, thereby linking module theory with monoidal bicategory structures. The results clarify how the module construction interacts with coproducts, pushouts, and inversions, and provide a robust foundation for enriched higher-categorical algebra. This yields a coherent blueprint for extending modules across enriched categories, with explicit universal properties and monoidal enhancements.
Abstract
The universal property for the Bénabou 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 $\CW$ into the bicategory $\CW\text{-}\mathrm{Mod}$ of $\CW$-enriched categories and modules between them does have a completion property with respect to freely adjoining lax colimits (collages); see \cite{85, CKW}. Here we are interested in the universal property of the construction of $\CW\text{-}\mathrm{Mod}$ from $\CW\text{-}\mathrm{Cat}$. What we have in mind is an objective version of the notion of {\em homological functor} used by André Joyal in 1985.