Homodular pseudofunctors and bicategories of modules

TL;DR

This paper explores the universal property of the bicategory of modules (distributors) in enriched category theory, clarifying its foundational role and completion properties within bicategorical structures.

Contribution

It formalizes the universal property of the bicategory of modules derived from enriched categories, providing a clearer foundation for their use in higher category theory.

Findings

Universal property of the bicategory of modules clarified.

Completion property with respect to lax colimits established.

Objective perspective on homological functors developed.

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.