Crossed Modules of Inverse Semigroups as Internal Categories

Abstract

A classical result of Spencer and Brown (1976) shows that the category of crossed modules of groups is equivalent to the category of 2-groups. We extend this equivalence to inverse semigroups. We construct a notion of crossed module of inductive groupoids, equipped with a groupoid action. Inductive groupoids are equivalent to inverse semigroups via the Ehresmann–Schein–Nambooripad Theorem (Schein, 1965). We show that crossed modules of inductive groupoids are equivalent to a category of categories internal to inductive groupoids, and consider examples of such crossed modules arising in the work of Dokuchaev, Khrypchenko, and Makuta (2022). The intuitions developed in proving the groupoid case allow us to show that crossed modules of inverse semigroups, defined using a semigroup action and modified crossed module axioms, are equivalent to a category of categories internal to inverse semigroups. Finally, we show that the two categories of crossed modules we have defined are equivalent.