MSGSS
Crossed Modules of Inverse Semigroups as Internal Categories
A crossed module is a group homomorphism $\partial:A \rightarrow B$ together with a group action of $B$ on $A$ that satisfies axioms relating the action and the homomorphism. Crossed modules were first discovered in algebraic topology in the 1940s (Whitehead, 1941, 1946), but they also appear naturally in group theory and cohomology theory (Brown, 1982). A classical result of Spencer and Brown (1976) shows that the category of crossed modules of groups is equivalent to the category of categories internal to groups, i.e., 2-groups. This equivalence enabled results such as the 2-dimensional Seifert–van Kampen theorem (Brown and Higgens, 1978) and is one of the earliest uses of higher-dimensional algebra.
In this thesis we extend the Spencer–Brown equivalence to the context of inverse semigroups, a generalization of groups that model partial symmetries. 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 class 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 class of categories internal to inverse semigroups. Finally, we show that the two categories of crossed modules we have defined are equivalent.