The Ehresmann-Schein-Nambooripad Theorem and its Successors

Abstract

The Ehremann--Schein--Nambooripad Theorem expresses the fundamental connection between the notions of inverse semigroups and inductive groupoids, which exists because these concepts provide two distinct approaches to the study of one-one partial transformations.  In the case of arbitrary partial transformations, the analogous two approaches are provided by restriction semigroups and inductive categories, the former being generalisations of inverse semigroups, and the latter of inductive groupoids.  There is indeed also a generalisation of the Ehremann--Schein--Nambooripad Theorem which encapsulates the connection between these two more general objects.  In this article, we will explore the origins of these theorems, and survey the basic theory surrounding them.