# Compact inverse categories

## TL;DR

This paper characterizes compact inverse categories as semilattices of compact groupoids, extending classical structure theorems for inverse monoids and categories with a focus on compactness and simplicity.

## Contribution

It provides a new structural description of compact inverse categories as semilattices of compact groupoids, simplifying their understanding and classification.

## Key findings

- One-object compact inverse categories are exactly commutative inverse monoids.
- Compact groupoids are characterized by 3-cocycles as per Baez-Lauda.
- The structure theorem extends classical results to the compact setting.

## Abstract

The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Jarek is that commutative inverse monoids become semilattices of abelian groups. It has also been categorified by DeWolf-Pronk to a structure theorem for inverse categories as locally complete inductive groupoids. We show that in the case of compact inverse categories, this takes the particularly nice form of a semilattice of compact groupoids. Moreover, one-object compact inverse categories are exactly commutative inverse monoids. Compact groupoids, in turn, are determined in particularly simple terms of 3-cocycles by Baez-Lauda.