Groupoids in categories with pretopology

TL;DR

This paper surveys the theory of groupoids and their morphisms within categories with pretopology, exploring assumptions, equivalences, and bicategory structures for a comprehensive understanding of their categorical properties.

Contribution

It provides a unified framework for groupoids, actions, and morphisms in categories with pretopology, including new insights into bicategory equivalences and generalizations of actors.

Findings

Both span-based and bibundle-based approaches yield equivalent bicategories.

Extra assumptions on pretopologies are necessary for the theory.

Generalized actors via bibundles form a new bicategory.

Abstract

We survey the general theory of groupoids, groupoid actions, groupoid principal bundles, and various kinds of morphisms between groupoids in the framework of categories with pretopology. We study extra assumptions on pretopologies that are needed for this theory. We check these extra assumptions in several categories with pretopologies. Functors between groupoids may be localised at equivalences in two ways. One uses spans of functors, the other bibundles (commuting actions) of groupoids. We show that both approaches give equivalent bicategories. Another type of groupoid morphisms, called actors, are closely related to functors between the categories of groupoid actions. We also generalise actors using bibundles, and show that this gives another bicategory of groupoids.