Internal categories, anafunctors and localisations

TL;DR

This paper reviews the theory of anafunctors and demonstrates how they can be used to form bicategorical localisations of internal categories and groupoids at weak equivalences, unifying various proofs with minimal assumptions.

Contribution

It shows that anafunctors can be used to localise 2-categories of internal categories and groupoids at weak equivalences, simplifying and unifying existing proofs.

Findings

Bicategorical localisation can be achieved using anafunctors.

The approach applies to subcanonical sites with minimal assumptions.

It unifies multiple proofs in the literature.

Abstract

In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2-categories of internal categories or groupoids at weak equivalences using anafunctors as 1-arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on S.