Exact completions and small sheaves

TL;DR

This paper establishes a unifying theorem for various notions of exact completion, demonstrating that k-ary exact categories form a reflective sub-2-category of k-ary sites, encompassing many known constructions.

Contribution

It introduces a general theorem linking k-ary exact categories and k-ary sites, unifying multiple existing concepts of exact completion under a common framework.

Findings

Proves k-ary exact categories are a reflective sub-2-category of k-ary sites.

Includes various known exact completions as special cases.

Defines a generalized notion of morphism of sites and relates k-ary sites to enhanced allegories.

Abstract

We prove a general theorem which includes most notions of "exact completion". The theorem is that "k-ary exact categories" are a reflective sub-2-category of "k-ary sites", for any regular cardinal k. A k-ary exact category is an exact category with disjoint and universal k-small coproducts, and a k-ary site is a site whose covering sieves are generated by k-small families and which satisfies a weak size condition. For different values of k, this includes the exact completions of a regular category or a category with (weak) finite limits; the pretopos completion of a coherent category; and the category of sheaves on a small site. For a large site with k the size of the universe, it gives a well-behaved "category of small sheaves". Along the way, we define a slightly generalized notion of "morphism of sites", and show that k-ary sites are equivalent to a type of "enhanced allegory".