Exact Completions and Small Sheaves

We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κ-ary exact categories” are a reflective sub-2-category of “κ-ary sites”, for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal κ-small coproducts, and a κ-ary site is a site whose covering sieves are generated by κ-small families and which satisfies a solution-set condition for finite limits relative to κ. In the unary case, this includes the exact completions of a regular category, of a category with (weak) finite limits, and of a category with a factorization system. When κ = ω, it includes the pretopos completion of a coherent category. And when κ = K is the size of the universe, it includes the category of sheaves on a small site, and the category of small presheaves on a locally small and finitely complete category. The K-ary exact completion of a large nontrivial site gives a well-behaved “category of small sheaves”. Along the way, we define a slightly generalized notion of “morphism of sites” and show that κ-ary sites are equivalent to a type of “enhanced allegory”. This enables us to construct the exact completion in two ways, which can be regarded as decategorifications of “representable profunctors” (i.e. entire functional relations) and “anafunctors”, respectively.