A -ary site is a site whose covering sieves are determined by -small covering families, and which has a very weak sort of finite limits. These conditions get weaker as gets larger, until when is the size of the universe, every small site is -ary.
-ary sites are a very general (perhaps the most general) appropriate input for -ary exact completion.
Let be an arity class.
A site is weakly -ary if for any covering sieve of an object in , there exists a -small family in such that (1) each , and (2) the sieve generated by is a covering sieve.
This definition can also be rephrased purely in terms of the covering families; see (Shulman).
A -ary site is a weakly -ary site which has all finite local -prelimits (i.e. whenever is a finite category).
Any limit is, in particular, a local -prelimit. Thus, any weakly -ary site with finite limits is -ary.
If the class of all cones over is -ary, then it is a local -prelimit. Thus, any -small and weakly -ary site is -ary. In particular, any small site is an infinitary site.
Generalizing the previous two examples, the class of all -small and effective-epic families on a κ-ary regular category (including a κ-ary exact category) is a -ary topology. This is called its -canonical topology.
The Zariski topology on is finitary.
contains, as a full reflective sub-2-category, the 2-category of κ-ary exact categories with their -canonical topologies. The reflector is called exact completion. When is the size of the universe, this reflector applied to a small (hence infinitary) site constructs its topos of sheaves.