A 2-category equipped with a coverage is called a 2-site.
If is a regular 2-category, then the collection of all singleton families , where is eso, forms a coverage called the regular coverage.
Likewise, if is a coherent 2-category, the collection of all finite jointly-eso families forms a coverage called the coherent coverage.
On , the canonical coverage consists of all families that are jointly essentially surjective on objects.
A pre-Grothendieck coverage on a 2-category is a coverage satisfying the following additional conditions:
If is an equivalence, then the one-element family is a covering family.
If is a covering family and for each , so is , then is also a covering family.
Now, a sieve on an object is defined to be a functor with a transformation which is objectwise fully faithful (equivalently, it is a fully faithful morphism in ). Equivalently, it may be defined as a subcategory of the slice 2-category which is closed under precomposition with all morphisms of .
Every family generates a sieve by defining to be the full subcategory of on those such that for some and some . The following observation is due to StreetCBS.
A 2-presheaf is a 2-sheaf for a covering family if and only if
is an equivalence, where is the sieve on generated by .
Therefore, just as in the 1-categorical case, it is natural to restrict attention to covering sieves. We define a Grothendieck coverage on a 2-category to consist of, for each object , a collection of sieves on called covering sieves, such that
If is a covering sieve on and is any morphism, then is a covering sieve on .
For each the sieve consisting of all morphisms into (the sieve generated by the singleton family ) is a covering sieve.
If is a covering sieve on and is an arbitrary sieve on such that for each in , is a covering sieve on , then is also a covering sieve on .
Here if is a sieve on and is a morphism, denotes the sieve on consisting of all morphisms into such that factors, up to isomorphism, through some morphism in .
As in the 1-categorical case, one can then show that every coverage generates a unique Grothendieck coverage having the same 2-sheaves.
If is a 1-category regarded as a 2-category with only identity 2-morphisms, then a coverage (pretopology, topology) on reduces to the usual notion of coverage, Grothendieck pretopology, or Grothendieck topology.
Strict 2-sites were considered in
Bicategorical 2-sites in
See also StreetCBS.
More discussion is in