A 2-sheaf is a higher sheaf of categories. More restrictive than this is a higher sheaf with values in groupoids, which would be a (2,1)-sheaf. Both these notions are often referred to as stack, or sometimes “stack of groupoids” and “stack of categories” for definiteness. But moreover, traditionally a stack (in either flavor) is considered only over a 1-site, whereas it makes sense to consider (2,1)-sheaves more generally over (2,1)-sites and 2-sheaves over 2-sites.
Therefore, saying “2-sheaf” serves to indicate the full generality of the notion of higher sheaves in 2-category theory, as opposed to various special cases of this general notion which have traditionally been considered.
We also have the double comma objects with projections , , and .
Now, a functor is called a 2-presheaf. It is 1-separated if
It is 2-separated if it is 1-separated and
It is a 2-sheaf if it is 2-separated and
there exists an object and isomorphisms such that for all the following square commutes:
A 2-sheaf, especially on a 1-site, is frequently called a stack. However, this has the unfortunate consequence that a 3-sheaf is then called a 2-stack, and so on with the numbering all offset by one. Also, it can be helpful to use a new term because of the notable differences between 2-sheaves on 2-sites and 2-sheaves on 1-sites. The main novelty is that and need not be invertible.
Note, though, they must be invertible as soon as is (2,1)-site: by definition and since an inverse is provided by , where is the symmetry equivalence.
If lacks finite limits, then in the definitions of “2-separated” and “2-sheaf” instead of the comma objects , we need to use arbitrary objects equipped with maps , , and a 2-cell . We leave the precise definition to the reader.
A 2-site is said to be subcanonical if for any , the representable functor is a 2-sheaf. When has finite limits, it is easy to verify that this is true precisely when every covering family is a (necessarily pullback-stable) quotient of its kernel 2-polycongruence?. In particular, the regular coverage on a regular 2-category is subcanonical, as is the coherent coverage on a coherent 2-category.
The 2-category of 2-sheaves on a small 2-site is, by definition, a Grothendieck 2-topos.
Over a 1-site, the Grothendieck construction says that 2-functors on the site are equivalent to fibered categories over the site. Hence in this case the theory of 2-sheaves can be entirely formulated in terms of fibered categories. See References – In terms of fibered categories.
This appears as (Bunge-Pare, corollary 2.6).
Moreover, over a 1-site the 2-topos of 2-sheaves ought to be equivalent to the (suitably defined) 2-category of internal categories in the underlying 1-topos. See References – In terms of internal categories.
This is discussed at 2-Topos – In terms of internal categories.
A classical class of examples for 2-sheaves are codomain fibrations over suitable sites, or rather their tangent categories. As discussed there, this includes the case of sheaves of categories of modules over sites of algebras.
This is for instance (Bunge-Pare, corollary 2.4).
Historically, the original definition of stack included the case of category-valued functors, hence of 2-sheaves, in:
and in section 3 of
The above text involves content transferred from
2-sites were earlier considered in