The notion of 2-sheaf is the generalization of the notion of sheaf to the higher category theory of 2-categories/bicategories. A 2-category of 2-sheaves forms a 2-topos.
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.
Let $C$ be a 2-site having finite 2-limits (for convenience). For a covering family $(f_i:U_i\to U)_i$ we have the comma objects
We also have the double comma objects $(f_i/f_j/f_k) = (f_i/f_j)\times_{U_j} (f_j/f_k)$ with projections $r_{i j k}:(f_i/f_j/f_k)\to (f_i/f_j)$, $s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k)$, and $t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k)$.
Now, a functor $X:C^{op} \to Cat$ 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 $x\in X(U)$ and isomorphisms $X(f_i)(x)\cong x_i$ such that for all $i,j$ 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 $\mu_{i j}$ and $\zeta_{i j}$ need not be invertible.
Note, though, they must be invertible as soon as $C$ is (2,1)-site: $\mu_{i j}$ by definition and $\zeta_{i j}$ since an inverse is provided by $\iota_{i j}^*(\zeta_{i j})$, where $\iota_{i j}\mapsto (f_i/f_j) \to (f_j/f_i)$ is the symmetry equivalence.
If $C$ lacks finite limits, then in the definitions of “2-separated” and “2-sheaf” instead of the comma objects $(f_i/f_j)$, we need to use arbitrary objects $V$ equipped with maps $p:V\to U_i$, $q:V\to U_j$, and a 2-cell $f_i p \to f_j q$. We leave the precise definition to the reader.
A 2-site is said to be subcanonical if for any $U\in C$, the representable functor $C(-,U)$ is a 2-sheaf. When $C$ 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 $2Sh(C)$ of 2-sheaves on a small 2-site $C$ is, by definition, a Grothendieck 2-topos.
If the underlying 2-site happens to be an (n,r)-site for $n$ and/or $r$ lower than 2, there may be other equivalent ways to think of 2-sheaves.
A 2-topos with a 2-site of definition that happens to be just a 1-site or (2,1)-site is 1-localic or (2,1)-localic.
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.
Also, over a 1-site a 2-sheaf is essentially a indexed category. Therefore stacks over 1-sites can also be discussed in this language, see notably the work (Bunge-Pare).
In particular, if the 1-site $C$ is a topos, then every topos over $C$ as its base topos (a $C$-topos) induces an indexed category.
If $C$ is a topos and $E$ is a $C$-topos, then (the indexed category corresponding to) $E$ is a 2-sheaf on $C$ with respect to the canonical topology.
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.
Over a (2,1)-site the 2-topos of 2-sheaves ought to be equivalent to the 2-category of internal (infinity,1)-categories in the corresponding (2,1)-topos.
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.
For $C$ an exact category with finite limits, the codomain fibration $Cod : C^I \to C$ or equivalently (under the Grothendieck construction), the self-indexing of $C$ is a 2-sheaf with respect to the canonical topology.
This is for instance (Bunge-Pare, corollary 2.4).
2-sheaf / stack
Historically, the original definition of stack included the case of category-valued functors, hence of 2-sheaves, in:
Jean Giraud, Cohomologie non abélienne, Grundlehren 179, Springer Verlag (1971) (doi:10.1007/978-3-662-62103-5)
Jean Giraud, Classifying topos, in: William Lawvere (ed.) Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, vol 274. Springer (1972) (doi:10.1007/BFb0073964)
Category-valued stacks as internal categories in the underlying sheaf topos have been considered in
and in section 3 of
A discussion of stacks over 1-sites in terms of their associated fibered categories is in
The above text involves content transferred from
2-sites were earlier considered in
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