nLab 2-sheaf

Idea

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.

Remark on terminology

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.

Definition

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

• For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$ and $a,b: x\to y$, if $X(f_i)(a) = X(f_i)(b)$ for all $i$, then $a=b$.

It is 2-separated if it is 1-separated and

• For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$, given $b_i:X(f_i)(x) \to X(f_i)(y)$ such that $\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x)$, there exists a (necessarily unique) $b:x\to y$ such that $b_i = X(f_i)(b)$.

It is a 2-sheaf if it is 2-separated and

• For any covering family $(f_i:U_i\to U)_i$ and any $x_i\in X(U_i)$ together with morphisms $\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(x_j)$ such that the following diagram commutes:
  $$\array{X(r_{i j k})X(p_{i j})(x_i) & \overset{X(r_{i j k})(\zeta_{i j})}{\to} & X(r_{i j k})X(q_{i j})(x_j) & \overset{\cong}{\to} & X(s_{i j k})X(p_{j k})(x_j)\\ ^\cong \downarrow && && \downarrow ^{X(s_{i j k})(\zeta_{j k})}\\ X(t_{i j k}) X(p_{i k})(x_i) & \underset{X(t_{i j k})(\zeta_{i k})}{\to} & X(t_{i j k}) X(q_{i k})(x_k) & \underset{\cong}{\to} & X(s_{i j k}) X(q_{j k})(x_k)}$$

  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:

  $$\array{ X(p_{i j})X(f_i)(X) & \overset{\cong}{\to} & X(p_{i j})(x_i)\\ ^{X(\mu_{i j})}\downarrow && \downarrow^{\zeta_{i j}}\\ X(q_{i j})X(f_j)(x) & \underset{\cong}{\to} & X(q_{i j})(x_j).}$$

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.

Properties

Characterization of over $(n,r)$-sites

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

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.

Proposition

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 – As 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.

Examples

Codomain fibrations / sheaves of modules

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.

Proposition

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).

Related concepts

• presheaf / sheaf / cosheaf

• 2-sheaf / stack

• (∞,1)-sheaf / ∞-stack

• (∞,2)-sheaf

• (∞,n)-sheaf

• descent

References

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 (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 274, Springer (1972) [doi:10.1007/BFb0073964]

In terms of categories internal to sheaf toposes

Category-valued stacks as internal categories in the underlying sheaf topos:

• Marta Bunge, Robert Paré, Stacks and equivalence of indexed categories, Cahiers de Top. et Géom. Diff. Catég 20 4 (1979) 373-399 [numdam:CTGDC_1979__20_4_373_0]

• Marta Bunge, Stack completions and Morita equivalence for categories in a topos, Cahiers de Top. et Géom. Diff. Catég 20 4, (1979) 401-436 [numdam, MR558106]

• André Joyal, Myles Tierney, section 3 of: Strong stacks and classifying spaces, in: Category Theory (Como, 1990), Lecture Notes in Mathematics 1488, Springer (1991) 213-236 [doi:10.1007/BFb0084222]

  (establishing the canonical model structure on Cat in the internal generality)

In terms of fibered categories

A discussion of stacks over 1-sites in terms of their associated fibered categories is in

• Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (pdf)

2-Sites

The above text involves content transferred from

• Michael Shulman, 2-site

2-sites were earlier considered in

• Ross Street, StreetCBS