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 special case is the notion of stack. See also derived stack.

The 2-category of 2-sheaves forms a 2-topos.

2-sheaves

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 $r_{ijk}:(f_i/f_j/f_k)\to (f_i/f_j)$, $s_{ijk}:(f_i/f_j/f_k)\to (f_j/f_k)$, and $t_{ijk}:(f_i/f_j/f_k)\to (f_i/f_k)$.

Now, a functor $X:C^{\mathrm{op}}\to \mathrm{Cat}$ 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 ${\mu }_{ij}(y)\circ X(p_{ij})(b_i)=X(q_{ij})(b_i)\circ {\mu }_{ij}(x)$, 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 ${\zeta }_{ij}:X(p_{ij})(x_i)\to X(q_{ij})(x_j)$ such that the following diagram commutes:

  $$\begin{array}{ccccc}X(r_{ijk})X(p_{ij})(x_i)& \stackrel{X(r_{ijk})({\zeta }_{ij})}{\to }& X(r_{ijk})X(q_{ij})(x_j)& \stackrel{\cong }{\to }& X(s_{ijk})X(p_{jk})(x_j)\\ {}^{\cong }\downarrow & & & & {\downarrow }^{X(s_{ijk})({\zeta }_{jk})}\\ X(t_{ijk})X(p_{ik})(x_i)& \underset{X(t_{ijk})({\zeta }_{ik})}{\to }& X(t_{ijk})X(q_{ik})(x_k)& \underset{\cong }{\to }& X(s_{ijk})X(q_{jk})(x_k)\end{array}$$\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:

  $$\begin{array}{ccc}X(p_{ij})X(f_i)(X)& \stackrel{\cong }{\to }& X(p_{ij})(x_i)\\ {}^{X({\mu }_{ij})}\downarrow & & {\downarrow }^{{\zeta }_{ij}}\\ X(q_{ij})X(f_j)(x)& \underset{\cong }{\to }& X(q_{ij})(x_j).\end{array}$$\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 ${\mu }_{ij}$ and ${\zeta }_{ij}$ need not be invertible.

Note, though, they must be invertible as soon as $C$ is (2,1)-site: ${\mu }_{ij}$ by definition and ${\zeta }_{ij}$ since an inverse is provided by ${\iota }_{ij}^{*}({\zeta }_{ij})$, where ${\iota }_{ij}\mapsto (f_i/f_j)\to (f_j/f_i)$ 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 $2\mathrm{Sh}(C)$ of 2-sheaves on a small 2-site $C$ is, by definition, a Grothendieck 2-topos?.

Related concepts

• presheaf / sheaf / cosheaf

• 2-sheaf / stack

• (∞,1)-sheaf / ∞-stack

• descent

  • cover

  • cohomological descent

  • descent morphism

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

References

The above involves content transferred from

• Michael Shulman, 2-site

Strict 2-sites were considered in

• Ross Street, Two-dimensional sheaf theory J. Pure Appl. Algebra 23 (1982), no. 3, 251-270, MR83d:18014, doi

Bicategorical 2-sites in

• Ross Street, Characterization of Bicategories of Stacks, MR0682967 (84d:18006), p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM 962, 1982