# Definition

A coverage on a 2-category $C$ consists of, for each object $U\in C$, a collection of families $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ of morphisms with codomain $U$, called covering families, such that

• If $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $\left({h}_{j}:{V}_{j}\to V{\right)}_{j}$ such that each composite $g{h}_{j}$ factors through some ${f}_{i}$, up to isomorphism.

This is the 2-categorical analogue of the 1-categorical notion of coverage introduced in the Elephant.

A 2-category equipped with a coverage is called a 2-site.

# Examples

• If $C$ is a regular 2-category, then the collection of all singleton families $\left(f:V\to U\right)$, where $f$ is eso, forms a coverage called the regular coverage.

• Likewise, if $C$ is a coherent 2-category, the collection of all finite jointly-eso families forms a coverage called the coherent coverage.

• On $\mathrm{Cat}$, the canonical coverage consists of all families that are jointly essentially surjective on objects.

# 2-sheaves

Let $C$ be a 2-site having finite limits (for convenience). For a covering family $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ we have the comma objects

We also have the double comma objects $\left({f}_{i}/{f}_{j}/{f}_{k}\right)=\left({f}_{i}/{f}_{j}\right){×}_{{U}_{j}}\left({f}_{j}/{f}_{k}\right)$ with projections ${r}_{ijk}:\left({f}_{i}/{f}_{j}/{f}_{k}\right)\to \left({f}_{i}/{f}_{j}\right)$, ${s}_{ijk}:\left({f}_{i}/{f}_{j}/{f}_{k}\right)\to \left({f}_{j}/{f}_{k}\right)$, and ${t}_{ijk}:\left({f}_{i}/{f}_{j}/{f}_{k}\right)\to \left({f}_{i}/{f}_{k}\right)$.

Now, a functor $X:{C}^{\mathrm{op}}\to \mathrm{Cat}$ is called a 2-presheaf. It is 1-separated if

• For any covering family $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ and any $x,y\in X\left(U\right)$ and $a,b:x\to y$, if $X\left({f}_{i}\right)\left(a\right)=X\left({f}_{i}\right)\left(b\right)$ for all $i$, then $a=b$.

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

• For any covering family $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ and any $x,y\in X\left(U\right)$, given ${b}_{i}:X\left({f}_{i}\right)\left(x\right)\to X\left({f}_{i}\right)\left(y\right)$ such that ${\mu }_{ij}\left(y\right)\circ X\left({p}_{ij}\right)\left({b}_{i}\right)=X\left({q}_{ij}\right)\left({b}_{i}\right)\circ {\mu }_{ij}\left(x\right)$, there exists a (necessarily unique) $b:x\to y$ such that ${b}_{i}=X\left({f}_{i}\right)\left(b\right)$.

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

• For any covering family $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ and any ${x}_{i}\in X\left({U}_{i}\right)$ together with morphisms ${\zeta }_{ij}:X\left({p}_{ij}\right)\left({x}_{i}\right)\to X\left({q}_{ij}\right)\left({x}_{j}\right)$ such that the following diagram commutes:

$\begin{array}{ccccc}X\left({r}_{ijk}\right)X\left({p}_{ij}\right)\left({x}_{i}\right)& \stackrel{X\left({r}_{ijk}\right)\left({\zeta }_{ij}\right)}{\to }& X\left({r}_{ijk}\right)X\left({q}_{ij}\right)\left({x}_{j}\right)& \stackrel{\cong }{\to }& X\left({s}_{ijk}\right)X\left({p}_{jk}\right)\left({x}_{j}\right)\\ {}^{\cong }↓& & & & {↓}^{X\left({s}_{ijk}\right)\left({\zeta }_{jk}\right)}\\ X\left({t}_{ijk}\right)X\left({p}_{ik}\right)\left({x}_{i}\right)& \underset{X\left({t}_{ijk}\right)\left({\zeta }_{ik}\right)}{\to }& X\left({t}_{ijk}\right)X\left({q}_{ik}\right)\left({x}_{k}\right)& \underset{\cong }{\to }& X\left({s}_{ijk}\right)X\left({q}_{jk}\right)\left({x}_{k}\right)\end{array}$\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 $x\in X\left(U\right)$ and isomorphisms $X\left({f}_{i}\right)\left(x\right)\cong {x}_{i}$ such that for all $i,j$ the following square commutes:

$\begin{array}{ccc}X\left({p}_{ij}\right)X\left({f}_{i}\right)\left(X\right)& \stackrel{\cong }{\to }& X\left({p}_{ij}\right)\left({x}_{i}\right)\\ {}^{X\left({\mu }_{ij}\right)}↓& & {↓}^{{\zeta }_{ij}}\\ X\left({q}_{ij}\right)X\left({f}_{j}\right)\left(x\right)& \underset{\cong }{\to }& X\left({q}_{ij}\right)\left({x}_{j}\right).\end{array}$\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 ${\iota }_{ij}^{*}\left({\zeta }_{ij}\right)$, where ${\iota }_{ij}maps\left({f}_{i}/{f}_{j}\right)\to \left({f}_{j}/{f}_{i}\right)$ is the symmetry equivalence.

If $C$ lacks finite limits, then in the definitions of “2-separated” and “2-sheaf” instead of the comma objects $\left({f}_{i}/{f}_{j}\right)$, 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 $C\left(-,U\right)$ 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 $2\mathrm{Sh}\left(C\right)$ of 2-sheaves on a small 2-site $C$ is, by definition, a Grothendieck 2-topos.

# Saturation conditions

A pre-Grothendieck coverage on a 2-category is a coverage satisfying the following additional conditions:

• If $f:V\to U$ is an equivalence, then the one-element family $\left(f:V\to U\right)$ is a covering family.

• If $\left({f}_{i}:{U}_{i}\to U{\right)}_{i\in I}$ is a covering family and for each $i$, so is $\left({h}_{ij}:{U}_{ij}\to {U}_{i}{\right)}_{j\in {J}_{i}}$, then $\left({f}_{i}{h}_{ij}:{U}_{ij}\to U{\right)}_{i\in I,j\in {U}_{i}}$ is also a covering family.

This is the 2-categorical version of a Grothendieck pretopology.

Now, a sieve on an object $U\in C$ is defined to be a functor $R:{C}^{\mathrm{op}}\to \mathrm{Cat}$ with a transformation $R\to C\left(-,U\right)$ which is objectwise fully faithful (equivalently, it is ff in $\left[{C}^{\mathrm{op}},\mathrm{Cat}\right]$). Every family $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ generates a sieve by defining $R\left(V\right)$ to be the full subcategory of $C\left(V,U\right)$ on those $g:V\to U$ such that $g\cong {f}_{i}h$ for some $i$ and some $h:V\to {U}_{i}$. The following observation is due to StreetCBS.

###### Lemma

A 2-presheaf $X:{C}^{\mathrm{op}}\to \mathrm{Cat}$ is a 2-sheaf for a covering family $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$ if and only if

$X\left(U\right)\simeq \left[{C}^{\mathrm{op}},\mathrm{Cat}\right]\left(C\left(-,U\right),X\right)\to \left[{C}^{\mathrm{op}},\mathrm{Cat}\right]\left(R,X\right)$X(U) \simeq[C^{op},Cat](C(-,U),X) \to [C^{op},Cat](R,X)

is an equivalence, where $R$ is the sieve on $U$ generated by $\left({f}_{i}:{U}_{i}\to U{\right)}_{i}$.

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 $C$ to consist of, for each object $U$, a collection of sieves on $U$ called covering sieves, such that

• If $R$ is a covering sieve on $U$ and $g:V\to U$ is any morphism, then ${g}^{*}\left(R\right)$ is a covering sieve on $V$.

• For each $U$ the sieve ${M}_{U}$ consisting of all morphisms into $U$ (the sieve generated by the singleton family $\left({1}_{U}\right)$) is a covering sieve.

• If $R$ is a covering sieve on $U$ and $S$ is an arbitrary sieve on $U$ such that for each $f:V\to U$ in $R$, ${f}^{*}\left(S\right)$ is a covering sieve on $V$, then $S$ is also a covering sieve on $U$.

Here if $R$ is a sieve on $U$ and $g:V\to U$ is a morphism, ${g}^{*}\left(R\right)$ denotes the sieve on $V$ consisting of all morphisms $h$ into $V$ such that $gh$ factors, up to isomorphism, through some morphism in $R$.

As in the 1-categorical case, one can then show that every coverage generates a unique Grothendieck coverage having the same 2-sheaves.

Revised on March 10, 2010 20:02:49 by Mike Shulman (128.135.197.116)