nLab
2-site

This entry is about the notion of site in 2-category theory. For the notion “bisite” of a 1-categorical site equipped with two coverages see instead separated presheaf.

Idea

The notion of 2-site is the generalization of the notion of site to the higher category theory of 2-categories (bicategories).

Over a 2-site one has a 2-topos of 2-sheaves.

Definition

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

• If $(f_i:U_i\to U{)}_i$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $(h_j:V_j\to V{)}_j$ 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 $(f:V\to U)$, 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.

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 $(f:V\to U)$ is a covering family.

• If $(f_i:U_i\to U{)}_{i\in I}$ is a covering family and for each $i$, so is $(h_{ij}:U_{ij}\to U_i{)}_{j\in J_i}$, then $(f_i{h}_{ij}:U_{ij}\to U{)}_{i\in I,j\in U_i}$ is also a covering family.

This is the 2-categorical version of a Grothendieck pretopology (minus the common condition of having actual pullbacks).

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 $R\to C(-,U)$ which is objectwise fully faithful (equivalently, it is a fully faithful morphism in $[C^{\mathrm{op}},\mathrm{Cat}]$). Equivalently, it may be defined as a subcategory of the slice 2-category $C/U$ which is closed under precomposition with all morphisms of $C$.

Every family $(f_i:U_i\to U{)}_i$ generates a sieve by defining $R(V)$ to be the full subcategory of $C(V,U)$ 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 $(f_i:U_i\to U{)}_i$ if and only if

$$X(U)\simeq [C^{\mathrm{op}},\mathrm{Cat}](C(-,U),X)\to [C^{\mathrm{op}},\mathrm{Cat}](R,X)$$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 $(f_i:U_i\to U{)}_i$.

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 $g^{*}(R)$ 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 $(1_U)$) 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 $R$, $f^{*}(S)$ 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, $g^{*}(R)$ 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.

Properties

• The 2-category of 2-sheaves on a 2-site is a Grothendieck 2-topos.

• If $C$ is a 1-category regarded as a 2-category with only identity 2-morphisms, then a coverage (pretopology, topology) on $C$ reduces to the usual notion of coverage, Grothendieck pretopology, or Grothendieck topology.

Related concepts

• (n,r)-site

• 1-site

• 2-site, (2,1)-site

• (∞,1)-site

  • model site, simplicial site

References

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, MR84d:18006, p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM 962, 1982

See also StreetCBS.

More discussion is in

• Michael Shulman, 2-site