2-site (changes) in nLab

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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.

Contents

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 CC consists of, for each object U∈CU\in C, a collection of families (f i:U i→U) i(f_i: U_i\to U)_i of morphisms with codomain UU, called covering families, such that

• If (f i:U i→U) i(f_i:U_i\to U)_i is a covering family and g:V→Ug:V\to U is a morphism, then there exists a covering family (h j:V j→V) j(h_j:V_j\to V)_j such that each composite gh jg h_j factors through some f if_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 CC is a regular 2-category, then the collection of all singleton families (f:V→U)(f:V\to U), where ff is eso, forms a coverage called the regular coverage.

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

• On CatCat, 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→Uf:V\to U is an equivalence, then the one-element family (f:V→U)(f:V\to U) is a covering family.

• If (f i:U i→U) i∈I(f_i:U_i\to U)_{i\in I} is a covering family and for each ii, so is (h ij:U ij→U i) j∈J i(h_{i j}:U_{i j} \to U_i)_{j\in J_i}, then (f ih ij:U ij→U) i∈I,j∈U i(f_i h_{i j}:U_{i j}\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∈CU\in C is defined to be a functor R:C op→CatR:C^{op}\to Cat with a transformation R→C(−,U)R\to C(-,U) which is objectwise fully faithful (equivalently, it is a fully faithful morphism in [C op,Cat][C^{op},Cat]). Equivalently, it may be defined as a subcategory of the slice 2-category C/UC/U which is closed under precomposition with all morphisms of CC.

Every family (f i:U i→U) i(f_i\colon U_i\to U)_i generates a sieve by defining R(V)R(V) to be the full subcategory of C(V,U)C(V,U) on those g:V→Ug:V\to U such that g≅f ihg \cong f_i h for some ii and some h:V→U ih:V\to U_i. The following observation is due to StreetCBS.

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 CC to consist of, for each object UU, a collection of sieves on UU called covering sieves, such that

• If RR is a covering sieve on UU and g:V→Ug:V\to U is any morphism, then g *(R)g^*(R) is a covering sieve on VV.

• For each UU the sieve M UM_U consisting of all morphisms into UU (the sieve generated by the singleton family (1 U)(1_U)) is a covering sieve.

• If RR is a covering sieve on UU and SS is an arbitrary sieve on UU such that for each f:V→Uf:V\to U in RR, f *(S)f^*(S) is a covering sieve on VV, then SS is also a covering sieve on UU.

Here if RR is a sieve on UU and g:V→Ug:V\to U is a morphism, g *(R)g^*(R) denotes the sieve on VV consisting of all morphisms hh into VV such that ghg h factors, up to isomorphism, through some morphism in RR.

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

• (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