Michael Shulman
2-site

Definition

A coverage on a 2-category C consists of, for each object UC, a collection of families (f i:U iU) i of morphisms with codomain U, called covering families, such that

  • If (f i:U iU) i is a covering family and g:VU is a morphism, then there exists a covering family (h j:V jV) j such that each composite gh 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:VU), 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 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 (f i:U iU) i we have the comma objects

Comma Square (f i/f j) U i U j U f i f j q ij p ij μ ij

We also have the double comma objects (f i/f j/f k)=(f i/f j)× U j(f j/f k) with projections r ijk:(f i/f j/f k)(f i/f j), s ijk:(f i/f j/f k)(f j/f k), and t ijk:(f i/f j/f k)(f i/f k).

Now, a functor X:C opCat is called a 2-presheaf. It is 1-separated if

  • For any covering family (f i:U iU) i and any x,yX(U) and a,b:xy, 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 iU) i and any x,yX(U), given b i:X(f i)(x)X(f i)(y) such that μ ij(y)X(p ij)(b i)=X(q ij)(b i)μ ij(x), there exists a (necessarily unique) b:xy 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 iU) i and any x iX(U i) together with morphisms ζ ij:X(p ij)(x i)X(q ij)(x j) such that the following diagram commutes:

    X(r ijk)X(p ij)(x i) X(r ijk)(ζ ij) X(r ijk)X(q ij)(x j) X(s ijk)X(p jk)(x j) X(s ijk)(ζ jk) X(t ijk)X(p ik)(x i) X(t ijk)(ζ ik) X(t ijk)X(q ik)(x k) X(s ijk)X(q jk)(x k)\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 xX(U) and isomorphisms X(f i)(x)x i such that for all i,j the following square commutes:

    X(p ij)X(f i)(X) X(p ij)(x i) X(μ ij) ζ ij X(q ij)X(f j)(x) X(q ij)(x j).\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 μ ij and ζ ij need not be invertible.

Note, though, they must be invertible as soon as C is (2,1)-site: μ ij by definition and ζ ij since an inverse is provided by ι ij *(ζ ij), where ι ijmaps(f i/f j)(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:VU i, q:VU j, and a 2-cell f ipf jq. We leave the precise definition to the reader.

A 2-site is said to be subcanonical if for any UC, 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.

Saturation conditions

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

  • If f:VU is an equivalence, then the one-element family (f:VU) is a covering family.

  • If (f i:U iU) iI is a covering family and for each i, so is (h ij:U ijU i) jJ i, then (f ih ij:U ijU) iI,jU i is also a covering family.

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

Now, a sieve on an object UC is defined to be a functor R:C opCat with a transformation RC(,U) which is objectwise fully faithful (equivalently, it is ff in [C op,Cat]). Every family (f i:U iU) i generates a sieve by defining R(V) to be the full subcategory of C(V,U) on those g:VU such that gf ih for some i and some h:VU i. The following observation is due to StreetCBS.

Lemma

A 2-presheaf X:C opCat is a 2-sheaf for a covering family (f i:U iU) i if and only if

X(U)[C op,Cat](C(,U),X)[C op,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 iU) 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:VU 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:VU 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:VU 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.

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