Michael Shulman
2-site

Definition

A coverage on a 2-category CC consists of, for each object UCU\in C, a collection of families (f i:U iU) i(f_i: U_i\to U)_i of morphisms with codomain UU, called covering families, such that

  • If (f i:U iU) i(f_i:U_i\to U)_i is a covering family and g:VUg:V\to U is a morphism, then there exists a covering family (h j:V jV) 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:VU)(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.

2-sheaves

Let CC be a 2-site having finite limits (for convenience). For a covering family (f i:U iU) i(f_i:U_i\to U)_i we have the comma objects

Comma Square ( f i / f j ) (f_i/f_j) U i U_i U j U_j U U f i f_i f j f_j q ij q_{i j} p ij p_{i j} μ ij \mu_{i j}

We also have the double comma objects (f i/f j/f k)=(f i/f j)× U j(f j/f k)(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)(f i/f j)r_{i j k}:(f_i/f_j/f_k)\to (f_i/f_j), s ijk:(f i/f j/f k)(f j/f k)s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k), and t ijk:(f i/f j/f k)(f i/f k)t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k).

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

  • For any covering family (f i:U iU) i(f_i:U_i\to U)_i and any x,yX(U)x,y\in X(U) and a,b:xya,b: x\to y, if X(f i)(a)=X(f i)(b)X(f_i)(a) = X(f_i)(b) for all ii, then a=ba=b.

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

  • For any covering family (f i:U iU) i(f_i:U_i\to U)_i and any x,yX(U)x,y\in X(U), given b i:X(f i)(x)X(f i)(y)b_i:X(f_i)(x) \to X(f_i)(y) such that μ ij(y)X(p ij)(b i)=X(q ij)(b i)μ ij(x)\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x), there exists a (necessarily unique) b:xyb:x\to y such that b i=X(f i)(b)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(f_i:U_i\to U)_i and any x iX(U i)x_i\in X(U_i) together with morphisms ζ ij:X(p ij)(x i)X(q ij)(x j)\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(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)x\in X(U) and isomorphisms X(f i)(x)x iX(f_i)(x)\cong x_i such that for all i,ji,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\mu_{i j} and ζ ij\zeta_{i j} need not be invertible.

Note, though, they must be invertible as soon as CC is (2,1)-site: μ ij\mu_{i j} by definition and ζ ij\zeta_{i j} since an inverse is provided by ι ij *(ζ ij)\iota_{i j}^*(\zeta_{i j}), where ι ijmaps(f i/f j)(f j/f i)\iota_{i j}\maps (f_i/f_j) \to (f_j/f_i) is the symmetry equivalence.

If CC lacks finite limits, then in the definitions of “2-separated” and “2-sheaf” instead of the comma objects (f i/f j)(f_i/f_j), we need to use arbitrary objects VV equipped with maps p:VU ip:V\to U_i, q:VU jq:V\to U_j, and a 2-cell f ipf jqf_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 UCU\in C, the representable functor C(,U)C(-,U) is a 2-sheaf. When CC 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)2Sh(C) of 2-sheaves on a small 2-site CC 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:VUf:V\to U is an equivalence, then the one-element family (f:VU)(f:V\to U) is a covering family.

  • If (f i:U iU) iI(f_i:U_i\to U)_{i\in I} is a covering family and for each ii, so is (h ij:U ijU i) jJ i(h_{i j}:U_{i j} \to U_i)_{j\in J_i}, then (f ih ij:U ijU) iI,jU 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.

Now, a sieve on an object UCU\in C is defined to be a functor R:C opCatR:C^{op}\to Cat with a transformation RC(,U)R\to C(-,U) which is objectwise fully faithful (equivalently, it is ff in [C op,Cat][C^{op},Cat]). Every family (f i:U iU) i(f_i: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:VUg:V\to U such that gf ihg \cong f_i h for some ii and some h:VU ih:V\to U_i. The following observation is due to StreetCBS.

Lemma

A 2-presheaf X:C opCatX:C^{op}\to Cat is a 2-sheaf for a covering family (f i:U iU) i(f_i:U_i\to U)_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 RR is the sieve on UU generated by (f i:U iU) i(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 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:VUg: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:VUf: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:VUg: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.

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