nLab
coverage

Context

Topos theory

Could not include topos theory - contents

Contents

Idea

A coverage on a category CC consists of, for each object UCU\in C, a collection of families {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} of morphisms with target UU to be thought of as covering families. The essential characteristic of these covering families is that they be “stable under pullback.” A number of other “saturation” conditions are frequently also imposed for convenience. A category equipped with a coverage is called a site.

One of the main purposes of a coverage is that it provides the minimum structure necessary to define a notion of sheaf (or more generally stack) on CC. A Grothendieck topos is defined to be the category of sheaves (of sets) on a small site. From this perspective, the example to keep in mind is the poset O(X)O(X) of open sets in some topological space (or locale) XX, where a morphism is an inclusion, and a family of inclusions {U iU}\{U_i \hookrightarrow U\} is a covering family iff U= iU iU = \bigcup_i U_i.

Another perspective on a coverage is that the covering families are “postulated well-behaved quotients.” That is, saying that {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} is a covering family means that we want to think of UU as a well-behaved quotient (i.e. colimit) of the U iU_i. Here “well-behaved” means primarily “stable under pullback.” In general, UU may or may not actually be a colimit of the U iU_i; if it always is we call the site subcanonical. From this perspective, the embedding of CC into its category of sheaves is “the free cocompletion of CC that takes covering families to well-behaved quotients”; compare how the Yoneda embedding of an arbitrary category CC into its category of presheaves is its free cocompletion, period.

The traditional name for a coverage, with the extra saturation conditions imposed, is a Grothendieck topology, and this is still widely used in mathematics. Following the Elephant, on this page we use coverage for a pullback-stable system of covering families and Grothendieck coverage if the extra saturation conditions are imposed. See Grothendieck topology for a discussion of the objections to that term.

A related notion is that of basis for a Grothendieck topology, which is similar to the notion of coverage, and similarly induces a Grothendieck topology, but assumes existence of pullbacks and closure of covering families under these pullbacks.

Definition

Definition

A coverage on a category CC consists of a function assigning to each object UCU\in C a collection of families of morphisms {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I}, called covering families, such that

  • If {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} is a covering family and g:VUg:V\to U is a morphism, then there exists a covering family {h j:V jV}\{h_j:V_j\to V\} such that each composite gh jg h_j factors through some f if_i.
V i U j h j f i V g U. \array{ V_i &\stackrel{\exists}{\to}& U_j \\ \downarrow^{h_j} && \downarrow^{f_i} \\ V &\underset{\forall g}{\to}& U } \,.
Definition

A site is a category equipped with a coverage.

Often sites are required to be small; see large site for complications that may otherwise arise.

Sheaves on a site

See sheaf, of course, but it seems appropriate to briefly recall the concept here. If {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} is a family of coterminal morphisms, a presheaf X:C opSetX:C^{op}\to Set is called a sheaf for this family if

  • For any collection of elements x iX(U i)x_i \in X(U_i) such that, whenever g:VU ig:V\to U_i and h:VU jh:V\to U_j are such that f ig=f jhf_i g = f_j h, we have X(g)(x i)=X(h)(x j)X(g)(x_i) = X(h)(x_j), then there exists a unique xX(U)x\in X(U) such that X(f i)(x)=x iX(f_i)(x)=x_i for all ii.

If CC is a site, a presheaf X:C opSetX:C^{op}\to Set is called a sheaf on CC if it is a sheaf for every covering family in CC. We call a site CC subcanonical if every representable functor C(,c):C opSetC(-,c):C^{op}\to Set is a sheaf.

The category of sheaves Sh(C)Sh(C) is a full subcategory of the category [C op,Set][C^{op},Set] of presheaves. If CC is subcanonical, then its Yoneda embedding C[C op,Set]C\to [C^{op},Set] factors through Sh(C)Sh(C). If CC is small, then Sh(C)Sh(C) is reflective in [C op,Set][C^{op},Set] and a Grothendieck topos.

Sites with pullbacks

If, as is frequently the case, CC has pullbacks, then it is natural to impose the following stronger condition:

  • If {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} is a covering family and g:VUg:V\to U is a morphism, then the family of pullbacks {g *(f i):g *U iV}\{g^*(f_i):g^*U_i\to V\} is a covering family of VV.

One can also impose the weaker condition that the pullbacks of covering families exist and are covering families, even if not all pullbacks exist in CC. The saturation conditions below imply that on a category with pullbacks, every coverage is equivalent to one satisfying this stronger condition, which perhaps we may call a cartesian coverage.

Likewise, when CC has pullbacks (of covering families), the condition for a presheaf XX to be a sheaf for a covering family {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} can be stated more simply (and probably more familiarly, to some readers), as the assertion that the following diagram is an equalizer:

(1)X(U) iIX(U i) i,jIX(U i× UU j). X(U) \to \prod_{i\in I} X(U_i) \rightrightarrows \prod_{i,j\in I} X(U_i\times_U U_j).

The generalization to stacks using cosimplicial objects is then straightforward.

Saturation conditions

The collection of covering families can be “closed up” under a number of convenient operations without changing the notion of sheaf.

  1. Any presheaf is a sheaf for the singleton family {1 U:UU}\{1_U:U\to U\}.

  2. Any presheaf which is a sheaf for a family {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} and also for some family {h ij:U ijU i} jJ i\{h_{i j}:U_{i j} \to U_i\}_{j\in J_i} for each ii is also a sheaf for the family of all composites {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}.

  3. Let CC be a site and {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I} a covering family, and suppose {g j:V jU} jJ\{g_j:V_j\to U\}_{j\in J} is a family of morphisms such that each f if_i factors through some g jg_j. Then any sheaf XX on CC is also a sheaf for the family {g j:V jU} jJ\{g_j:V_j\to U\}_{j\in J}. (NB: for this condition, it is essential that {f i}\{f_i\} be part of a coverage and that XX be a sheaf for the entire coverage, not just for {f i}\{f_i\}.)

  4. For any family {f i:U iU} iI\{f_i:U_i\to U\}_{i\in I}, the sieve it generates is the family of all morphisms g:VUg:V\to U which factor through some f if_i. A presheaf XX is a sheaf for {f i}\{f_i\} iff it is a sheaf for the sieve it generates.

Grothendieck coverages

Grothendieck originally considered only coverages that are closed under some or all of the above saturation conditions.

Because of the final condition, we may choose to consider only covering sieves. Incorporating the other saturation conditions as well, we define a Grothendieck coverage (commonly called a Grothendieck topology) to be a collection of sieves called covering sieves, satisfying the following pullback-stability and saturation conditions. (If RR is a sieve on UU and g:VUg:V\to U is a morphism, we define g *(R)g^*(R) to be the sieve on VV consisting of all morphisms hh into VV such that ghg h factors through some morphism in RR.)

  • 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.

One can then show that for every coverage, there is a unique Grothendieck coverage having the same sheaves. When CC is small, then Grothendieck coverages on CC are also in bijective correspondence with Lawvere-Tierney topologies on its presheaf topos [C op,Set][C^{op},Set], and thus in bijection with subtoposes of [C op,Set][C^{op},Set]. For more on this see category of sheaves.

On the other hand, it is often useful to consider only pullback-stable covering families, without needing to close them up into sieves satisfying the saturation conditions. For instance, in many cases the generating covering families will be finite and easy to describe. As we saw above, the notion of sheaf can also be defined more explicitly in terms of covering families, especially when CC has pullbacks.

Frequently, though, these covering families will satisfy at least some of the saturation conditions. The name Grothendieck pretopology or basis for a Grothendieck topology is commonly used for a coverage (often of the stronger sort requiring pullbacks) that also satisfies

  • Every isomorphism 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.

Examples

  • For XX a topological space and Op(X)Op(X) its category of open subsets, the collection of open covers is a subcanonical coverage on Op(X)Op(X). I.e. a covering family on an open subset UXU \subset X is a collection of further open subsets {U iX}\{U_i \subset X\} such that their union (in XX) is UU: iU i=U\cup_i U_i = U.

    This is the standard choice of coverage on Op(X)Op(X). Sheaves for this coverage are the usual notion of sheaf on a topologcal space.

    A basis for the topology on XX is also a coverage on XX, generating the same Grothendieck topology but in general not being closed under pullbacks (which in Op(X)Op(X) is intersection of open subsets). Notice that thence a basis for a topology on XX is not what is called a basis for a Grothendieck topology on Op(X)Op(X).

  • Similarly on (any small version of) the category Top or Diff or similar categories of topological spaces possibly with extra structure, open covers form a coverage.

    Another choice of coverage is given by taking covering families to consist of étale maps, i.e. of local homeomorphisms. Notice that every open cover {U iX}\{U_i \to X\} consist of local homeomorphisms and in addition gives the local homeomorphism out of the coproduct iU iX\coprod_i U_i \to X.

  • On Diff also good open covers form an equivalent coverage.
    While good open covers are not stable under pullback in Diff, every pullback of a good open cover gives an open cover that may be refined by a good open cover. This is all we need in the definition of coverage.

  • There are many interesting coverages on the category of schemes; it was these examples which originally motivated Grothendieck to consider the notion. See fpqc topology, etc.

  • On any category there is the trivial coverage which has no covering families at all. Every presheaf is a sheaf for this coverage (and in particular, it is subcanonical). The corresponding Grothendieck coverage consists of all sieves that contain a split epimorphism. (Note that every presheaf is a sheaf for any family containing a split epic.)

  • On any regular category there is a coverage, called the regular coverage, whose covering families are the singletons {f:VU}\{f:V\to U\} where ff is a regular epimorphism. It is subcanonical.

  • On any coherent category there is a a coverage, called the coherent coverage, whose covering families are the finite families {f i:U iU} 1in\{f_i:U_i \to U\}_{1\le i\le n} the union of whose images is all of UU. It is subcanonical. Likewise there is a geometric coverage on any infinitary-coherent category.

  • On any extensive category there is a coverage, called the extensive coverage, whose covering families are the inclusions into a (finite) coproduct. It is subcanonical. The coherent coverage on an extensive coherent category is generated by the union of the regular coverage and the extensive one.

  • Any category has a canonical coverage, defined to be the largest subcanonical one. (Hence the name “subcanonical” = “contained in the canonical coverage.”) The covering sieves for the canonical coverage are precisely those which are universally effective-epimorphic, meaning that their target is their colimit and this colimit is preserved by pullback.

  • The canonical coverage on a Grothendieck topos coincides with its geometric coverage, and moreover every sheaf for this coverage is representable. That is, a Grothendieck topos is a (large) site which is equivalent to its own category of sheaves.

Applications

In addition to the construction of sheaves and stacks, other (not unrelated) applications of coverages include:

In higher category theory

The notion of site and hence that of Grothendieck topology, Grothendieck pretopology and coverage typically has its straightforward analogs in higher category theory.

in (∞,1)-category theory the corresponding notion is that of (∞,1)-site. Such an (,1)(\infty,1)-site has correspondingly its (∞,1)-category of (∞,1)-sheaves. A discussion of a model category presentation of this in terms of localization at a coverage is at model structure on simplicial presheaves in the section Localization at a coverage.

References

Revised on July 17, 2014 22:01:48 by Urs Schreiber (82.136.246.44)