A coverage on a category $C$ consists of, for each object $U\in C$, a collection of families $\{f_i:U_i\to U\}_{i\in I}$ of morphisms with target $U$ 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 $C$. 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 $O(X)$ of open sets in some topological space (or locale) $X$, where a morphism is an inclusion, and a family of inclusions $\{U_i \hookrightarrow U\}$ is a covering family iff $U = \bigcup_i U_i$.
Another perspective on a coverage is that the covering families are “postulated well-behaved quotients.” That is, saying that $\{f_i:U_i\to U\}_{i\in I}$ is a covering family means that we want to think of $U$ as a well-behaved quotient (i.e. colimit) of the $U_i$. Here “well-behaved” means primarily “stable under pullback.” In general, $U$ may or may not actually be a colimit of the $U_i$; if it always is we call the site subcanonical. From this perspective, the embedding of $C$ into its category of sheaves is “the free cocompletion of $C$ that takes covering families to well-behaved quotients”; compare how the Yoneda embedding of an arbitrary category $C$ 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.
A coverage on a category $C$ consists of a function assigning to each object $U\in C$ a collection of families of morphisms $\{f_i:U_i\to U\}_{i\in I}$, called covering families, such that
The logic here is: $\forall f, \forall g, \exists h, \forall j, \exists k, \exists i, =$.
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.
See sheaf, of course, but it seems appropriate to briefly recall the concept here. If $\{f_i:U_i\to U\}_{i\in I}$ is a family of morphisms with codomain $U$, a presheaf $X:C^{op}\to Set$ is called a sheaf for this family if
If $C$ is a site, a presheaf $X:C^{op}\to Set$ is called a sheaf on $C$ if it is a sheaf for every covering family in $C$. We call a site $C$ subcanonical if every representable functor $C(-,c):C^{op}\to Set$ is a sheaf.
The category of sheaves $Sh(C)$ is a full subcategory of the category $[C^{op},Set]$ of presheaves. If $C$ is subcanonical, then its Yoneda embedding $C\to [C^{op},Set]$ factors through $Sh(C)$. If $C$ is small, then $Sh(C)$ is reflective in $[C^{op},Set]$ and a Grothendieck topos.
If, as is frequently the case, $C$ has pullbacks, then it is natural to impose the following stronger condition:
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 $C$. 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 $C$ has pullbacks (of covering families), the condition for a presheaf $X$ to be a sheaf for a covering family $\{f_i:U_i\to U\}_{i\in I}$ can be stated more simply (and probably more familiarly, to some readers), as the assertion that the following diagram is an equalizer:
The generalization to stacks using cosimplicial objects is then straightforward.
The collection of covering families can be “closed up” under a number of convenient operations without changing the notion of sheaf.
Any presheaf is a sheaf for the singleton family $\{1_U:U\to U\}$.
Any presheaf which is a sheaf for a family $\{f_i:U_i\to U\}_{i\in I}$ and also for some family $\{h_{i j}:U_{i j} \to U_i\}_{j\in J_i}$ for each $i$ is also a sheaf for the family of all composites $\{f_i h_{i j}:U_{i j}\to U\}_{i\in I, j\in U_i}$.
Let $C$ be a site and $\{f_i:U_i\to U\}_{i\in I}$ a covering family, and suppose $\{g_j:V_j\to U\}_{j\in J}$ is a family of morphisms such that each $f_i$ factors through some $g_j$. Then any sheaf $X$ on $C$ is also a sheaf for the family $\{g_j:V_j\to U\}_{j\in J}$. (NB: for this condition, it is essential that $\{f_i\}$ be part of a coverage and that $X$ be a sheaf for the entire coverage, not just for $\{f_i\}$.)
For any family $\{f_i:U_i\to U\}_{i\in I}$, the sieve it generates is the family of all morphisms $g:V\to U$ which factor through some $f_i$. A presheaf $X$ is a sheaf for $\{f_i\}$ iff it is a sheaf for the sieve it generates.
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 $R$ is a sieve on $U$ and $g:V\to U$ is a morphism, we define $g^*(R)$ to be the sieve on $V$ consisting of all morphisms $h$ into $V$ such that $g h$ factors through some morphism in $R$.)
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 $U$.
One can then show that for every coverage, there is a unique Grothendieck coverage having the same sheaves. When $C$ is small, then Grothendieck coverages on $C$ are also in bijective correspondence with Lawvere-Tierney topologies on its presheaf topos $[C^{op},Set]$, and thus in bijection with subtoposes of $[C^{op},Set]$. 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 $C$ 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 $\{f_i:U_i\to U\}_{i\in I}$ is a covering family and for each $i$, so is $\{h_{i j}:U_{i j} \to U_i\}_{j\in J_i}$, then $\{f_i h_{i j}:U_{i j}\to U\}_{i\in I, j\in U_i}$ is also a covering family.
For $X$ a topological space and $Op(X)$ its category of open subsets, the collection of open covers is a subcanonical coverage on $Op(X)$. I.e. a covering family on an open subset $U \subset X$ is a collection of further open subsets $\{U_i \subset X\}$ such that their union (in $X$) is $U$: $\cup_i U_i = U$.
This is the standard choice of coverage on $Op(X)$. Sheaves for this coverage are the usual notion of sheaf on a topologcal space.
A basis for the topology on $X$ is also a coverage on $X$, generating the same Grothendieck topology but in general not being closed under pullbacks (which in $Op(X)$ is intersection of open subsets). Notice that thence a basis for a topology on $X$ is not what is called a basis for a Grothendieck topology on $Op(X)$.
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 $\{U_i \to X\}$ consist of local homeomorphisms and in addition gives the local homeomorphism out of the coproduct $\coprod_i U_i \to X$.
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 $\{f:V\to U\}$ where $f$ 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 $\{f_i:U_i \to U\}_{1\le i\le n}$ the union of whose images is all of $U$. 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.
In addition to the construction of sheaves and stacks, other (not unrelated) applications of coverages include:
The definition of internal anafunctors.
The construction of model structures for internal categories.
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 $(\infty,1)$-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.
Peter Johnstone, Sketches of an Elephant , especially section C2.1.
Anders Kock, Postulated colimits and left exactness of Kan-extensions