This post is the first goal in a series on sieves (subobjects of representable presheaves); I will give another generalisation in the next two posts. In the first post of the series, I defined sieves and gave basic examples, and last week I showed how the sheaf condition on a site can be stated in terms of sieves:
Corollary. Let be a (small) site. For a set of morphisms with the same target, write for the presheaf image of . Then a presheaf is a sheaf if and only if for every covering in , the inclusion induces an isomorphism
Thus, if is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category of sheaves purely in terms of sieves. This is the notion of a Grothendieck topology that we describe at the end of this post.
Before giving the definition, note that any morphism in gives a pullback taking to its inverse image under (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to defined uniquely up to unique isomorphism). Thus, is itself a presheaf (it takes values in since is small).
Also note the following method for producing sieves: if is a presheaf, a subpresheaf, and a section over some , we get a sieve by
By the Yoneda lemma, this is just the inverse image of along the morphism classifying . Note that is the maximal sieve if and only if .
Definition. Let be a small category. Then a Grothendieck topology on consists of a subpresheaf such that
- For all , the maximal sieve is in .
- If and with , then .
- If is a sieve such that , then (equivalently, then is the maximal sieve ).
The sieves are called covering sieves. Since is a presheaf, we see that for any and any covering sieve , the pullback is covering. Condition 2 says that any sieve containing a covering sieve is covering. In the presence of condition 1, conditions 2 and 3 together are equivalent to the local character found in SGA IV, Exp. II, Def. 1.1:
- If with , such that for every morphism the inverse image of along is in , then .
Indeed, applying this criterion when immedately shows if , since the inverse image of along is the maximal sieve . Thus the local character implies criterion 2. The local character says that if contains a covering sieve , then is covering. Assuming criterion 2, the sieve contains a covering sieve if and only if is itself covering, so the local character is equivalent to criterion 3.
Remark. One property that follows from the axioms is that is closed under binary intersection, i.e. if then . Indeed, if for some , then
so . Axioms 2 and 3 give .
Example. Let be a pretopology on the (small) category ; see Tag 00VH for a list of axioms. For each , define the subset as those that contain a sieve of the form for some covering in . (See the corollary at the top for the definition of .) Concretely, this means that there exists a covering such that for all , i.e. is covered by morphisms that are in the given sieve .
Lemma. The association is a topology. It is the coarsest topology on for which each for is a covering sieve.
Proof. We will use the criteria of Tag 00VH. If , then there exists with . If is any morphism in , then by criterion 3 of Tag 00VH. But , because a morphism factors through if and only if factors through . Thus, , so , and is a subpresheaf of .
Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose satisfies . Then there exists with . This means that for all , i.e. for all . Thus, for each there exists in such that , i.e. for all and all . Thus, if denotes , then we get . But is a covering by criterion 2 of Tag 00VH, so .
If is any other Grothendieck topology for which each for is covering, then contains by criterion 2.
To state the obvious (hopefully), the notion of sheaf can therefore be defined on a Grothendieck topology in a way that coincides with the usual notion for a Grothendieck pretopology:
Definition. Let be a small category, and let be a Grothendieck topology. Then a presheaf is a sheaf if for any and any , the map induces an isomorphism
Thus, a Grothendieck topology is an internal characterisation (inside ) of which morphisms one needs to localise to get . In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.