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.