Grothendieck pretopologies feature prominently in algebraic geometry, but the more beautiful concept of Grothendieck topologies is rarely touched upon. In a series of short posts, I aim to introduce some of these ideas, show how key concepts like the sheaf condition get very nice categorical descriptions in this language, and give examples of why topoi have much better formal properties than sites.
Let be a small category, and write for the functor category .
Definition. A sieve on an object is a subpresheaf of the representable presheaf .
Concretely, this means that each is a set of morphisms with the property that if is any morphism, then is in . Thus, this is like a “right ideal in “. Since is small, we see that sieves on form a set, which we will denote .
Lemma. Let and be small categories, an object, and a functor. Then there is a pullback map
If and is the forgetful functor, then gives a bijection
Proof. If is a sieve, then so is since and implies , so . For the second statement, given a sieve on , define the sieve on by
Then is a sieve on , and is the unique sieve on such that .
Beware that the notation could also mean the presheaf pullback , but we won’t use it as such.
Remark. In particular, it suffices to study the case where has a terminal object, which we will denote by (in analogy with the small Zariski and étale sites of a scheme , which have as a terminal object). We are thus interested in studying the subobjects of the terminal presheaf . We will do so both in the case of presheaves and in the case of sheaves. Note that is a sheaf: for any set (empty or not), the product is a singleton, so the diagrams
are vacuously equalisers whenever is a covering (or any collection of morphisms).
Definition. A property on a set is a function to the power set of a point . The property holds for if , and fails if .
Given a property on the objects of a small category , we say that is left closed if for any morphism , the implication holds. (This terminology is my own. Below, we confusingly prove that these are equivalent to what we described earlier as “right ideals”. This change of orientation arises from the fact that diagrams are drawn in the opposite direction compared to composition of morphisms.)
If is a site (a small category together with a Grothendieck pretopology), we say that is local if it is left closed, and for any covering in , if holds for all , then holds.
Lemma. Let be a small category with a terminal object .
- Giving a subpresheaf of is equivalent to giving a left closed property on the objects of .
- If is a site, then giving a subsheaf of the presheaf is equivalent to a giving a local property .
A homotopy theorist might say that a local property is a -truncated sheaf [of spaces] on .
Proof. 1. The terminal presheaf takes on values at every , thus any subpresheaf takes on the values and , hence is a property on the objects of . The presheaf condition means that for every morphism , there is a map , which is exactly the implication since there are no maps .
Alternatively, one notes immediately from the definition that a sieve on an object is the same thing as a subcategory of which is left closed.
2. Being a subpresheaf translates to a left closed property by 1. Then is a sheaf if and only if, for every covering in , the diagram
is an equaliser. If one is empty, then so is since is left closed, so the diagram is always an equaliser.
Thus, in the sheaf condition, we may assume for all , i.e. holds for all . Since is left closed, this implies that for all , so the two arrows agree on , and the diagram is an equaliser if and only if . Running over all coverings in , this is exactly the condition that is local.