Lemma 7.48.1. Let \mathcal{C} be a site with coverings \text{Cov}(\mathcal{C}). For every object U of \mathcal{C}, let J(U) denote the set of sieves S on U with the following property: there exists a covering \{ f_ i : U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C}) so that the sieve S' generated by the f_ i (see Definition 7.47.3) is contained in S.
7.48 The topology defined by a site
Suppose that \mathcal{C} is a category, and suppose that \text{Cov}_1(\mathcal{C}) and \text{Cov}_2(\mathcal{C}) are sets of coverings that define the structure of a site on \mathcal{C}. In this situation it can happen that the categories of sheaves (of sets) for \text{Cov}_1(\mathcal{C}) and \text{Cov}_2(\mathcal{C}) are the same, see for example Lemma 7.8.7.
It turns out that the category of sheaves on \mathcal{C} with respect to some topology J determines and is determined by the topology J. This is a nontrivial statement which we will address later, see Theorem 7.50.2.
Accepting this for the moment it makes sense to study the topology determined by a site.
Proof. To prove the first assertion we just note that axioms (1), (2) and (3) of the definition of a site (Definition 7.6.2) directly imply the axioms (3), (2) and (1) of the definition of a topology (Definition 7.47.6). As an example we prove J has property (2). Namely, let U be an object of \mathcal{C}, let S, S' be sieves on U such that S \in J(U), and such that for every V \to U in S(V) we have S' \times _ U V \in J(V). By definition of J(U) we can find a covering \{ f_ i : U_ i \to U\} of the site such that S the image of h_{U_ i} \to h_ U is contained in S. Since each S'\times _ U U_ i is in J(U_ i) we see that there are coverings \{ U_{ij} \to U_ i\} of the site such that h_{U_{ij}} \to h_{U_ i} is contained in S' \times _ U U_ i. By definition of the base change this means that h_{U_{ij}} \to h_ U is contained in the subpresheaf S' \subset h_ U. By axiom (2) for sites we see that \{ U_{ij} \to U\} is a covering of U and we conclude that S' \in J(U) by definition of J.
Let \mathcal{F} be a presheaf. Suppose that \mathcal{F} is a sheaf in the topology J. We will show that \mathcal{F} is a sheaf on the site as well. Let \{ f_ i : U_ i \to U\} _{i\in I} be a covering of the site. Let s_ i \in \mathcal{F}(U_ i) be a family of sections such that s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j} for all i, j. We have to show that there exists a unique section s \in \mathcal{F}(U) restricting back to the s_ i on the U_ i. Let S \subset h_ U be the sieve generated by the f_ i. Note that S \in J(U) by definition. In stead of constructing s, by the sheaf condition in the topology, it suffices to construct an element
Take \alpha \in S(T) for some object T \in \mathcal{U}. This means exactly that \alpha : T \to U is a morphism which factors through f_ i for some i\in I (and maybe more than 1). Pick such an index i and a factorization \alpha = f_ i \circ \alpha _ i. Define \varphi (\alpha ) = \alpha _ i^* s_ i. If i', \alpha = f_ i \circ \alpha _{i'}' is a second choice, then \alpha _ i^* s_ i = (\alpha _{i'}')^* s_{i'} exactly because of our condition s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j} for all i, j. Thus \varphi (\alpha ) is well defined. We leave it to the reader to verify that \varphi , which in turn determines s is correct in the sense that s restricts back to s_ i.
Let \mathcal{F} be a presheaf. Suppose that \mathcal{F} is a sheaf on the site (\mathcal{C}, \text{Cov}(\mathcal{C})). We will show that \mathcal{F} is a sheaf for the topology J as well. Let U be an object of \mathcal{C}. Let S be a covering sieve on U with respect to the topology J. Let
We have to show there is a unique element in \mathcal{F}(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) which restricts back to \varphi . By definition there exists a covering \{ f_ i : U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C}) such that f_ i : U_ i \in U belongs to S(U_ i). Hence we can set s_ i = \varphi (f_ i) \in \mathcal{F}(U_ i). Then it is a pleasant exercise to see that s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j} for all i, j. Thus we obtain the desired section s by the sheaf condition for \mathcal{F} on the site (\mathcal{C}, \text{Cov}(\mathcal{C})). Details left to the reader. \square
Definition 7.48.2. Let \mathcal{C} be a site with coverings \text{Cov}(\mathcal{C}). The topology associated to \mathcal{C} is the topology J constructed in Lemma 7.48.1 above.
Let \mathcal{C} be a category. Let \text{Cov}_1(\mathcal{C}) and \text{Cov}_2(\mathcal{C}) be two coverings defining the structure of a site on \mathcal{C}. It may very well happen that the topologies defined by these are the same. If this happens then we say \text{Cov}_1(\mathcal{C}) and \text{Cov}_2(\mathcal{C}) define the same topology on \mathcal{C}. And if this happens then the categories of sheaves are the same, by Lemma 7.48.1.
It is usually the case that we only care about the topology defined by a collection of coverings, and we view the possibility of choosing different sets of coverings as a tool to study the topology.
Remark 7.48.3. Enlarging the class of coverings. Clearly, if \text{Cov}(\mathcal{C}) defines the structure of a site on \mathcal{C} then we may add to \mathcal{C} any set of families of morphisms with fixed target tautologically equivalent (see Definition 7.8.2) to elements of \text{Cov}(\mathcal{C}) without changing the topology.
Remark 7.48.4. Shrinking the class of coverings. Let \mathcal{C} be a site. Consider the set
where P(\text{Arrows}(\mathcal{C})) is the power set of the set of morphisms, i.e., the set of all sets of morphisms. Let \mathcal{S}_\tau \subset \mathcal{S} be the subset consisting of those (T, U) \in \mathcal{S} such that (a) all \varphi \in T have target U, (b) the collection \{ \varphi \} _{\varphi \in T} is tautologically equivalent (see Definition 7.8.2) to some covering in \text{Cov}(\mathcal{C}). Clearly, considering the elements of \mathcal{S}_\tau as the coverings, we do not get exactly the notion of a site as defined in Definition 7.6.2. The structure (\mathcal{C}, \mathcal{S}_\tau ) we get satisfies slightly modified conditions. The modified conditions are:
\text{Cov}(\mathcal{C}) \subset P(\text{Arrows}(\mathcal{C})) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}),
If V \to U is an isomorphism then (\{ V \to U\} , U) \in \text{Cov}(\mathcal{C}).
If (T, U) \in \text{Cov}(\mathcal{C}) and for f : U' \to U in T we are given (T_ f, U') \in \text{Cov}(\mathcal{C}), then setting T' = \{ f \circ f' \mid f \in T,\ f' \in T_ f\} , we get (T', U) \in \text{Cov}(\mathcal{C}).
If (T, U) \in \text{Cov}(\mathcal{C}) and g : V \to U is a morphism of \mathcal{C} then
U' \times _{f, U, g} V exists for f : U' \to U in T, and
setting T' = \{ \text{pr}_2 : U' \times _{f, U, g} V \to V \mid f : U' \to U \in T\} for some choice of fibre products we get (T', V) \in \text{Cov}(\mathcal{C}).
And it is easy to verify that, given a structure satisfying (0') – (3') above, then after suitably enlarging \text{Cov}(\mathcal{C}) (compare Sets, Section 3.11) we get a site. Obviously there is little difference between this notion and the actual notion of a site, at least from the point of view of the topology. There are two benefits: because of condition (0') above the coverings automatically form a set, and because of (0') the totality of all structures of this type forms a set as well. The price you pay for this is that you have to keep writing “tautologically equivalent” everywhere.
Comments (2)
Comment #370 by Baptiste Calmès on
Comment #371 by Pieter Belmans on