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 7.8.5.

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 $\mathcal{F}(U) = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F})$ 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.

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## Comments (2)

Comment #370 by Baptiste Calmès on

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