Grothendieck topologies (topologies 4/6)

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 $\mathscr C$ be a (small) site. For a set of morphisms $\mathscr U = \{U_i \to U\}_{i \in I}$ with the same target, write $S_{\mathscr U} \subseteq h_U$ for the presheaf image of $\coprod_{i\in I} h_{U_i} \to h_U$ . Then a presheaf $\mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ is a sheaf if and only if for every covering $\mathscr U = \{U_i \to U\}_{i \in I}$ in $\mathscr C$ , the inclusion $S_{\mathscr U} \hookrightarrow h_U$ induces an isomorphism

$\operatorname{Hom}(h_U,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S_{\mathscr U},\mathscr F).$

Thus, if $\mathscr C$ is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category $\mathbf{Sh}(\mathscr C) \subseteq \mathbf{PSh}(\mathscr C)$ 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 $f \colon Y \to X$ in $\mathscr C$ gives a pullback $\mathbf{Siv}(X) \to \mathbf{Siv}(Y)$ taking $S \subseteq h_X$ to its inverse image under $h_f \colon h_Y \to h_X$ (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to $h_Y$ defined uniquely up to unique isomorphism). Thus, $\mathbf{Siv}$ is itself a presheaf $\mathscr C^{\operatorname{op}} \to \mathbf{Set}$ (it takes values in $\mathbf{Set}$ since $\mathscr C$ is small).

Also note the following method for producing sieves: if $\mathscr F$ is a presheaf, $\mathscr G \subseteq \mathscr F$ a subpresheaf, and $s \in \mathscr F(X)$ a section over some $X \in \mathscr C$ , we get a sieve $(s \in \mathscr G) \in \mathbf{Siv}(X)$ by

$(s \in \mathscr G)(Y) = \left\{f \colon Y \to X\ \big|\ f^*(s) \in \mathscr G(Y)\right\}.$

By the Yoneda lemma, this is just the inverse image of $\mathscr G \subseteq \mathscr F$ along the morphism $h_X \to \mathscr F$ classifying $s$ . Note that $(s \in \mathscr G)$ is the maximal sieve $h_X$ if and only if $s \in \mathscr G(X)$ .

Definition. Let $\mathscr C$ be a small category. Then a Grothendieck topology on $\mathscr C$ consists of a subpresheaf $J \subseteq \mathbf{Siv}$ such that

For all $X \in \mathscr C$ , the maximal sieve $h_X \subseteq h_X$ is in $J(X)$ .
If $S \in J(X)$ and $S' \in \mathbf{Siv}(X)$ with $S \subseteq S'$ , then $S' \in J(X)$ .
If $S \in \mathbf{Siv}(X)$ is a sieve such that $(S \in J) \in J(X)$ , then $S \in J(X)$ (equivalently, then $(S \in J)$ is the maximal sieve $h_X$ ).

The sieves $S \in J(X)$ are called covering sieves. Since $J$ is a presheaf, we see that for any $f \colon Y \to X$ and any covering sieve $S \subseteq h_X$ , the pullback $f^*S \subseteq h_Y$ 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 $_1$ , Exp. II, Def. 1.1:

If $S, S' \in \mathbf{Siv}(X)$ with $S \in J(X)$ , such that for every morphism $h_Y \to S$ the inverse image of $S' \subseteq h_X$ along $h_Y \to S \to h_X$ is in $J(Y)$ , then $S' \in J(X)$ .

Indeed, applying this criterion when $S \subseteq S'$ immedately shows $S' \in J(X)$ if $S \in J(X)$ , since the inverse image of $S' \subseteq h_X$ along $h_Y \to S \to h_X$ is the maximal sieve $h_Y$ . Thus the local character implies criterion 2. The local character says that if $(S' \in J)$ contains a covering sieve $S$ , then $S'$ is covering. Assuming criterion 2, the sieve $(S' \in J)$ contains a covering sieve if and only if $(S' \in J)$ is itself covering, so the local character is equivalent to criterion 3.

Remark. One property that follows from the axioms is that $J(X)$ is closed under binary intersection, i.e. if $S, T \in J(X)$ then $(S \cap T) \in J(X)$ . Indeed, if $f \in S(Y)$ for some $f \colon Y \to X$ , then

$f^*(S \cap T) = f^*S \cap f^*T = h_Y \cap f^*T = f^*T \in J(Y),$

so $S \subseteq ((S \cap T) \in J)$ . Axioms 2 and 3 give $(S \cap T) \in J(X)$ .

Example. Let $\mathcal Cov(\mathscr C)$ be a pretopology on the (small) category $\mathscr C$ ; see Tag 00VH for a list of axioms. For each $X \in \mathscr C$ , define the subset $J(X) \subseteq \mathbf{Siv}(X)$ as those $S \subseteq h_X$ that contain a sieve of the form $S_{\mathscr U}$ for some covering $\mathscr U = \{U_i \to X\}$ in $\mathcal Cov(\mathscr C)$ . (See the corollary at the top for the definition of $S_{\mathscr U}$ .) Concretely, this means that there exists a covering $\{f_i \colon U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C)$ such that $f_i \in S(U_i)$ for all $i \in I$ , i.e. $X$ is covered by morphisms $f_i \colon U_i \to X$ that are in the given sieve $S$ .

Lemma. The association $X \mapsto J(X)$ is a topology. It is the coarsest topology on $\mathscr C$ for which each $S_{\mathscr U}$ for $\mathscr U \in \mathcal Cov(\mathscr C)$ is a covering sieve.

Proof. We will use the criteria of Tag 00VH. If $S \in J(X)$ , then there exists $\mathscr U = \{U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C)$ with $S_{\mathscr U} \subseteq S$ . If $f \colon Y \to X$ is any morphism in $\mathscr C$ , then $f^*\mathscr U = \{U_i \times_X Y \to Y\}_{i \in I} \in \mathcal Cov(\mathscr C)$ by criterion 3 of Tag 00VH. But $S_{f^*\mathscr U} = f^*S_{\mathscr U}$ , because a morphism $g \colon U \to Y$ factors through $U_i \times_X Y$ if and only if $fg \colon U \to X$ factors through $U_i$ . Thus, $S_{f^*\mathscr U} = f^*S_{\mathscr U} \subseteq f^*S$ , so $f^*S \in J(Y)$ , and $J$ is a subpresheaf of $\mathbf{Siv}$ .

Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose $S \in \mathbf{Siv}(X)$ satisfies $(S \in J) \in J(X)$ . Then there exists $\mathscr U = \{f_i \colon U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C)$ with $S_{\mathscr U} \subseteq (S \in J)$ . This means that $f_i \in (S \in J)(U_i)$ for all $i$ , i.e. $f_i^*S \in J(U_i)$ for all $i$ . Thus, for each $i \in I$ there exists $\mathscr V_i = \{g_{ij} \colon V_{ij} \to U_i\}_{j \in J_i}$ in $\mathcal Cov(\mathscr C)$ such that $S_{\mathscr V_i} \subseteq f_i^*S$ , i.e. $f_ig_{ij} \in S(X)$ for all $i \in I$ and all $j \in J_i$ . Thus, if $\mathscr V$ denotes $\{f_ig_{ij} \colon V_{ij} \to X\}_{i \in I, j \in J_i}$ , then we get $S_{\mathscr V} \subseteq S$ . But $\mathscr V$ is a covering by criterion 2 of Tag 00VH, so $S \in J(X)$ .

If $J'$ is any other Grothendieck topology for which each $S_{\mathscr U}$ for $\mathscr U \in \mathcal Cov(\mathscr C)$ is covering, then $J'$ contains $J$ by criterion 2. $\qedsymbol$

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 $\mathscr C$ be a small category, and let $J \subseteq \mathbf{Siv}$ be a Grothendieck topology. Then a presheaf $\mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ is a sheaf if for any $X \in \mathscr C$ and any $S \in J(X)$ , the map $S \hookrightarrow h_X$ induces an isomorphism

$\operatorname{Hom}(h_X,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S,\mathscr F).$

Thus, a Grothendieck topology is an internal characterisation (inside $\mathbf{PSh}(\mathscr C)$ ) of which morphisms $S \to h_X$ one needs to localise to get $\mathbf{Sh}(\mathscr C,J)$ . In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.

Lovely little lemmas

Or maybe ‘amusing observations’

Grothendieck topologies (topologies 4/6)

Leave a Reply Cancel reply