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

  1. For all X \in \mathscr C, the maximal sieve h_X \subseteq h_X is in J(X).
  2. If S \in J(X) and S' \in \mathbf{Siv}(X) with S \subseteq S', then S' \in J(X).
  3. 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.

Leave a Reply

Your email address will not be published.