Subterminal presheaves and sheaves (topologies 1/6)

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 \mathscr C be a small category, and write \mathbf{PSh}(\mathscr C) for the functor category [\mathscr C^{\operatorname{op}},\mathbf{Set}].

Definition. A sieve on an object X \in \mathscr C is a subpresheaf S \subseteq h_X of the representable presheaf h_X = \operatorname{Hom}(-,X).

Concretely, this means that each S(U) is a set of morphisms f \colon U \to X with the property that if g \colon V \to U is any morphism, then fg \colon V \to X is in S(V). Thus, this is like a “right ideal in \operatorname{Hom}(-,X)“. Since \mathscr C is small, we see that sieves on X form a set, which we will denote \mathbf{Siv}(X).

Lemma. Let \mathscr D and \mathscr C be small categories, X \in \mathscr D an object, and F \colon \mathscr D \to \mathscr C a functor. Then there is a pullback map

    \[F^* \colon \mathbf{Siv}(F(X)) \to \mathbf{Siv}(X)\]

defined by

    \[F^*S(U) = \big\{f \colon U \to X\ \big|\ F(f) \in S(F(U))\big\}.\]

If \mathscr D = \mathscr C/X and F is the forgetful functor, then F^* gives a bijection

    \[\mathbf{Siv}(X) \stackrel\sim\to \mathbf{Siv}(X \stackrel{\operatorname{id}}\to X).\]

Proof. If S is a sieve, then so is F^*S since f \in F^*S(U) and g \in \operatorname{Hom}(V,U) implies F(fg) = F(f)F(g) \in S(F(V)), so fg \in F^*S(V). For the second statement, given a sieve T on X \to X, define the sieve S on X by

    \[S(U) = \left\{ f \colon U \to X\ \ \left|\ \ \left(\begin{array}{ccccc}\!\!U\!\!\!\!\! & & \!\!\!\!\!\stackrel f\longrightarrow\!\!\!\!\! & & \!\!\!\!\!X\!\! \\ & \!\!\!{\underset{f}{}}\!\!\searrow\!\!\!\! & & \!\!\!\!\swarrow\!\!{\underset{\operatorname{id}}{}}\!\!\!\! & \\[-.3em] & & X.\! & & \end{array}\right) \in T\left(U \stackrel f\to X\right)\right\}\right..\]

Then S is a sieve on X, and is the unique sieve on X such that F^*S = T. \qedsymbol

Beware that the notation F^*S could also mean the presheaf pullback S \circ F, but we won’t use it as such.

Remark. In particular, it suffices to study the case where \mathscr C has a terminal object, which we will denote by X (in analogy with the small Zariski and étale sites of a scheme X, which have X as a terminal object). We are thus interested in studying the subobjects of the terminal presheaf \mathbf{1}_X. We will do so both in the case of presheaves and in the case of sheaves. Note that \mathbf 1_X is a sheaf: for any set I (empty or not), the product \prod_{i \in I} \{*\} is a singleton, so the diagrams

    \[\mathscr F(U) \to \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i,j\in I} \mathscr F\left(U_i \underset U\times U_j\right)\]

are vacuously equalisers whenever \{U_i \to U\}_{i \in I} is a covering (or any collection of morphisms).

Definition. A property \mathcal P on a set A is a function \mathcal P \colon A \to P(\{*\}) to the power set of a point \{*\}. The property \mathcal P holds for a \in A if \mathcal P(a) = \{*\}, and fails if \mathcal P(a) = \varnothing.

Given a property \mathcal P on the objects of a small category \mathscr C, we say that \mathcal P is left closed if for any morphism f \colon U \to V, the implication \mathcal P(V) \Rightarrow \mathcal P(U) 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 \mathscr C is a site (a small category together with a Grothendieck pretopology), we say that \mathcal P is local if it is left closed, and for any covering \{U_i \to U\}_{i \in I} in \mathscr C, if \mathcal P(U_i) holds for all i \in I, then \mathcal P(U) holds.

Lemma. Let \mathscr C be a small category with a terminal object X.

  1. Giving a subpresheaf of \mathbf 1_X is equivalent to giving a left closed property \mathcal P on the objects of \mathscr C.
  2. If \mathscr C is a site, then giving a subsheaf of the presheaf \mathbf 1_X is equivalent to a giving a local property \mathcal P.

A homotopy theorist might say that a local property is a (-1)-truncated sheaf [of spaces] on \mathscr C.

Proof. 1. The terminal presheaf \mathbf 1_X takes on values \{*\} at every U \in \mathscr C, thus any subpresheaf \mathscr F takes on the values \varnothing and \{*\}, hence is a property \mathcal P on the objects of \mathscr C. The presheaf condition means that for every morphism f \colon U \to V, there is a map f^* \colon \mathscr F(V) \to \mathscr F(U), which is exactly the implication \mathcal P(V) \Rightarrow \mathcal P(U) since there are no maps \{*\} \to \varnothing.

Alternatively, one notes immediately from the definition that a sieve on an object X \in \mathscr C is the same thing as a subcategory of \mathscr C/X which is left closed.

2. Being a subpresheaf translates to a left closed property \mathcal P by 1. Then \mathscr F is a sheaf if and only if, for every covering \{U_i \to U\}_{i \in I} in \mathscr C, the diagram

    \[\mathscr F(U) \to \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i, j \in I} \mathscr F\Big(U_i \underset U\times U_j\Big)\]

is an equaliser. If one \mathscr F(U_i) is empty, then so is \mathscr F(U) since \mathcal P is left closed, so the diagram is always an equaliser.

Thus, in the sheaf condition, we may assume \mathscr F(U_i) = \{*\} for all i \in I, i.e. \mathcal P(U_i) holds for all i \in I. Since \mathcal P is left closed, this implies that \mathscr F(U_i \times_U U_j) = \{*\} for all i, j \in I, so the two arrows agree on \prod_i \mathscr F(U_i), and the diagram is an equaliser if and only if \mathscr F(U) = \{*\}. Running over all coverings \{U_i \to U\} in \mathscr C, this is exactly the condition that \mathcal P is local. \qedsymbol

Leave a Reply

Your email address will not be published.