Lawvere–Tierney topologies (topologies 6/6)

In the post of two weeks ago, we showed how Grothendieck topologies form a neat framework for the categorical aspects of the more familiar (at least to algebraic geometers) Grothendieck pretopologies. In this final post of the series, we take this one step further, to the notion of a Lawevere–Tierney topology on an arbitrary elementary topos \mathscr E.

Definition. An elementary topos is a category \mathscr E that has finite limits, is Cartesian closed, and has a subobject classifier (see last week’s post).

The only example we’re interested in today is the category \mathscr E = [\mathscr C^{\operatorname{op}},\mathbf{Set}] of presheafs (of sets) on a small category \mathscr C, which we saw last week has subobject classifiers. But in fact, any Grothendieck topos (sheaves of sets on a (small) site) is an example of an elementary topos, so one could even introduce further topologies on those.

Example. The category \mathbf{Set}_{\text{fin}} of finite sets is an elementary topos, but not a Grothendieck topos since it is neither complete nor cocomplete.

Before giving the definition, we need to define one more structure on \Omega: the meet. Recall that the intersection (or meet) of two monomorphisms U \hookrightarrow X \hookleftarrow V is the fibre product

    \[U \wedge V := U \underset X\times V \hookrightarrow X.\]

The intersection of \mathbf 1 \times \Omega \hookrightarrow \Omega \times \Omega and \Omega \times \mathbf 1 \hookrightarrow \Omega \times \Omega is the monomorphism \mathbf 1 \hookrightarrow \Omega \times \Omega given by (t,t), which is classified by a map \wedge \colon \Omega \times \Omega \to \Omega. Since \mathbf 1 \to \Omega is the universal monomorphism, we see that \mathbf 1 \hookrightarrow \Omega \times \Omega is the universal intersection of two subobjects, i.e. if U \to X and V \to X are classified by f \colon X \to \Omega and g \colon X \to \Omega respectively, then U \wedge V is classified by the composition

    \[X \overset{(f,g)}\longrightarrow \Omega \times \Omega \overset\wedge\longrightarrow \Omega.\]

(If we denote this simply by f \wedge g \colon X \to \Omega, then \wedge \colon \Omega \times \Omega \to \Omega is \operatorname{pr}_1 \wedge \operatorname{pr}_2.)

Definition. Let \mathscr E be an elementary topos with subobject classifier t \colon \mathbf 1 \to \Omega. Then a Lawvere–Tierney topology on \mathscr E is a morphism j \colon \Omega \to \Omega such that the following diagrams commute:

    \[\begin{array}{ccc}\mathbf 1\!\! & \stackrel t\to\!\! & \!\!\!\Omega \\ & \!\!\underset{t\!\!\!}{}\searrow\!\! & \downarrow j\!\! \\[-.2em] & & \!\!\!\Omega,\!\!\end{array}\qquad\qquad\begin{array}{ccc}\Omega\!\! & \stackrel j\to\!\! & \!\!\!\Omega \\ & \!\!\underset{j\!\!\!}{}\searrow\!\! & \downarrow j\!\! \\[-.3em] & & \!\!\!\Omega,\!\!\end{array}\qquad\qquad\quad\begin{array}{ccc}\Omega \times \Omega\!\!\! & \stackrel\wedge\to\!\! & \!\!\!\Omega \\ \!\!\!\!\!\!\!\!\!\!\!\!j\times j \downarrow & & \downarrow j\!\! \\ \Omega \times \Omega\!\!\! &\stackrel\wedge\to\!\! & \!\!\!\Omega.\!\!\end{array}\]

We saw two weeks ago that a Grothendieck topology is a certain subpresheaf J \subseteq \mathbf{Siv}, and last week that \mathbf{Siv} is a subobject classifier \Omega on [\mathscr C^{\operatorname{op}},\mathbf{Set}]. Thus a subpresheaf J \subseteq \Omega is classified by a morphism j \colon \Omega \to \Omega, which we saw last week is given by S \mapsto (S \in J).

Lemma. The subpresheaf J \subseteq \Omega is a Grothendieck topology on \mathscr C if and only if j \colon \Omega \to \Omega is a Lawvere–Tierney topology on [\mathscr C^{\operatorname{op}},\mathbf{Set}]. In particular, Grothendieck topologies on \mathscr C are in bijective correspondence with Lawvere–Tierney topologies on [\mathscr C^{\operatorname{op}},\mathbf{Set}].

Thus Lawvere–Tierney topologies are an internalisation of the notion of Grothendieck topology to an arbitrary elementary topos \mathscr E.

Proof of Lemma. By definition of the morphism j, we have a pullback square

    \[\begin{array}{ccc}J & \to & \mathbf 1 \\ \downarrow & & \downarrow \\ \Omega & \stackrel j\to & \Omega.\!\end{array}\]

The first commutative diagram in the definition above means that the top arrow has a section \mathbf 1 \to J such that the composition \mathbf 1 \to J \hookrightarrow \Omega is t, i.e. \mathbf 1 \subseteq J as subobjects of \Omega. Since t \colon \mathbf 1 \to \Omega is the map taking 1 \in \mathbf 1(X) to the maximal sieve h_X \subseteq h_X for any X \in \mathscr C, this means exactly that h_X \in J(X) for all X \in \mathscr C, which is condition 1 of a Grothendieck topology. For the second, consider the pullback

    \[\begin{array}{ccccc}J' & \to & J & \to & \mathbf 1 \\ \downarrow & & \downarrow & & \downarrow \\ \Omega & \stackrel j\to & \Omega & \stackrel j\to & \Omega.\!\end{array}\]

The condition jj=j means that J' \cong J as subobjects of \Omega. We already saw that \mathbf 1 \subseteq J for a Grothendieck or Lawvere–Tierney topology, so pulling back along j gives J \subseteq J'. Thus the second diagram in the definition of a Lawvere–Tierney topology commutes if and only if J' \subseteq J, i.e. if S \in \Omega(X) = \mathbf{Siv}(X) with j_X(S) \in J(X), then S \in J(X). But j_X \colon \Omega(X) \to \Omega(X) is given by S \mapsto (S \in J), so this is exactly axiom 3 of a Grothendieck topology.

For the third diagram, we first claim that j_X \colon \Omega(X) \to \Omega(X) is monotone for all X \in \mathscr C if and only if J satisfies axiom 2 of a Grothendieck topology. Indeed, if j is monotone and S, S' \in \Omega(X) satisfy S \subseteq S' and S \in J(X), then the inclusion h_X = (S \in J) \subseteq (S' \in J) shows (S' \in J) = h_X, so S' \in J(X) by axiom 3. Conversely, if J satisfies axiom 2 and S,S' \in \Omega(X) satisfy S \subseteq S', then for any f \colon Y \to X we have f^*S \subseteq f^*S', so f^*S \in J(Y) \Rightarrow f^*S' \in J(Y), i.e. (S \in J) \subseteq (S' \in J).

The third diagram in the definition above says that the map j_X \colon \Omega(X) \to \Omega(X) given by S \mapsto (S \in J) is a morphism of meet semilattices. This implies in particular that j_X is monotone, as S \subseteq S' if and only if S \wedge S' = S, so the third diagram above implies axiom 2 of a Grothendieck topology.

Conversely, if J is a Grothendieck topology, then axiom 2 implies that j_X \colon \Omega \to \Omega is monotone. In particular, j_X(S \cap T) \subseteq j_X(S) \cap j_X(T) for any S, T \in \Omega(X), since S \cap T \subseteq S, T. For the reverse implication, if f \colon Y \to X satisfies f \in (S \in J)(Y) \cap (T \in J)(Y), then f^*S \in J(Y) and f^*T \in J(Y), so the remark of two weeks ago shows that f^*(S \cap T) \in J(Y), i.e. f \in ((S \cap T) \in J)(Y). We see that j_X(S \cap T) = j_X(S) \cap j_X(T), showing that j_X is a morphism of meet semilattices. \qedsymbol

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