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. $\qedsymbol$

Lovely little lemmas

Or maybe ‘amusing observations’

Lawvere–Tierney topologies (topologies 6/6)

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