April | 2022 | Lovely little lemmas

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. $\qedsymbol$

In the first post of this series, we saw how subobjects of representable presheaves $h_U \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ correspond to sieves on $U$ . Last week, we saw how sieves give a convenient language for defining Grothendieck topologies on a small category. In next week’s (hopefully) final instalment of this series, we will generalise this even further to Lawvere–Tierney topologies on an arbitrary topos. Today’s post defines the last object we need to do this, which we will show generalises the presheaf $\mathbf{Siv}$ from last week:

Definition. Let $\mathscr C$ be a (possibly large) category with a terminal object $\mathbf{1}$ . Then a subobject classifier on $\mathscr C$ is a monomorphism $\mathbf{1} \to \Omega$ in $\mathscr C$ such that for every monomorphism $U \to X$ in $\mathscr C$ , there exists a unique arrow $X \to \Omega$ such that there is a pullback diagram

$\begin{array}{ccc}U & \to & \mathbf{1} \\ \downarrow & & \downarrow \\ X & \to & \Omega.\!\end{array}$

That is, $\mathbf{1} \to \Omega$ is the “universal” monomorphism in $\mathscr C$ , i.e. the pair $(\Omega,\mathbf{1} \hookrightarrow \Omega)$ represents the (possibly large) presheaf $X \mapsto \{\text{monomorphisms } U \hookrightarrow X\}/\cong$ , where $\cong$ denotes isomorphism in the slice category $\mathscr C/X$ . It is an easy exercise to show that any representative $(\Omega, T \hookrightarrow \Omega)$ of this presheaf actually has the form described above, i.e. $T$ is a terminal object (apply the uniqueness property above to the identity monomorphism $X \hookrightarrow X$ , and use the pullback square

$\begin{array}{ccc}T & = & T \\ \shortparallel & & \downarrow \\ T & \to & \Omega\end{array}$

coming from the hypothesis that $T \hookrightarrow \Omega$ is a monomorphism).

Example. If $\mathscr C = \mathbf{Set}$ , then the two-point set $\Omega = \{0,1\}$ with its natural inclusion $\mathbf 1 \to \Omega$ given by $1 \mapsto 1$ is a subobject classifier: the monomorphism $U \subseteq X$ corresponds to the indicator function $\delta_U \colon X \to \Omega$ that is $1$ on $U$ and $0$ on its complement. (In other situations I would denote this by $\mathbf 1_U$ , but that notation was already used in this series to denote the representable presheaf $h_U$ .)

It’s even more natural to take $\Omega$ to be the power set $\{\varnothing,\mathbf 1\}$ of $\mathbf{1}$ . As in the first post of this series, we think of $\mathbf 1$ representing “true” and $\varnothing$ representing “false”. The generalisation of the power set of $\mathbf{1}$ to presheaf categories is the presheaf $\mathbf{Siv}$ of subpresheaves of $h_X$ defined last week:

Lemma. Let $\mathscr C$ be a small category. Then the presheaf $\mathbf{Siv}$ together with the map $\mathbf 1 \to \mathbf{Siv}$ taking the unique section $1 \in \mathbf 1(X)$ to the maximal sieve $h_X \subseteq h_X$ for any $X \in \mathscr C$ is a subobject classifier in $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ .

Proof. Note that the prescribed map $\mathbf 1 \to \mathbf{Siv}$ is a morphism of presheaves, since the inverse image of the maximal sieve $h_X$ under any morphism $f \colon Y \to X$ in $\mathscr C$ is the maximal sieve $h_Y$ . Again using the notation from last week, if $\mathscr G \hookrightarrow \mathscr F$ is any monomorphism of presheaves, we get a morphism of presheaves $\phi \colon \mathscr F \to \Omega$ defined on $X \in \mathscr C$ by

$\begin{align*}\mathscr F(X) &\to \mathbf{Siv}(X) \\s &\mapsto (s \in \mathscr G).\end{align*}$

If $f \colon Y \to X$ is a morphism in $\mathscr C$ , then for any $Z \in \mathscr C$ we have

$\begin{align*}\big(f^*(s \in \mathscr G)\big)(Z) &= \{g \colon Z \to Y\ |\ fg \in (s \in \mathscr G)(Z)\} \\&= \{g \colon Z \to Y\ |\ (fg)^*(s) \in \mathscr G(Z)\} \\&= \{g \colon Z \to Y\ |\ g^*(f^*(s)) \in \mathscr G(Z)\} = (f^*s \in \mathscr G)(Z),\end{align*}$

showing that $f^*\phi(s) = \phi(f^*s)$ , so $\phi$ is indeed a natural transformation. We already noted last week that $(s \in \mathscr G) = h_X$ for $s \in \mathscr F(X)$ if and only if $s \in \mathscr G(X)$ , so $\mathscr G$ is the pullback

$\begin{array}{ccc}\mathscr G & \to & \mathbf{1} \\ \downarrow & & \downarrow \\ \mathscr F & \to & \mathbf{Siv}.\!\end{array}$

Conversely, if $\psi \colon \mathscr F \to \mathbf{Siv}$ is any morphism with this property and $s \in \mathscr F(X)$ , then $s \in \mathscr G(X)$ if and only if $\psi(s) = h_X$ , which together with naturality of $\psi$ gives

$\begin{align*}(s \in \mathscr G)(Y) &= \{f \colon Y \to X\ |\ f^*s \in \mathscr G(Y)\} \\&= \{f \colon Y \to X\ |\ \psi(f^*s) = h_Y\} \\&= \{f \colon Y \to X\ |\ f^*\psi(s) = h_Y\} \\&= \left\{f \colon Y \to X\ |\ \operatorname{id}_Y \in \big(f^*\psi(s)\big)(Y)\right\} \\&= \{f \colon Y \to X\ |\ f \circ \operatorname{id}_Y \in (\psi(s))(Y)\} = (\psi(s))(Y),\end{align*}$

so $\psi(s) = (s \in \mathscr G)$ . $\qedsymbol$

We will discuss some other properties of subobject classifiers in future posts.

Lovely little lemmas

Or maybe ‘amusing observations’

Monthly Archives: April 2022

Lawvere–Tierney topologies (topologies 6/6)

Subobject classifiers on presheaf categories (topologies 5/6)