Subobject classifiers on presheaf categories (topologies 5/6)

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.

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