In the first post of this series, we saw how subobjects of representable presheaves correspond to sieves on . 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 from last week:
Definition. Let be a (possibly large) category with a terminal object . Then a subobject classifier on is a monomorphism in such that for every monomorphism in , there exists a unique arrow such that there is a pullback diagram
That is, is the “universal” monomorphism in , i.e. the pair represents the (possibly large) presheaf , where denotes isomorphism in the slice category . It is an easy exercise to show that any representative of this presheaf actually has the form described above, i.e. is a terminal object (apply the uniqueness property above to the identity monomorphism , and use the pullback square
coming from the hypothesis that is a monomorphism).
Example. If , then the two-point set with its natural inclusion given by is a subobject classifier: the monomorphism corresponds to the indicator function that is on and on its complement. (In other situations I would denote this by , but that notation was already used in this series to denote the representable presheaf .)
It’s even more natural to take to be the power set of . As in the first post of this series, we think of representing “true” and representing “false”. The generalisation of the power set of to presheaf categories is the presheaf of subpresheaves of defined last week:
Lemma. Let be a small category. Then the presheaf together with the map taking the unique section to the maximal sieve for any is a subobject classifier in .
Proof. Note that the prescribed map is a morphism of presheaves, since the inverse image of the maximal sieve under any morphism in is the maximal sieve . Again using the notation from last week, if is any monomorphism of presheaves, we get a morphism of presheaves defined on by
If is a morphism in , then for any we have
showing that , so is indeed a natural transformation. We already noted last week that for if and only if , so is the pullback
Conversely, if is any morphism with this property and , then if and only if , which together with naturality of gives
We will discuss some other properties of subobject classifiers in future posts.