I’m currently reading Sheaves in Geometry and Logic, by MacLane and Moerdijk. Possibly, subobject classifiers are going to be important, so here are some notes to make sure I understand this stuff correctly. The text of the book is pretty clear, but a lot of details are glossed over.
Recall that in any category, a subobject of some object is an equivalence class of monic arrows , where two such arrows are said to be equivalent if and only if there is an isomorphism between their domain that makes the obvious triangle commute. Notice that because we’re talking about monic arrows, if there is such an isomorphism making the triangle commute, then that isomorphism is necessarily the only one making the triangle commute.
Now suppose our ambient category has all finite limits (including the limit of the empty diagram, that is, the terminal object ). In this context only, a subobject classifier is a monic arrow such that, for any monic arrow , there exists a unique which makes the following diagram into pullback diagram (cartesian square):
An important thing to notice here is that the “characteristic function” must actually be the same arrow for all representatives of a single subobject. Indeed, suppose is another monic such that there exists an isomorphism such that (that is, suppose and are both representatives of the same subobject of ). There exists a having the pullback square property of the previous definition with respect to . Now one can show that (caution: we use as the lower morphism) is a pullback square. Hence by the unicity clause of the definition, we must have .
Recall that is the class of all subobject of . By the preceeding remark, we have a well-defined function which sends a subobject to its “characteristic function” . In fact, this is a bijection! It is a surjection because the pullback of (or more generally any monic) along any morphism is a monic arrow, and so represents a subobject of . It is an injection because any two pullbacks of are isomorphic. Consequently, if the ambient category is locally small, then it is also well-powered (recall this means is a set for each object ). With the category being locally small, we have even more: the collection of all bijections assemble into a natural isomorphism of (contravariant) functors In other words, the functor is representable. Let’s prove this claim. Recall that the action of the functor on morphisms is by “pulling back”: the arrow is defined to be the set function which sends a subobject (represented by) to the subobject represented by , the pullback of along . This is well-defined (independant of the choice of representative for a given subobject). We can paste two pullback squares to obtain a bigger pullback square (rectangle): By the unicity clause in the definition of the subobject classifier, we must have . But this equation means exactly that the following diagram is commutative: Hence the isomorphism is natural as claimed, so is a representable functor.
In light of the previous discussion, the obvious, reciprocal, question is: for a category to have a subobject classifier, is it enough for the subobject functor to be representable? The fact that this is true is Proposition 1 at page 33 of SGL:
Proposition. A locally small category with all finite limits has a subobject classifier if and only if the subobject functor is representable. When that is the case, the category is well-powered.
Proof. The discussion in previous paragraphs shows the necessity of the representability of the subobject functor. To show its sufficience, suppose there exists a representation for some representative object . We need to show there exists a subobject classifier. Let be the “universal element” for the representation, that is, is the subobject corresponding to the identity arrow . By the Yoneda lemma, any arrow corresponds to the subobject . Therefore, for any subobject of , there exists a unique arrow such that . Because the action of the subobject functor on an arrow is to pullback along it, the arrow is the unique arrow making the following a cartesian square: We are almost done. For to be a subobject classifier, it is not enough that any monic arrow is the pullback of a unique characteristic function, as is the case here. Additionally, we need to be the terminal object in the category: there needs to be exactly one arrow from any object into . The pullback square associated with (i.e. seen as a subobject of itself) gives us an arrow for any object , so we know there’s always at least one. Suppose we have two arrows . Then the two following squares are pullback squares: Therefore , which implies by unicity. Since is a monic arrow, this yields . Consequently there is at most one arrow from any object to . Since we’ve already shown there’s at least one, this means is a (the) terminal object, hence is a subobject classifier.
As is true for any representation, there is an isomorphism between any two representatives and , and it is the unique isomorphism which commutes with the representations. More precisely, pre-composition with yields a commutative diagram of functors and natural isomorphisms Let and be universal elements for representations by and , respectively. Then, following both paths where the element goes yields . This means we have a pullback square In the proof above, we saw that and are actually both the terminal object in our ambient category. Hence the top arrow in the previous diagram is an isomorphism. Therefore, a subobject classifier is unique up to (unique) isomorphism. From now on, we will say the subobject classifier, when it exists.