Exercise 4

Theorem: An equivalence of categories preserves a subobject classifier

Let and be equivalent, with a subobject classifier in . Let and be the equivalence. We may choose and to be an adjoint equivalence, so that , and and are natural isomorphisms.

What we wish to show is that – this means precisely that is a subobject classifier in ( is a representing object for the subobject functor). But since and are adjoint, the right-hand functor is isomorphic to . So, it is sufficient to show that .

Indeed, we will define a natural transformation as . Notice that this assigns to an arrow precisely its transpose under the adjunction . This shows that each is an isomorphism (it is precisely the homset bijection of the adjunction), so we need only show that the transformation is natural. Keeping in mind that operates on arrows “by pullback”, we have the naturality diagram for :

Take a subobject at the top-right. Following the top left path, we pullback along to obtain , and then take the transpose to get . Following the other direction, we take the transpose to obtain , and then we must pull this back along . To say that that these are the same amounts to saying that the outer rectangle of this combined diagram is a pullback:

Indeed, the top square is a pullback because it is the image under of a pullback, and viewing as a right adjoint (after inverting the unit and counit) means it preserves limits. The bottom square can be shown a pullback directly by considering another cone . Simply define . We have immediately. For the top:

Uniqueness of is immediate from the definition.

The pasting lemma then confirms that the outer rectangle is a pullback, and thus we have shown that the is natural and thus a natural isomorphism, completing the proof.