Motivation: The definition of an elementary topos requires both a subobject clasiffier and either power objects or exponentiation. But also, if a category has power objects and a terminal object $\mathbf{1}$ then (according to Kock and Mikkelsen) it has a subobject classifier and it is $P\mathbf{1}$.
Question: Is the opposite true? Does a subobject classifier imply the existence of power objects? I wouldn't know how to start such a proof. The $\Omega$-axiom of the subobject classifier doesn't seem strong enough to enable a construction of some object $Px$ and morphism $\in_x:x\times Px\to\Omega$ such that… (here I would use the definition of power objects by Goldblatt. Kock and Mikkelsen use an alternative one, also explored by Goldblatt, which doesn't use a subobject classifier)
I can't find much online or in the reference books
Thanks in advance!
Best Answer
A simple counterexample is the category of countable sets. This has a subobject classifier (the usual 2-element set) but a countably infinite set does not have a power object since its power object would have to have uncountably many points (since it has uncountably many subobjects).