Counterexample in topos theory

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

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).

• Thanks for the answer. I've been thinking about it and I think it's not at all obvious that it is correct, in the sense that we would have to prove that 1) the usual 2-element set is a subobject classifier and 2) the power object corresponds to the set of subsets in this "new" category. It could be possible that this "new" category $\textbf{NumSet}$ doesn't inherit these properties from $\textbf{Set}$. But it is obviously the best hint and there are easy proofs replicating those of $\textbf{Set}$. Thanks! Commented Sep 27, 2023 at 8:57

An interesting counter-example is the category of pointed sets whose objects are non-empty sets equipped with a designated element, and whose morphisms are functions that send the designated element of one set to the designated element of another set.

Subsets containing the designated element correspond bijectively to functions to $$\{0,1\}$$ mapping the designated element to $$1$$ in $$\{0,1\}$$. Thus, $$\{0,1\}$$ with $$1$$ a designated element is a subobject classifier $$\Omega$$ for pointed sets.

However, the only power object is (up to isomorphism) the subobject classifier itself. To see this, first note that the universal property of the power object $$PX$$ is that of an exponential object $$[X,Y]$$ where $$Y=\Omega$$: a morphism $$\epsilon_{X,Y}\colon X\times[X,Y]\to Y$$ so that any morphism $$f\colon X\times Z\to Y$$ factors as $$\epsilon_{X,Y}\circ X\times\bar f\colon X\times Y\to X\times[X,Y]\to Y$$ for a unique $$\bar f\colon Z\to Y$$. T

In particular, when $$Z$$ is a terminal object, so that the projection $$X\times Z\to X$$ is an isomorphism, we have that morphisms $$X\to Y$$ correspond bijectively to o morphisms $$X\times Z\to Y$$, and hence bijectively to morphisms $$Z\to[X,Y]$$.

But singletons are not only terminal objects in the category of pointed sets and also a initial objects: any pointed set admits a unique morphism from a singleton sending the singleton's necessarily designated element to the designated element of the pointed set.

This in a pointed category with products (where a terminal object is initial), an exponential object $$[X,Y]$$ exists only if there is a unique morphism $$X\to Y$$. In particular, in the presence of a subobject classifier, a power object $$PX\cong[X,\Omega]$$ exists only if there is a unique morphism $$X\to\Omega$$, i.e. only if $$X$$ has a unique subobject. But because the category is pointed, any $$X$$ admits two monomorphisms: $$1\to X$$ and $$\mathrm{id}_X\colon X\to X$$. These determine the same object if and only if $$1\to X$$ is an isomorphism, i.e. $$X$$ is terminal. Thus the only power objects that a pointed category has are those of the terminal/initial object, i.e. subobject classifiers.

This example is interesting because the set of subsets containing the designated element of a pointed set is itself pointed by making the imporper subset a designated element. More generally, the set of pointed functions between two pointed sets is pointed by taking the function sending all elements to the designated element as a designated function.

In fact, what is happening is that the category of pointed sets is an example of a monoidal closed category with a subobject classifier, whereas a topos is a cartesian closed category with a subobject classifier.

• This is such a great answer. Prime example of the type of thing you can only find on math.stackexchange! Commented Feb 14 at 17:37
• Actually, is that true? Do you know of any references for this? Commented Feb 14 at 17:38
• Sorry, don't know any references for this. Commented Feb 14 at 17:53