I'm reading Aluffi - Algebra Chapter 0. In problem 3.10, he describes subobject classifiers: for some category $C$, a subobject classifier is an $\Omega$ with the property that for all $A$ in $Obj(C)$, the set of morphisms $Hom_C(A, \Omega)$ are in one-to-one correspondence with the "subobjects" (i.e. for $Set$, the elements) of $A$. He asks to find the subobject classifier for $Set$.
My issue with this is as follows: it would seem to be the case that $|A \to \Omega| = |A|$ for all $A \in Obj(C)$ (i.e. the number of functions from $A$ to $\Omega$ must equal the number of elements in $A$ -- otherwise, how could this be a one-to-one correspondence?). But if that's the case, then it seems like there is no candidate set. Consider the following argument:
- If $\Omega$ is an infinite set, there are an infinite number of morphisms $A \to \Omega$ for any $A \in Obj(C)$. Then no infinite set can be a subobject classifier (since, trivially, it will fail for any finite set).
- If $\Omega$ is a finite set of $n$ elements, then there are $n^{|A|}$ functions from $A$ to $\Omega$. Clearly there is no choice of $n$ such that $\forall m \in \mathbb{N}, n^m = m$. Then no finite set can be a subobject classifier.
Then since every object in $Set$ is either a finite or infinite set, it would seem to be the case that there is no subobject classifier $\Omega \in Obj(Set)$, which would disprove exactly what the reader is asked to prove.
It's possible that my argument is flawed, but I think more likely is that my understanding of the definition of a subobject classifier is off. What is it that I'm missing?
Keep in mind that I'm new to category theory and abstract algebra (I'm reading through Aluffi on my own), so plain-spoken responses are appreciated :)
A subobject of an object $A$ is (represented by) some monomorphism $B\to A$.
Monomorphisms in $Set$ are exactly the injective functions, moreover two monomorphisms are regarded as the same subobject iff their range coincide, so they correspond to subsets, not elements.
Either following your thought about the cardinalities, or by a direct guess, you can find the subobject classifier of $Set$ easily.
And, indeed, your argument is correct that there is no 'element classifier' for $Set$.