I'm looking at translations of statements involving functions into the language of category theory, and it seems that there are at least two category-theoretic notions collecting functions between objects $A$ and $B$: exponential objects $B^A$, and $\mathrm{Hom}(A,B)$.

In $\mathbf{Set}$ these notions coincide, which leads me to wonder if the same happens in other categories. Since exponential objects remain in the same category, this seems only to be a sensible question in categories enriched over themselves. I know of two other such categories: $\mathbf{Cat}$, small categories, in which I believe the notions still coincide (although I haven't checked the details, I assume the notation $\mathcal D^\mathcal C$ for functor categories is no accident), and $\mathbf{Ab}$, abelian groups.

Now I'm a little stuck because hom-sets in $\mathbf{Ab}$ do have an abelian-group structure, but they aren't exponential objects, and indeed $\mathbf{Ab}$ has almost no exponentials. My guess is that this is essentially because the evaluation map $(f,x) \mapsto f(x)$ isn't a homomorphism of groups.

My question is: are there examples in which hom-objects and exponentials both exist, but are non-isomorphic objects of the same category?

  • 1
    $\begingroup$ For $\bf{Ab}$ instead of the direct product, one usually considers the tensor product for the 'monoidal structure'. Then $f\otimes x \mapsto f(x)$ will be a homomorphism, and the 'exponentials' will be present as right adjoint. $\endgroup$
    – Berci
    Jan 27, 2013 at 0:03

1 Answer 1


Sure. Let $G$ be a non-trivial abelian group. Then the category of sets with a left $G$-action, $\mathbf{B} G$, is a symmetric monoidal closed category in two different ways. First, $\mathbf{B} G$ is a topos, so it is cartesian closed. The exponential in $\mathbf{B} G$ turns out to be essentially the same as in $\textbf{Set}$, in the sense that the forgetful functor $\mathbf{B} G \to \textbf{Set}$ preserves exponentials. In particular, the underlying set of the exponential $Y^X$ in $\mathbf{B} G$ is in general not the hom-set $\textrm{Hom}_G(X, Y)$!

Nonetheless, the hom-set $\textrm{Hom}_G(X, Y)$ does have a left $G$-action, namely the pointwise action inherited from $Y$. (This is where we use the fact that $G$ is abelian.) Moreover:

  • The left $G$-action on $\textrm{Hom}_G(X, Y)$ is natural in $X$ and $Y$.
  • $\textrm{Hom}_G(X, -)$ has a left adjoint, $- \otimes_G X$.
  • $\otimes_G$ and $\textrm{Hom}_G$ together make $\mathbf{B} G$ into a symmetric monoidal closed category.
  • $\begingroup$ Hmm. Any input on which is the "correct" notion of function in this case? (I realise this is a vague question, which is why it's only a comment) $\endgroup$ Jan 26, 2013 at 15:56
  • $\begingroup$ The internal logic of toposes is better understood than the internal logic of general symmetric monoidal closed categories, as far as I know. $\endgroup$
    – Zhen Lin
    Jan 26, 2013 at 18:12
  • $\begingroup$ Hmm.. My intuition would lead towards the $\otimes$ version for finding the 'right internal hom'.. $\endgroup$
    – Berci
    Jan 27, 2013 at 0:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.