In the category of R-Modules, for a given ring R, The set of all morphism between two objects is an object in the category. Obviously this is also true in the category of sets.

Are there more categories in which this is true?

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    $\begingroup$ Any closed monoidal category $C$ has internal hom-objects coincide with the "external hom set", i.e., the $\mathrm{hom}_{C}(X,Y)$ is an object in $C$. $\endgroup$ – Alex Nelson Jul 28 '17 at 14:02
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    $\begingroup$ @user well, surely you need to qualify the statement for modules some more. If $_RM_S$ is a $(R, S)$ bimodule, and $_RN_T$ is an $(R,T)$ bimodule, then $Hom(_RM, _RN)$ has a natural $(S,T)$ bimodule structure, but no inherent $R$ module structure. I guess you were only thinking of commutative $R$, perhaps? $\endgroup$ – rschwieb Jul 28 '17 at 14:33

As Alex Nelson mentioned in a comment, categories with this property are called closed monoidal categories. Closed monoidal categories have an axiomatic characterization; naturally, some categories have this extra structure and some don't.

Firstly, a monoidal category is a category $C$ together with a functor $- \otimes - : C \times C \to C$ and distinguished object $1$. These must satisfy axioms along the lines of "$\otimes$ is associative and $1$ is the identity of $\otimes$." (You have to think through the details of how to write that down, since $\otimes$ is a functor on a category rather than an operation on a set; details at the link.)

There are plenty of examples of monoidal categories:

  • If $C$ has a terminal object and finite products, you can use that structure for $\otimes$ and 1 (this gives lots of examples)
  • Modules / vector spaces with tensor product
  • Modules / vector spaces with the direct product
  • Pointed sets with the smash product
  • Pointed sets with the direct product
  • Given any category $C$, the category of functors $C \to C$ forms a monoidal category where $1$ is the identity functor and $\otimes$ is composition of functors.
  • Given a group $G$, there is a category with a single object $\star$, where the morphisms $\star \to \star$ are the elements of the group and composition is given by multiplication in the group. Exercise: if $G$ is Abelian, there is a natural way to view this as a monoidal category.

A monoidal category is closed if there is a moreover a functor $[-,-] : C^{\operatorname{op}} \times C \to C$ such that \begin{equation*} \operatorname{Hom}(X,[Y,Z]) \cong \operatorname{Hom}(X \otimes Y,Z) \end{equation*} for all objects $X$, $Y$, and $Z$. (The bijection has to be natural in these objects, but that usually isn't hard to verify.)

Note that it is possible to prove that, if such a functor $[-,-]$ exists, it is unique up to isomorphism, so this is really a property that monoidal categories can have, not extra structure. Try and see if you can figure out which of the above examples are closed.

If $[-,-]$ exists, it serves as an "internal $\operatorname{Hom}$" object: in any monoidal category, we have a functor $\operatorname{Hom}(1,-) : C \to \operatorname{Set}$ that sends an object to its "elements." If $C$ is closed, then the "elements" of $[X,Y]$ are just the morphisms $X \to Y$: \begin{align*} \operatorname{Hom}(1,[X,Y]) &\cong \operatorname{Hom}(1 \otimes X, Y) \\ &\cong \operatorname{Hom}(X,Y) \end{align*}

Accordingly, being a closed monoidal category is the correct notion for when you can say "the hom-sets are an object of the category." I hope this exposition has cleared things up and given you more to think about!

  • $\begingroup$ @user269559 Note that in most situations where you want to view hom-sets as objects, you already have a forgetful functor $U:C \to \mathbf{Set}$ (for $R$-modules for example this is just the usual underlying-set functor). So you have to make sure that this $U$ coincide with $\hom(1,-)$ (which is the case for modules, or sets, but can fail in other usual cases like group representations). $\endgroup$ – Pece Jul 29 '17 at 6:00

In addition to the information provided by Mike Haskel about closed monoidal categories (and the more general notion of monoidal categories), you should also look into the more general notion of closed categories. These have an internal Hom functor $[-,-]$ but need not have a left adjoint $-\otimes A$ for $[A,-]$. So they match more closely the exact formulation in the question. The theories of both closed and monoidal categories were presented in considerable detail by Eilenberg and Kelly in a paper "Closed categories" from 1966; Fred Linton's review on MathSciNet at MR0225841 (37 #1432) gives a good overview of the central ideas.


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