commutative algebra - Hom functor in an abelian category?

This is a very basic question about definitions, but I haven't been able to find the answer to it online. If we let $\mathscr{A}$ be an Abelian category, then for any object $A\in\mathscr{A}$, we can define the Hom functor $\mathrm{Hom}_\mathscr{A}(A,-)$.

What category does this take values in? I know that it can take values in $\mathbf{Ab}$, but in the case where $\mathscr{A}=R\hbox{-}\mathbf{Mod}$, the functor seems to take values in $\mathscr{A}$. Which convention holds in general?

2 Answers

  1. Leander- Reply

    2019-11-14

    By definition, it takes values in $Ab$.

    The special thing about $\mathscr{A}=_RMod$ is that it is an enriched category, enriched over itself, see this wonderful page or Kelly's book.

  2. Lee- Reply

    2019-11-14

    In a $V$-enriched category, the natural "hom object" of natural transformations between functors $F$ and $G$ is the end $\int V(F,G)$. As a special limit in $V$, it is an object of $V$.

    In the Ab-enriched case, modules are functors into Ab, and so hom-objects between them (that is, module homeomorphisms) also form an abelian group.

    They also admit a module structure over the center of the domain.

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