Is the category of modules over an abstract monoid enriched?

Question

If $A$ is an algebra over $\mathbb C$, or, in other words, a monoid in the closed monoidal category $_{\mathbb C}\operatorname{Vect}$ of all vector spaces over $\mathbb C$, then, clearly, the category $_A\operatorname{Vect}$ of all left $A$-modules is enriched over the category $_{\mathbb C}\operatorname{Vect}$.

Is the same true if we replace $_{\mathbb C}\operatorname{Vect}$ by an arbitrary monoidal closed category $V$?

I.e.,

if we consider an arbitrary closed monoidal category $V$ and take an arbitrary monoid $A$ in $V$, will the category $_A V$ of all left $A$-modules be an enriched category over $V$?

Answer 1

You need that $V$ has equalizers (but for many basic stuff in the theory of $V$-enriched categories we need $V$ to be complete and cocomplete anyway). I denote internal Homs by $\underline{\mathrm{Hom}}$.

If $\underline{M} = (M, h : A \otimes M \to M)$, $\underline{N} = (N , k : A \otimes N \to N)$ are left $ A$-modules, define $\underline{\mathrm{Hom}}_A(\underline{M},\underline{N})$ as the equalizer of

$$\underline{\mathrm{Hom}}(M, N) \xrightarrow{h^*} \underline{\mathrm{Hom}}(M \otimes A,N)$$

and $$\underline{\mathrm{Hom}}(M,N) \xrightarrow{(\check{k})_*} \underline{\mathrm{Hom}}(M,\underline{\mathrm{Hom}}(A,N)) \xrightarrow{\cong} \underline{\mathrm{Hom}}(M \otimes A,N).$$

A reference for this is Proposition 1.2.17 in F. Marty, Des ouverts Zariski et des morphismes lisses en géométrie relative.

Answer 2

Fact: an homomorphism of $A$-modules (for a monoid $A$ in a monoidal category $V$) is a map $f : M \to N$ such that $f(a.m)=a.fm$, or in purely diagrammatic form $f\circ \rho = \rho'\circ (A\otimes f)$, where $\rho : A \otimes M \to M$ and $\rho' : A\otimes N \to N$ are the actions that make $M,N$ two $A$-modules.

But then, the subobject of "homomorphisms of modules" is just the subobject of the hom-object $V(M,N)$ defined as the equaliser of two maps $$ \rho^* : V(M,N) \to V(A\otimes M,N) $$ and $$ \rho'_* \circ (A\otimes \_) : V(M,N) \to V(A\otimes M,A\otimes N) \to V(A\otimes M,N) $$

This definition gives you precisely what you want when the hom-objects are sets, but id "elementless".

Also, I'm using a general construction in a particular case: $A$-modules are algebras for the monad $T_A=A\otimes \_$, and $A$-module homomorphisms are $T_A$-algebra morphisms. If $T$ is any monad on a category $V$, the subobject of $T$-algebra morphisms between two $T$-algebras $(M, m : TM \to M)$ and $(N, n : TN \to N)$ is the equaliser of the pair of morphisms $$ m^* : V(M,N) \to V(TM,N) \qquad n_* \circ T_{MN} : V(M,N) \to V(TM,TN) \to V(TM, N) $$