# Is the category of modules over an abstract monoid enriched?

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 $$V$$?

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.

• Martin, excuse me, for constructing this morphism $$\operatorname{Hom}(M,N)\to \operatorname{Hom}(M,\operatorname{Hom}(A,N)),$$ I think we need the transformation of symmetry, $$X\otimes Y\to Y\otimes X,$$ is that true? So your construction seems to work only for symmetric monoidal categories? (Or, I don't know, perhaps, all closed monoidal categories are assumed to be symmetric?) – Sergei Akbarov Apr 30 at 9:49
• No, one uses (see my notation) the morphism $N \to \underline{\mathrm{Hom}}(A,N)$ which is adjoint to $A \otimes N \to N$. A symmetry is not necessary. – Martin Brandenburg Apr 30 at 15:28
• (At least I think that it is not necessary. But I work with symmetric monoidal categories most of the time, so better doublecheck this.) – Martin Brandenburg Apr 30 at 15:50

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 $$m^* : V(M,N) \to V(TM,N) \qquad n_* \circ T_{MN} : V(M,N) \to V(TM,TN) \to V(TM, N)$$

• Fosco, I don't understand this detail: $$V(M,N)\to V(A\otimes M,A\otimes N).$$ To construct this morphism, we need a morphism $A\otimes M\to M$ and a morphism $N\to A\otimes N$, what are these two morphisms? Or I am missing something? – Sergei Akbarov Apr 30 at 7:40
• @SergeiAkbarov No, one uses that $A \otimes -$ is a $V$-functor. – Martin Brandenburg Apr 30 at 8:08
• @MartinBrandenburg: Martin, do you mean a functor between enriched categories? – Sergei Akbarov Apr 30 at 8:19
• @SergeiAkbarov Yes. – Martin Brandenburg Apr 30 at 15:29
• First lines of page 9: tac.mta.ca/tac/reprints/articles/10/tr10.pdf – fosco Apr 30 at 16:18