# Every Abelian group is canonically a $\mathbb{Z}$-module. Is this just a coincidence?

Every Abelian group is canonically a $\mathbb{Z}$-module, where $\mathbb{Z}$ is the initial monoid in the monoidal category of Abelian groups. And every Abelian monoid is canonically an $\mathbb{N}$-(semi-)module, where $\mathbb{N}$ is the initial monoid in the monoidal category of Abelian monoids.

Are these two examples special cases of a more general principle? Or is the similarity between them just a coincidence?

• In what sense do you mean 'coincidence'? – zibadawa timmy Aug 1 '14 at 10:05
• @zibadawatimmy, I mean 'coincidence' in the sense of 'not special cases of a more general principle.' I have edited to clarify. – goblin Aug 1 '14 at 10:56
• Well, abelian groups and $\mathbb Z$-modules are equivalent, in an obvious way. The other likely holds with an equally obvious equivalence, though I've not verified it in any way. – zibadawa timmy Aug 1 '14 at 11:04
• I think the question is : is there a general notion of "module" such that any monoidal category with initial monoid $M$ is equivalent to the category of $M$-modules ? (I'm not qualified to answer though.) – Pece Aug 1 '14 at 12:23
• @Pece Every pointed fusion category over $k=\mathbb C$ has the field as its zero object. And these are precisely $\mathrm{Vec}_G^\omega$ for $G$ a finite group and $\omega\in H^3(G,k)$. And every group can be viewed as a category with one object, and conversely. These would be counterexamples, yes? – zibadawa timmy Aug 1 '14 at 14:11

If $C$ is a closed monoidal category and $R$ is a monoid object in $C$, then a left $R$-module is the same as an object $M \in C$ together with a homomorphism of monoids $R \to \underline{\mathrm{End}}(M)$. If $R=\mathbf{1}_C$ is the initial monoid, it follows that every object of $C$ has a unique $R$-module structure. If $C$ is not closed, it can be also checked directly that every object has a unique left $\mathbf{1}_C$-module structure.
For $C=(\mathsf{CMon},\otimes,\mathbb{N},\dotsc)$ we see that every commutative monoid has a unique $\mathbb{N}$-module structure, and for $C=(\mathsf{Ab},\otimes,\mathbb{Z},\dotsc)$ we see that every abelian group has a unique $\mathbb{Z}$-module structure.