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?

  • 3
    $\begingroup$ In what sense do you mean 'coincidence'? $\endgroup$ – zibadawa timmy Aug 1 '14 at 10:05
  • $\begingroup$ @zibadawatimmy, I mean 'coincidence' in the sense of 'not special cases of a more general principle.' I have edited to clarify. $\endgroup$ – goblin Aug 1 '14 at 10:56
  • $\begingroup$ 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. $\endgroup$ – zibadawa timmy Aug 1 '14 at 11:04
  • 2
    $\begingroup$ 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.) $\endgroup$ – Pece Aug 1 '14 at 12:23
  • $\begingroup$ @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? $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.