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Let $\mathsf{Grp}$ be the category of groups. A bifunctor $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an addition bifunctor if:

  • $A(C_n,C_m) \simeq C_{n+m}$,
  • $A(C_0,G) \simeq A(G,C_0) \simeq G$,

for every group $G$ and every $n,m$, with $C_n$ the cyclic group of $n$ elements if $n>0$, and $C_0 \simeq \mathbb{Z}$.

Question: Is there an addition bifunctor for the category of groups?
(or for the subcategory of countable groups)

Stronger question: Is there an addition bifunctor providing a monoidal structure?

This post is inspired by that one (without knowing whether there is an explicit link).

Multiplicative analogous: Existence of a multiplication bifunctor for the category of groups.

Sebastien Palcoux
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    How are you defining your category of cyclic groups (what are its morphisms), and how are you defining your addition and multiplication functors (what does each do to those morphisms)? – user44191 Jan 29 '20 at 05:53
  • @user44191 Your comment is relevant! – Sebastien Palcoux Jan 29 '20 at 08:15
  • @user44191 I should replace "category" by "set" and "functor" by "map" because my motivation here is not in category theory and I don't see a natural way to define these addition and multiplication functors; do you? – Sebastien Palcoux Jan 29 '20 at 08:56
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    I'm...strongly doubtful that there's a natural "addition" functor, and weakly doubtful that there's a natural "multiplication" functor. But if you do go with "set" and "map", then from what I can see, there's no interesting "structure" to $\tilde{A}$ - the extension can be defined completely arbitrarily. – user44191 Jan 29 '20 at 09:24
  • @SebastienPalcoux When your motivation is not category theory, I suggest to rephrase the question accordingly and also remove the ct-tag. – Martin Brandenburg Jan 29 '20 at 10:07
  • @MartinBrandenburg I am in a dilemma: On one hand I don't see an easy/natural way to define such an addition (or multiplication) functor, but it can still exist. On the other hand, removing the category structure would allow arbitrary answers, which is not interesting. Something more subtle is required, and the category theory could help. – Sebastien Palcoux Jan 29 '20 at 10:26
  • It was not really true to say << my motivation here is not in category theory >> because my motivation is in fact an eventual extension to subfactor theory, which is stronlgy related to tensor/fusion category theory. – Sebastien Palcoux Jan 29 '20 at 11:49
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    @MartinBrandenburg: We could ask whether the category $\mathcal{Grp}$ (or the subcategory of countable groups) admits a monoidal structure with unit $I \simeq C_0$ and $C_n \otimes C_m \simeq C_{n+m}$ (or with unit $I \simeq C_1$ and $C_n \otimes C_m \simeq C_{nm}$). – Sebastien Palcoux Jan 29 '20 at 12:06
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    Ok, I understand. A monoidal structure is even stronger than your initial requirement, but also more interesting. Now you got me ... ^^ – Martin Brandenburg Jan 29 '20 at 16:13

1 Answers1

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The answer is no. Notice that $A(-,C_1)$ is a functor $F : \mathsf{Grp} \to \mathsf{Grp}$ with $F(C_n) \cong C_{n+1}$. But there is no such $F$. There is a split monomorphism $C_1 \to C_2$, hence $F$ would induce a split monomorphism $C_2 \to C_3$, contradiction.

Martin Brandenburg
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