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There is a nice list here describing the group objects in various categories : https://ncatlab.org/nlab/show/group+object.

But there is no description of the group objects in :

  • the category $\mathsf{Lie}_k$ of Lie algebras over a field $k$.
  • the category $\mathsf{Mod}_R$ of left modules over a ring $R$.
  • the category $\mathsf{mod}_R$ of finitely generated left modules over a ring $R$.

What would these look like ? When $R = \Bbb Z$, group objects in $\mathsf{Mod}_R$ are just abelian groups. I would guess that in general, group objects in $\mathsf{Mod}_R$ are just left $R$-modules. Is that true? What about the other categories?

Thank you !

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  • $\begingroup$ All modules have a group operation, namely the operation of the underlying abelian group. Thus all objects in $mod_R$ becomes group objects when endowed with these maps. Also, note that a group object is not just an object, but also comes with the data of the multiplication, inverse, and identity maps. $\endgroup$ – MCT Mar 2 '18 at 22:02
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In a category of modules (or more generally in any additive category), every object has a unique group object structure. Indeed, a group object structure on a module $M$ consists of just homomorphisms $e:0\to M$, $i:M\to M$, and $\mu:M\times M\to M$ such that the binary operation $\mu$ satisfies the group axioms with $e(0)=0$ as its unit and $i$ as the inverse map. The fact that $\mu$ is a homomorphism forces it to just be the module addition operation in $M$: $$\mu(a,b)=\mu(0+a,b+0)=\mu(0,b)+\mu(a,0)=b+a$$ (this trick is known as the "Eckmann-Hilton argument"). So the only possible group object structure is the one given by $\mu(a,b)=a+b$ and $i(a)=-a$. Conversely, since these maps are indeed homomorphisms, they do give any object $M$ a group structure.

For Lie algebras, we can use the exact same argument to show that any Lie algebra $A$ has at most one group object structure, given by $\mu(a,b)=a+b$ and $i(a)=-a$. However, in this case, these maps usually are not actually homomorphisms at all. Indeed, in order for $\mu$ to be a homomorphism, we must have $$[a,b]=\mu([a,b],0)=\mu([a,b],[a,a])=[\mu(a,a),\mu(b,a)]=[2a,b+a]=2[a,b]$$ so $[a,b]=0$ for all $a,b\in A$ and $A$ is abelian. Conversely, if $A$ is abelian, then $\mu$ and $i$ are indeed homomorphisms. So, a Lie algebra has a group object structure iff it is abelian, and the group object structure is unique.

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