The nLab-article on internalization describes how to define certain algebraic structures internally to a category-with-structure. This gives, for example, the notion of a group object in a category with finite products. Now I am wondering: Can cyclic groups be defined internally to a category with finite products (i.e. as group objects with extra structure)?
Alternatively, what is a categorical way of looking at cyclic groups? One probably can think of them as quotient objects of the monoidal unit in the monoidal category of abelian groups. Is there a different conceptual way of looking at cyclic groups (or more generaly cyclic modules)?
Imo the most sensible way to give a purely categorical description of cyclic groups is in terms of their universal property. That is: they are the quotients by subgroups of the free abelian group on one generator $\mathbb Z$. Note that this universal property doesn't really require that the subgroups of $\mathbb Z$ are themselves isomorphic to $\mathbb Z$, so general cyclic groups could turn out very strange in an arbitrary category $\mathcal{C}$.
This description needs quite a bit of ambient structure to work. Afaik group objects in a category are only definable with a cartesian monoidal structure. Furthermore, we want to have a left adjoint to the forgetful functor $U:\mathsf{Ab}(\mathcal{C}) \rightarrow \mathcal{C}$. Finally, we need quotients by subobjects in $\mathsf{Ab}(\mathcal{C})$, which basically translates into the existence of certain colimits in $\mathcal{C}$.