What are cogroups? are they related to the dual notion of groups internal to a category? I know what is a group internal to a category with finite products. I am curious to see what is the dual notion of a group or any other algebraic structure for that matter, internal to a category with finite co-products? I would be very pleased to see links to reading materials in the answers. Thank you.
 A: A cogroup in a category $\mathcal{C}$ is an object together with a factorization of $\hom(X,-):  \mathcal{C} \to \mathsf{Set}$ over $\mathsf{Grp}$. Similarly, for every concrete category $V \to \mathsf{Set}$ (for example a category of algebraic structures) one defines co-$V$-objects in $\mathcal{C}$. If $\mathcal{C}$ has coproducts (including an initial object $0$), these objects can be defined more explicitly. For example, a cogroup is an object $X$ equipped with morphisms $X \to 0$ ("counit"), $X \to X \sqcup X$ ("comultiplication"), $X \to X$ ("coinverse") such that the five obvious diagrams commute. This concept has been introduced by Kan, who also classified cogroups in $\mathsf{Grp}$. Cogroups arise naturally in algebraic topology. For example the group structure on the $n$th homotopy group $\pi_n(-):=\hom(S^n,-)$ of spaces is nothing else than a cogroup structure on the sphere $S^n$ (in the pointed homotopy category), which is cocommutative for $n>1$.
The three canonical references are:

  
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*Daniel M. Kan, On monoids and their dual, Bol. Soc. Mat. Mexicana (2) 3 (1958), pp. 52-61.
  
*Peter Freyd, Algebra-valued functors in general and tensor products in particular, Colloq. Math. (14), pp. 89-106, 1966.
  
*George M. Bergman, Adam O. Hausknecht, Cogroups and corings in categories of associative rings, American Mathematical Society, Mathematical Surveys and Monographs # 45, 1996.
  

Additional links online:


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*MO/3740: cogroup objects

*nlab: cogroup objects
A: The obvious notion of "cogroup object in a category $\mathcal{C}$ with finite coproducts"  is "group object in $\mathcal{C}^\mathrm{op}$". This happens to include useful examples, such as the pointed spheres $(S^n, *)$ in the pointed homotopy category, as well as the tautological example of $\mathbb{Z}$ in $\mathbf{Grp}$ and in $\mathbf{Ab}$. More generally, given a representable functor $\mathcal{C} \to \mathbf{Set}$ where $\mathcal{C}$ has finite coproducts, each factorisation through the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$ corresponds to a cogroup structure on the representing object.
There are also less useful notions. For instance if one merely turns the arrows around but keeps $\times$ as $\times$, one ends up with something trivial: every object is automatically equipped with such a structure. (See here.)
