ct.category theory - Cogroup Objects

This is less of an answer and more of a bit of bickering about terminology. I think that the word "group object" rightly means the following:

A group object G in a Cartesian category is a map G x G → G, satisfying an associativity axiom, such that there exists a map 1 → G satisfying the unit axiom (it is then necessarily unique) and a map G → G satisfying the inversion axiom (it is then necessarily unique).

If you want, you can consider the unit and inversion maps as part of the data. What's important in the word "group object" is that it is in a Cartesian category, i.e. a category with products. The object 1 is any choice of terminal object for the category.

From this perspective, the most obvious notion of "cogroup object" is:

A cogroup object G in a Cartesian category is a map G → G x G, satisfying a coassociativity axiom, such that there exists a map G → 1 satisfying the counit axiom (it is then necessarily unique) and a map G → G satisfying the inversion axiom (it is then necessarily unique).

But this is a rather problemmatic definition. First of all, by definition there is a unique map G → 1 for any object G. Second, there is a unique coassiciative map G → G x G; it is given by the diagonal map, which is the element of Hom(G,GxG) corresponding to (id,id) ∈ Hom(G,G) x Hom(G,G). (Remember that the definition of Cartesian product is that for any H there is a natural isomorphism Hom(H,GxG) = Hom(H,G) x Hom(H,G).) So every object of a Cartesian category is a "counital cosemigroup" in a unique way. Finally, it's rather hard to write down exactly what the inversion axiom should say.

Well, so the problem is that we're trying to use the Cartesian structure. So Wikipedia and elsewhere adopt the only nontrivial definition, which is to say that for cogroups you should use the coproduct, not the product, so you should ask for a map G → G + G. I think this is the question you're asking about. I guess I would word it "are the any natural categories C with interesting group objects in C^op?", since this definition of "cogroup object" really is "group object in the opposite category".

Finally, I'll mention that certain related constructions do show up all the time. There are good words "algebra" and "coalgebra", which refer to objects in categories with designated monoidal structures (e.g. tensor product in VECT). What's cool is that to make sense of the inverse map's axiom when you aren't using either the product (for groups) or the coproduct (for cogroups), you actually need all the data of a "bialgebra". So the correct notions of "group" and "cogroup" coincide when you give up on categorical (co)products, and become the word "Hopf algebra".