The defining property of a cogroup object is that morphisms out of it form a group. Specifically, if is a category, then is a cogroup object in if is a group for any object in (and the group structure must be natural in ).
There are many examples of cogroup objects. Perhaps the most well-known are the n-sphere in the homotopy category of pointed topological spaces, . Then the fact that is a cogroup object in is precisely the statement that the homotopy group for is indeed a [[group9], naturally in , for all topological spaces .
The basic definition is as follows.
Let be a category. To give an object of a cogroup structure in is to give the functor a lift from to .
A cogroup object in is an object together with a choice of cogroup structure.
A morphism of cogroup objects is a morphism in between the underlying objects of the such that the natural transformation lifts to a natural transformation of functors into .
Thus cogroup objects and their morphisms can be thought of as the category of representable functors from to .
Providing has enough coproducts of (the th copowers to be precise), the concept of a cogroup structure on can be internalised.
To give an object of a cogroup structure is equivalent to choosing morphisms , , and satisfying the diagrams for associativity, unit, and inverse but the other way around.
Here, the phrase “the other way around” means: take the normal diagrams for a group object that express the properties of associativity, unit, and inverses, invert all the arrows, and replace products by coproducts.
A cogroup object in a category, say , is nothing more than a group object in the opposite category: . However, morphisms in the cogroup category go the other way around. That is to say, with the obvious notation:
Of course, there is nothing special about groups here. The same style of definition works for any variety of algebras in the sense of universal algebra, where .
Some terminological care should be taken in the case of comonoid, which makes sense in any monoidal category, not just cocartesian monoidal categories which is the general default environment for discussing co--algebras. Thus, check with the author to see which monoidal product is meant; in the case of comonoid it’s likely that it’s not the cocartesian notion that is intended. Whereas in the case of cogroups, confusion is not so likely: one needs the cocartesian structure (codiagonals, etc.) essentially because the axioms of a group involve duplication of variables, whereas this is not the case for axioms of a monoid. (Cf. the distinction between operad and Lawvere theory, where the latter can be viewed as a kind of “cartesian operad”.)
The higher spheres are actually abelian cogroup objects, as demonstrated by the fact that is abelian for .
More generally, any suspension is a cogroup object with the “pinch” map as the comultiplication. See at suspensions are H-cogroup objects. (Since the -sphere is not a suspension in , but only in , it need not be a cogroup and in fact is not.) This is dual to, and equivalent to, the statement that (based) loop spaces are group objects in since there is an adjunction, internal to :
There are examples of spaces that are cogroups in that are not suspensions, see Bernstein & Harper Cogroups which are not suspensions. Note that cogroups in are the same as co-H-spaces which are additionally (co-)associative and have (co-)inverses.
On the other hand, every abelian group is again an abelian cogroup since is self-enriched. Indeed, in an abelian category every object is simultaneously an abelian group object and an abelian cogroup object. In , the abelian cogroup object structure is unique, with comultiplication given by the diagonal morphism.
In Set, the only cogroup object (abelian or otherwise) is the empty set. This is because the counit map must be a morphism from to the terminal object of the opposite category. In the case of , this is the empty set.
This extends further: any category with a faithful functor to which preserves an initial object will have no non-trivial cogroup objects. In particular, the category Top of unbased topological spaces has only the empty space as a cogroup object.
The case of cogroups, and some other co-things, in certain other varieties of algebras has been extensively studied by Bergman and Hausknecht in Co-groups and co-rings in categories of associative rings, (MR1387111)
In particular, a co-group in the category of (unital) commutative rings is a commutative Hopf ring? and a cogroup in the category of (unital) commutative -algebras is a commutative Hopf -algebra; a fact highlighted in homotopy theory by Haynes Miller (in view of his generalization to commutative Hopf algebroids as cogroupoids in commutative algebra) in the context of discussion of dual Steenrod algebras, see (Ravenel 86, appendix A) for review.
Discussion of commutative Hopf algebras as cogroups is in
Cogroups in the category of (graded or not) associative algebras are very rare (unlike Hopf algebras) – in fact the underlying algebras are free; this has been clear since
This fact is later observed in bigger generality in