Cogroup objects

# Cogroup objects

## Idea

Cogroup objects are group object in an opposite category, and often one takes the opposite category of group object in an opposite category to be the category of cogroup objects.

The defining property of a cogroup object is that morphisms out of it form a group. Specifically, if $C$ is a category, then $G$ is a cogroup object in $C$ if $\operatorname{Hom}(G,X)$ is a group for any object $X$ in $C$ (and the group structure must be natural in $X$).

There are many examples of cogroup objects. Perhaps the most well-known are the n-sphere in the homotopy category of pointed topological spaces, $\operatorname{hTop}_*$. Then the fact that $S^n$ is a cogroup object in $\operatorname{hTop}$ is precisely the statement that the homotopy group $\pi_n(X)$ for $n \geq 1$ is indeed a group, naturally in $X$, for all topological spaces $X$.

## Definition

The basic definition is as follows.

###### Definition

Let $C$ be a category. To give an object $G$ of $C$ a cogroup structure in $C$ is to give the functor $\operatorname{Hom}(G,-)$ a lift from $\operatorname{Set}$ to $\operatorname{Grp}$.

A cogroup object in $C$ is an object $G$ together with a choice of cogroup structure.

A morphism of cogroup objects $G_1 \to G_2$ is a morphism in $C$ between the underlying objects of the $G_i$ such that the natural transformation $\operatorname{Hom}(G_2,-) \to \operatorname{Hom}(G_1,-)$ lifts to a natural transformation of functors into $\operatorname{Grp}$.

Thus cogroup objects and their morphisms can be thought of as the category of representable functors from $C$ to $\operatorname{Grp}$.

Providing $C$ has enough coproducts of $G$ (the $0,1,2,3$th copowers to be precise), the concept of a cogroup structure on $G$ can be internalised.

###### Theorem

To give an object $G$ of $C$ a cogroup structure is equivalent to choosing morphisms $\mu \colon G \to G \amalg G$, $\eta \colon G \to 0_C$, and $\iota \colon G \to G$ 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.

## Relationship To Group Objects

A cogroup object in a category, say $C$, is nothing more than a group object in the opposite category: $C^{op}$. However, morphisms in the cogroup category go the other way around. That is to say, with the obvious notation:

$C\operatorname{coGrp} = (C^{op}\operatorname{Grp})^{op}$

## Relationship to Other Objects

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 $C coAlg_T \coloneqq (C^{op}Alg_T)^{op}$.

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-$T$-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”.)

## Examples

1. All n-spheres for all $n$ are cogroup objects in the homotopy category of based topological spaces, $\operatorname{hTop}_*$.

This is the origin of the group structure on homotopy groups. it is also crucial in the structure of the Brown representability theorem.

The higher spheres are actually abelian cogroup objects, as demonstrated by the fact that $\pi_n(X)$ is abelian for $n \ge 2$.

2. More generally, any suspension is a cogroup object with the “pinch” map as the comultiplication. See at suspensions are H-cogroup objects. (Since the $0$-sphere is not a suspension in $\operatorname{hTop}_*$, but only in $\operatorname{hTop}$, 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 $\operatorname{hTop}_*$ since there is an adjunction, internal to $\operatorname{hTop}_*$:

$\operatorname{Hom}(\Sigma X,Y) \cong \operatorname{Hom}(X,\Omega Y)$
3. There are examples of spaces that are cogroups in $\operatorname{hTop}_*$ that are not suspensions, see Bernstein & Harper Cogroups which are not suspensions. Note that cogroups in $\operatorname{hTop}_*$ are the same as co-H-spaces which are additionally (co-)associative and have (co-)inverses.

4. Cogroup objects in the category of groups are free groups, and to give a free group the structure of a cogroup object is the same a choosing a generating set. This is an old result of D.M. Kan’s.

5. On the other hand, every abelian group is again an abelian cogroup since $\operatorname{Ab}$ is self-enriched. Indeed, in an abelian category every object is simultaneously an abelian group object and an abelian cogroup object. In $\operatorname{Ab}$, the abelian cogroup object structure is unique, with comultiplication given by the diagonal morphism.

6. In Set, the only cogroup object (abelian or otherwise) is the empty set. This is because the counit map must be a morphism from $X$ to the terminal object of the opposite category. In the case of $\operatorname{Set}$, this is the empty set.

7. This extends further: any category with a faithful functor to $\operatorname{Set}$ 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.

8. 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 $k$-algebras is a commutative Hopf $k$-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

• Israel Berstein, On cogroups in the category of graded algebras. Trans. Amer. Math. Soc. 115 (1965), 257–269 jstor

This fact is later observed in bigger generality in

• Benoit Fresse, Cogroups in algebras over an operad are free algebras, Commen. Math. Helv. 73:4, 1998, 637–676

doi