Cogroup objects


Cogroup objects are sort of dual to group objects. The defining property of a cogroup object is that morphisms out of it form a group. Specifically, if CC is a category, then GG is a cogroup object in CC if Hom(G,X)\operatorname{Hom}(G,X) is a group for any object XX in CC (and the group structure must be natural in XX).

There are many examples of cogroup objects. Perhaps the most well-known are the spheres in the homotopy category of based topological spaces, hTop *\operatorname{hTop}_*. Then the fact that S nS^n is a cogroup object in hTop\operatorname{hTop} is precisely the statement that the homotopy group π n(X)\pi_n(X) is a group, naturally in XX, for all topological spaces XX. (Note that this fails for n=0n = 0.)


The basic definition is as follows.


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

A cogroup object in CC is an object GG together with a choice of cogroup structure.

A morphism of cogroup objects G 1G 2G_1 \to G_2 is a morphism in CC between the underlying objects of the G iG_i such that the natural transformation Hom(G 2,)Hom(G 1,)\operatorname{Hom}(G_2,-) \to \operatorname{Hom}(G_1,-) lifts to a natural transformation of functors into Grp\operatorname{Grp}.

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

Providing CC has enough coproducts of GG (the 0,1,2,30,1,2,3th copowers to be precise), the concept of a cogroup structure on GG can be internalised.


To give an object GG of CC a cogroup structure is equivalent to choosing morphisms μ:GG⨿G\mu \colon G \to G \amalg G, η:G0 C\eta \colon G \to 0_C, and ι:GG\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 CC, is nothing more than a group object in the opposite category: C opC^{op}. However, the morphisms go the other way around. That is to say, with the obvious notation:

CGrp c=(C opGrp) op C\operatorname{Grp}^c = (C^{op}\operatorname{Grp})^{op}

Relationship to Other Objects

Of course, there is nothing special about groups here. The same definition works for any variety of algebras in the sense of universal algebra.


  1. As mentioned in the introduction, spheres are cogroup objects in the homotopy category of based topological spaces, hTop *\operatorname{hTop}_*. More generally, any suspension is a cogroup object with the “pinch” map as the comultiplication. (Since the 00-sphere is not a suspension in hTop *\operatorname{hTop}_*, but only in hTop\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 hTop *\operatorname{hTop}_* since there is an adjunction, internal to hTop *\operatorname{hTop}_*:

    Hom(ΣX,Y)Hom(X,ΩY) \operatorname{Hom}(\Sigma X,Y) \cong \operatorname{Hom}(X,\Omega Y)

    The higher spheres are actually abelian cogroup objects, as demonstrated by the fact that π n(X)\pi_n(X) is abelian for n2n \ge 2.

  2. There are examples of spaces that are cogroups in hTop\operatorname{hTop} that are not suspensions. Note that cogroups in hTophTop are the same as co-H-spaces which are additionally (co-)associative and have (co-)inverses.

  3. 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.

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

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

  6. This extends further: any category with a faithful functor to Set\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.

  7. 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)

Revised on December 16, 2009 22:01:12 by Toby Bartels (