The n-Category Café

In a 1996 discussion on the cat-list André Joyal writes

Peter Hilton and Beno Eckmann…proved that any cogroup in the category of groups is free.

I would like to say that the younger generation is not always aware that Eckmann and Hilton have fundamental contributions to category theory. They have provided basic examples of objects equipped with algebraic structures in categories. Consequently opening the road to further abstractions, like the concept of algebraic theory in the sense of Lawvere.

Among other things, Eckmann and Hilton were interested in identifying all cogroups in the homotopy category $hTop_*$ of pointed topological spaces. A basic example of such cogroup is the circle $S^1$. It explains why the functor $\pi_1(-)=[S^1,-]: hTop_* \to Sets$ has a natural group structure. If $G$ is a cogroup in $hTop_*$ then so is the smash product $X \wedge G$ for any pointed topological space $X$. This is because $X \wedge (-)$ preserves coproduct since it has a right adjoint (here we are supposing that Top is a convenient category of topological space). In particular, the spheres $S^{n+1} = S^n \wedge S^1$ have a cogroup structure. Any wedge (topologists call the coproduct the wedge) of cogroups is obviously a cogroup. In particular, any wedge of spheres of dimension $n \geq 1$ has a co-group structure.

All the known examples of cogroups in $hTop_*$ are obtained by taking a smash $X \wedge S^1$. It was conjectured by Eckmann and Hilton that all cogroups in $hTop^*$ are of the form $X \wedge S^1$. They observe that $\pi_1(G)$ is a cogroup in $Groups$ when $G$ is a cogroup in $hTop_*$. This is because the functor $\pi_1: hTop_* \to Groups$ preserves coproducts by Van Kampen theorem. In support to their conjecture they proved that any cogroup in $Groups$ is free. It follows that all cogroups in $Groups$ are of the form $\pi_1(X \wedge S^1)$ where $X$ is a pointed set.