Last time, we were discussing the category {\mathbf{PT}} whose objects are pointed topological spaces and whose morphisms are pointed homotopy classes of basepoint-preserving maps. It turns out that the homotopy groups are functors from this category {\mathbf{PT}} to the category of groups.

The homotopy group functors {\pi_n} are, however, representable. They are representable by {(S^n, s_0)}, where {s_0} is a base-point; this is equivalently {I^n/\partial I^n, \partial I^n} for {I^n} the {n}-cube and {\partial I^n} the boundary. The fact that these are functors to the category of groups is equivalent to saying that {(S^n, s_0)} is a cogroup object in {\mathbf{PT}}.

But why should {(S^n, s_0)} be a cogroup object? To answer this question, let us consider a pair of adjoint functors on {\mathbf{PT}}.