An Eckmann-Hilton dual to the \(\Pi\)-algebras of homotopy theory (Q1766841)

The \textit{cohomology operations category}, \(\mathcal H\), is the category of finite products of Eilenberg-MacLane spaces over \(\mathbb Z\) and the point, with morphisms the homotopy classes of maps between them. An \(H-\) \textit{algebra} is a functor \(Z:\mathcal H \rightarrow \mathcal{SET}_*\) sending products to products and the point to \(0\), the pointed set with one element. This is a Hilton-Eckmann dual to the \(\Pi\)-algebra via the fact that the Eilenberg-MacLane space is dual to a Moore space. The Moore spaces over \(\mathbb Z\) are spheres, and the product of \(\mathbb Z\)-Eilenberg-MacLane spaces are dual to the wedges of spheres. The \(\Pi\)-algebra is a contravariant functor from the category of finite wedges of spheres to pointed sets, so the \(\Pi\)-algebra is dual to the \(H\)-algebra. The \(H\)-algebra is a model of primary integral cohomology operations, stable and unstable. The definition of products of \(H\)-algebras is made and relating this concept to the \(H\)-algebras of wedges of spaces gives rise to a sequence of examples, resulting in two spaces which have the same \(H\)-algebra structure and yet are not homotopy equivalent.