The Steenrod algebra and theories associated to Hopf algebras (Q1431358)

A theory \textbf{T} is a small category with products and a model of \textbf{T} is a functor \(M : {\mathbf T} \rightarrow\) \textbf{Set} which carries products in \textbf{T} into products of sets. Let \textbf{model}(\textbf{T}) be the category of models. This paper shows the following results. (1) Let \(p\) be a prime. Then the category \textbf{model}\(({\mathbf K}({\mathbb Z}/p))\) is equivalent to the category of connected unstable algebras over the Steenrod algebra \({\mathcal A}_{(p)}\). (2) The Steenrod algebra \({\mathcal A}_{(p)}\) has an unstable structure \((B,D)\) which determines algebraically the \(\Omega\)-algebra \({\mathbf K}({\mathbb Z}/p)\). As a corollary to the second result it follows that the universal Toda bracket is defined algebraically in terms of the Steenrod algebra.