# Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy

## TL;DR

This paper introduces a hierarchy of higher distributivity laws to describe the composition of maps between Eilenberg--MacLane spectra, revealing new algebraic structures in the topological Steenrod algebra and constructing a derivation of it.

## Contribution

It defines higher distributivity laws up to homotopy and proves their satisfaction by the topological Steenrod algebra, offering new insights into its algebraic structure.

## Key findings

- Topological Steenrod algebra satisfies all higher distributivity laws.
- Higher distributivity laws are homotopy invariant.
- Constructs a derivation of degree -2 of the mod 2 Steenrod algebra.

## Abstract

Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sense. As an application of $2$-distributivity, we provide a new construction of a derivation of degree $-2$ of the mod $2$ Steenrod algebra.