A note on infinite loop space multiplications

Abstract:

A monoid $M$ is known to be abelian iff its multiplication $M \times M \to M$ is a homomorphism. We prove the corresponding result for homotopy-everything $H$-spaces, e.g. infinite loop spaces: For a homotopy-everything $H$space $X$ each $n$-ary operation ${X^n} \to X$ is a homotopy homomorphism, i.e. a homomorphism up to homotopy and all higher coherence conditions.