Homotopy normal maps

Abstract

A group property made homotopical is a property of the corresponding classifying space. This train of thought can lead to a homotopical definition of normal maps between topological groups (or loop spaces).

In this paper we deal with such maps, called homotopy normal maps, which are topological group maps $N\to G$ being “normal” in that they induce a compatible topological group structure on the homotopy quotient $G∕∕N:=EN{\times }_{N}G$. We develop the notion of homotopy normality and its basic properties and show it is invariant under homotopy monoidal endofunctors of topological spaces, eg localizations and completions. In the course of characterizing normality, we define a notion of a homotopy action of a loop space on a space phrased in terms of Segal’s $1$–fold delooping machine. Homotopy actions are “flexible” in the sense they are invariant under homotopy monoidal functors, but can also rigidify to (strict) group actions.