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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 → G being “normal” in that they induce a compatible
topological group structure on the homotopy quotient G ∕ ∕ N := EN ×NG. 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.
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