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
being “normal” in that they induce a compatible topological group structure on the homotopy
quotient
.
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
–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.
Keywords
normal subgroup, Segal space, bar construction,
localization, completion, homotopy monoidal functor
