Николай Иваныч (nivanych) wrote in category_theory,
Николай Иваныч

Sylow theorems for ∞-groups

Натолкнулся на забавное —
Sylow theorems for ∞-groups
Matan Prasma, Tomer M. Schlank
(Submitted on 14 Feb 2016)
Viewing Kan complexes as ∞-groupoids implies that pointed and connected Kan complexes are to be viewed as ∞-groups.
A fundamental question is then: to what extent can one "do group theory" with these objects?
In this paper we develop a notion of a finite ∞-group:
an ∞-group with finitely many non-trivial homotopy groups which are all finite.
We prove a homotopical analog of the Sylow theorems for finite ∞-groups.
We derive two corollaries:
the first is a homotopical analog of the Burnside's fixed point lemma for p-groups
and the second is a "group-theoretic" characterization of (finite) nilpotent spaces.
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