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May recognition theorem

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Group Theory

Contents

Idea

The original May recognition theorem (May 72) characterizes the homotopy types of nn-fold iterated loop spaces as “grouplike” En-algebras in the classical homotopy theory of spaces (∞Grpd), hence as ∞-groups with (n1)(n-1)-fold abelian structure (braided ∞-groups for n=2n = 2, sylleptic ∞-groups for n=3n=3, etc.)

The statement generalizes from ∞Grpd to any (∞,1)-topos (Lurie 09b, 1.3, Lurie 17, 5.2.6).

Statement

Proposition

(Recognition of group objects in an (∞,1)-topos)

Let H\mathbf{H} be an (∞,1)-topos Then the operation Ω\Omega of forming loop space objects constitutes an equivalence of (∞,1)-categories

Groups(H)BΩH 1 */ Groups(\mathbf{H}) \underoverset {\underset{\;\;\mathbf{B}\;\;}{\longrightarrow}} {\overset{\Omega}{\longleftarrow}} {\simeq} \mathbf{H}^{\ast/}_{\geq 1}

between

  1. ∞-groups (i.e. group objects) in H\mathbf{H}

  2. pointedconnected objects in H\mathbf{H}.

The inverse equivalence is the delooping B\mathbf{B}, see at looping and delooping.

This is Lurie 09a, Theorem 7.2.2.11.

More generally:

Proposition

(Recognition of abelian group objects in an (∞,1)-topos)

Let H\mathbf{H} be an (∞,1)-topos and nn \in \mathbb{N}, n1n \geq 1. Then the operation Ω n\Omega^n of forming nn-fold iterated loop space objects constitutes an equivalence of (∞,1)-categories

Groups E n(H)B nΩ nH n */ Groups_{E_n}(\mathbf{H}) \underoverset {\underset{\;\;\mathbf{B}^n\;\;}{\longrightarrow}} {\overset{\Omega^n}{\longleftarrow}} {\simeq} \mathbf{H}^{\ast/}_{\geq n}
  1. ∞-groups (i.e. group objects) in H\mathbf{H} with En-algebra-structure,

  2. pointedn-connected objects in H\mathbf{H}.

This is Lurie 09b, Theorem 1.3.6, Lurie 17, Theorem 6.2.6.15.

References

Created on August 3, 2020 at 07:24:46. See the history of this page for a list of all contributions to it.