# nLab May recognition theorem

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

# Contents

## Idea

The original May recognition theorem (May 72) characterizes the homotopy types of $n$-fold iterated loop spaces as “grouplike” En-algebras in the classical homotopy theory of spaces (∞Grpd), hence as ∞-groups with $(n-1)$-fold abelian structure (braided ∞-groups for $n = 2$, sylleptic ∞-groups for $n=3$, 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 $\mathbf{H}$ be an (∞,1)-topos Then the operation $\Omega$ of forming loop space objects constitutes an equivalence of (∞,1)-categories

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

between

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

2. pointed$\;$connected objects in $\mathbf{H}$.

The inverse equivalence is the delooping $\mathbf{B}$, 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 $\mathbf{H}$ be an (∞,1)-topos and $n \in \mathbb{N}$, $n \geq 1$. Then the operation $\Omega^n$ of forming $n$-fold iterated loop space objects constitutes an equivalence of (∞,1)-categories

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

2. pointed $(n-1)$-connected objects in $\mathbf{H}$.