nLab May recognition theorem

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 $\mathbf{H}$.

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

Related concepts

• looping and delooping

• Quillen equivalence between simplicial groups and reduced simplicial sets

• recognition of relative loop spaces

References

• Peter May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Springer 1972 (doi:10.1007/BFb0067491, pdf)

• Jacob Lurie, Higher Topos Theory (2009)

• Jacob Lurie, Derived Algebraic Geometry VI: $E_k$ Algebras (arXiv:0911.0018)

• Jacob Lurie, Higher Algebra (2017)