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The original May recognition theorem (May 72) characterizes the homotopy types of -fold iterated loop spaces as “grouplike” En-algebras in the classical homotopy theory of spaces (∞Grpd), hence as ∞-groups with -fold abelian structure (braided ∞-groups for , sylleptic ∞-groups for , etc.)
The statement generalizes from ∞Grpd to any (∞,1)-topos (Lurie 09b, 1.3, Lurie 17, 5.2.6).
(Recognition of group objects in an (∞,1)-topos)
Let be an (∞,1)-topos Then the operation of forming loop space objects constitutes an equivalence of (∞,1)-categories
between
∞-groups (i.e. group objects) in
The inverse equivalence is the delooping , see at looping and delooping.
This is Lurie 09a, Theorem 7.2.2.11.
More generally:
(Recognition of abelian group objects in an (∞,1)-topos)
Let be an (∞,1)-topos and , . Then the operation of forming -fold iterated loop space objects constitutes an equivalence of (∞,1)-categories
∞-groups (i.e. group objects) in with En-algebra-structure,
This is Lurie 09b, Theorem 1.3.6, Lurie 17, Theorem 6.2.6.15.
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: Algebras (arXiv:0911.0018)
Jacob Lurie, Higher Algebra (2017)
Last revised on March 21, 2023 at 12:28:36. See the history of this page for a list of all contributions to it.