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In simplicial homotopy theory there is a standard construction (Kan 58), traditionally denoted $G$, for the simplicial analog of the homotopy type of the loop space $\Omega X$ of a connected topological space, now incarnated as a simplicial group: It is the left adjoint in an adjunction
between pointed simplicial sets and simplicial groups, whose right adjoint is the simplicial classifying space construction.
This adjunction is a Quillen adjunction for the Kan–Quillen model structure on pointed simplicial sets and its transferred model structure on simplicial groups.
When restricted to reduced simplicial sets, this Quillen adjunction becomes a Quillen equivalence.
The generalization of this construction to non-reduced simplicial sets and simplicial groupoids is the Dwyer-Kan loop groupoid functor.
(Quillen equivalence between simplicial groups and reduced simplicial sets)
The simplicial classifying space-construction $\overline{W}(-)$ (Def. ) is the right adjoint in a Quillen equivalence between the projective model structure on simplicial groups and the injective model structure on reduced simplicial sets.
The left adjoint $\Omega$ is the simplicial loop space-construction.
(e.g. Goerss & Jardine 09, V Prop. 6.3)
For a more complete list of references, see the article simplicial delooping functor.
Daniel Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042) (Sections 10–11.)
Peter May, chapter VI of Simplicial objects in algebraic topology, Van Nostrand, Princeton (1967) (ISBN:9780226511818, djvu)
Paul Goerss, J. F. Jardine, Section V.4 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)
More details and relation to décalage:
Last revised on May 21, 2022 at 06:25:12. See the history of this page for a list of all contributions to it.