Under the identification of homotopy types of topological spaces with simplicial sets/Kan complexes (see at homotopy hypothesis) there is a standard construction (Kan 58) traditionally denoted $G$, for the loop space $\Omega X$ of a connected topological space as a simplicial group. It is the left adjoint in an adjunction
between reduced simplicial sets and simplicial groups, whose right adjoint is the simplicial classifying space construction.
The generalization of this construction to non-reduced simplicial sets and simplicial groupoids is the Dwyer-Kan loop groupoid functor.
Daniel Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53
Peter May, chapter VI of Simplicial objects in algebraic topology, Van Nostrand, Princeton (1967) (ISBN:9780226511818, djvu)
More details and relation to décalage:
Last revised on October 26, 2020 at 06:19:30. See the history of this page for a list of all contributions to it.