nLab
simplicial loop space

Idea

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

$$(G \dashv \overline{W}) \;\colon\; sSet_0 \underoverset {\underset{\overline{W}}{\longleftarrow}} {\overset{G}{\longrightarrow}} {\bot} sGrp$$

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.

References

• 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:

• Danny Stevenson, Décalage and Kan’s simplicial loop group functor, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 (arXiv:1112.0474, tac:26-28)