simplicial loop space




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 GG, for the loop space ΩX\Omega X of a connected topological space as a simplicial group. It is the left adjoint in an adjunction

(GW¯):sSet 0W¯GsGrp (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.


  • 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.