On twisting functions

Abstract

In this work, we unify different constructions of Kan's loop group $GX$ for a reduced simplicial set $X$ topologically, by identifying its geometric realization $|GX|$ as different submonoids of $Ω|X|$, the monoid of based Moore loops on $|X|$. Then we construct a cubical subcomplex $|CX|\subset |GX|$ as a submonoid and prove that after inverting all elements of degree $0$ in $CX$, the inclusion $|S^{-1}CX|\subset |GX|$ is a (weak) homotopy equivalence. Our construction is functorial and explicit, without using inductions.