2025/12/16 by Li Cai, Cai, Li
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2512.14084
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|⊂ |GX| as a submonoid and prove that after inverting all elements of degree 0 in CX, the inclusion |S-1CX|⊂ |GX| is a (weak) homotopy equivalence. Our construction is functorial and explicit, without using inductions.