The Szczarba map and the cubical cobar construction

TL;DR

This paper proves that Szczarba operators induce a simplicial map from the cubical cobar construction of a simplicial set to a simplicial group, confirming a key algebraic property.

Contribution

It provides a detailed proof that Szczarba operators induce a simplicial map, offering a conceptual proof of the comultiplicativity of a related dga map.

Findings

Szczarba operators induce a simplicial map from cubical cobar construction to G

Confirms a result by Minichiello-Rivera-Zeinalian

Shows the dga map is comultiplicative

Abstract

We consider a twisting function from a 1-reduced simplicial set $X$ to a simplicial group $G$. We prove in detail that the associated Szczarba operators induce a simplicial map from the triangulation of the cubical cobar construction of $X$ to $G$. This confirms a result due to Minichiello-Rivera-Zeinalian and gives, as pointed out by these authors, a conceptual proof of the fact that the dga map $\Omega\,C(X) \to C(G)$ induced by Szczarba's twisting cochain is comultiplicative.