Homotopy Coherent Representations

Summary: Our research demonstrates that Szczarba's twisting cochain for a twisted Cartesian product is essentially equivalent to the one created by Shih. Specifically, by utilizing a 'reversed' version of the classical Eilenberg-Mac Lane homotopy for the Eilenberg-Zilber contraction, Szczarba's twisting cochain can be derived via the basic perturbation lemma. Along the way, we establish several new identities involving these homotopies.