Let X be a simplicial set and G a simplicial group. Any group morphism from the Kan loop group ΩX to G is determined by a twisting function τ : X → G. In 1961, Szczarba gave an explicit construction of a twisting cochain t : C(X) → C(G) out of a twisting function X → G. Such a twisting cochain induces a multiplicative map from the cobar construction Ω C(X) to C(G).

Recently I proved that the map induced by Szczarba’s twisting cochain is also comultiplicative; the coproduct on Ω C(X) is defined in terms of homotopy Ger- stenhaber operations on X. Shortly afterwards, Minichiello–Rivera–Zeinalian gave a conceptual explanation of this fact, based on the idea of triangulating the cubical cobar construction of X. In this talk I want to elucidate the properties of Szczarba’s twisting cochain that make this construction possible