An explicit non-inductive twisting function is constructed for the twisted tensor product from any twisted cartesian product of simplicial sets by selecting a specific monoid morphism from Kan's loop group to Moore loop spaces.
The Szczarba map and the cubical cobar construction
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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.
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On twisting functions in twisted cartesian products and twisted tensor products
An explicit non-inductive twisting function is constructed for the twisted tensor product from any twisted cartesian product of simplicial sets by selecting a specific monoid morphism from Kan's loop group to Moore loop spaces.