Twisting structures and strongly homotopy morphisms

TL;DR

This paper develops a framework for twisted composition products of symmetric sequences in chain complexes, introduces P-co-rings from cooperad morphisms, and characterizes categories of P-algebras up to strong homotopy, with applications to Koszul resolutions.

Contribution

It generalizes twisting structures and constructs P-co-rings from cooperad morphisms, linking them to categories of P-algebras up to strong homotopy.

Findings

Kleisli category for K(g) is isomorphic to P-algebras up to strong homotopy

Classifying morphisms for strict and homotopy P-algebras are provided

Existence theorem for parametrized morphisms of chain (co)algebras up to strong homotopy

Abstract

In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. Let Q be a cooperad and let BP be the bar construction on the operad P. To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras. Parametrized morphisms of (co)associative chain (co)algebras up to strong homotopy are also introduced and studied, and a general existence theorem is proved. In the appendix, we study the particular case of the two-sided Koszul resolution of the associative operad.