Operadic Twisting -- with an application to Deligne's conjecture

TL;DR

This paper investigates the operadic twisting functor Tw, establishing it as a comonad, and applies this framework to demonstrate that solutions to Deligne's conjecture can be homotopically adjusted to be compatible with twisting.

Contribution

It introduces the operadic twisting functor Tw as a comonad and explores its properties, providing new insights into operad theory and Deligne's conjecture.

Findings

Tw is a comonad on the category of operads.

Coalgebras of Tw include classical and infinity operads.

Solutions to Deligne's conjecture can be homotopically modified to respect twisting.

Abstract

We study categorial properties of the operadic twisting functor Tw. In particular, we show that Tw is a comonad. Coalgebras of this comonad are operads for which a natural notion of twisting by Maurer-Cartan elements exists. We give a large class of examples, including the classical cases of the Lie, associative and Gerstenhaber operads, and their infinity-counterparts L-infinity, A-infinity, G-infinity. We also show that Tw is well behaved with respect to the homotopy theory of operads. As an application we show that every solution of Deligne's conjecture is homotopic to a solution that is compatible with twisting.