# Operadic twisting – with an application to Deligne's conjecture

Dolgushev, Vasily; Willwacher, Thomas (2015). Operadic twisting – with an application to Deligne's conjecture. Journal of Pure and Applied Algebra, 219(5):1349-1428.

## 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 Lie$_\infty$, As$_\infty$, Ger$_\infty$. 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.

## 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 Lie$_\infty$, As$_\infty$, Ger$_\infty$. 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.

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