Operads up to Homotopy and Deformations of Operad Maps

TL;DR

This paper develops an $L_$-algebra framework for understanding deformations of operad maps, unifying several existing approaches and providing a new algebraic perspective on operad deformation theory.

Contribution

It constructs an $L_$-algebra controlling operad map deformations from cofibrant resolutions, unifying multiple deformation theories.

Findings

Defines an $L_$-algebra structure on the total space of a cooperad

Controls deformations of operad maps via $L_$-algebras

Unifies existing approaches to operad deformation theory

Abstract

From the `cofree' cooperad $T'(A[-1])$ on a collection $A$ together with a differential, we construct an $L_\infty$-algebra structure on the total space $\bigoplus_nA(n)$ that descends to coinvariants. We use this construction to define an $L_\infty$-algebra controlling deformations of the operad $P$ under $Q$ from a cofibrant resolution for an operad $Q$, and an operad map $Q\longrightarrow P$. Starting from a diffent cofibrant resolution one obtains a quasi isomorohic $L_\infty$-algebra. This approach unifies Markl's cotangent cohomology of operads and the approaches to deformation of $Q$-algebras by Balavoine, and Kontsevich and Soibelman.