Operads up to Homotopy and Deformations of Operad Maps

Abstract

From the `cofree' cooperad $T^{\prime }(A[-1])$ on a collection $A$ together with a differential, we construct an $L_{\infty }$-algebra structure on the total space ${⨁}_{n}A(n)$ 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 $Q⟶P$. 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.

26 pages