Operads up to Homotopy and Deformations of Operad Maps

Details Operads up to Homotopy and Deformations of Operad Maps Operads up to Homotopy and Deformations of Operad Maps

2002-08-06

arxiv.org/abs/math/0208041v1

Mathematics

Quantum Algebra

26 pages

Scientific paper

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.

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