Crossed modules and cohomology of algebras over an operad

Abstract

We introduce a general definition of a $n$-crossed module of $P$-algebras over an algebraic operad $P$, which coincides with historical definitions in the cases of the operads As and Lie and $n=1$. We establish a natural isomorphism between the abelian group of equivalence classes of $n$-crossed modules over a pair $(A,M)$ for an operad $P$ and the $(n+1{)}^{\text{th}}$ operadic cohomology group of $A$ with coefficients in $M$.

22 pages