Crossed modules and cohomology of algebras over an operad

TL;DR

This paper generalizes the concept of crossed modules for P-algebras over an operad, establishing a link between their classification and operadic cohomology, thus extending classical algebraic structures.

Contribution

It introduces a unified definition of n-crossed modules for P-algebras and connects their classification to operadic cohomology groups.

Findings

Defined n-crossed modules for P-algebras.

Proved isomorphism with operadic cohomology groups.

Unified classical and modern algebraic structures.

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 $M$.