Crossed modules and cohomology of algebras over an operad

TL;DR

We define a general notion of $n$-crossed modules of $\mathscr{P}$-algebras and prove that their equivalence classes over a fixed pair $(A,M)$ are naturally in bijection with the $(n+1)$-st operadic cohomology group $\mathrm{H}^{n+1}_\mathscr{P}(A;M)$. The approach combines the Homotopy Transfer Theorem to produce cocycles from crossed modules with the Rectification Theorem to realize cocycles as crossed modules, establishing a two-way correspondence. A Baer-sum construction shows that these classes form an abelian group, with addition corresponding to cocycle addition, thereby linking higher crossed module data to operadic cohomology. The framework recovers classical results for $\mathscr{P}=\mathcal{A}ss$ and $\mathscr{P}=\mathcal{L}ie$ (via $n=1$) and integrates into the broader operadic (co)homology theory, with potential extensions to properads.

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 $M$.

Theorems & Definitions (42)

• Theorem : cf. \ref{['main_theorem']}
• Proposition 1.1: Loday:2012
• Proposition 1.2: The Rosetta Stone Loday:2012
• Theorem 1.3: Homotopy Transfer Theorem
• Theorem 1.4: Rectification Theorem
• proof
• Definition 2.2: $n$-crossed module
• Remark 2.3
• Remark 2.5
• Definition 2.6: Equivalence of $n$-crossed modules