Crossed modules and cohomology of algebras over an operad

Leray, Johan; Rivière, Salim; Wagemann, Friedrich

doi:10.2140/agt.2026.26.2635

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ISSN (electronic): 1472-2739

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Crossed modules and cohomology of algebras over an operad

Johan Leray, Salim Rivière and Friedrich Wagemann

Algebraic & Geometric Topology 26 (2026) 2635–2658

DOI: 10.2140/agt.2026.26.2635

Abstract

We introduce a general definition of an $n$ -crossed module of $𝒫$ -algebras over an algebraic operad $𝒫$ , which coincides with historical definitions in the cases of the operads $𝒜 s$ or $ℒ i e$ and $n = 1$ . Over a field of characteristic $0$ , 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 $(n + 1)$ -st operadic cohomology group of $A$ with coefficients in $M$ .

Keywords

crossed module, operadic cohomology, algebra over an operad

Mathematical Subject Classification

Primary: 18M60, 18M70

Secondary: 18G50, 18N25

References

Publication

Received: 7 February 2025

Revised: 17 June 2025

Accepted: 7 July 2025

Published: 1 September 2026

Authors

Johan Leray
Laboratoire de Mathématiques Jean Leray Nantes Université Nantes France

Salim Rivière
Laboratoire de Mathématiques Jean Leray Nantes Université Nantes France

Friedrich Wagemann
Laboratoire de Mathématiques Jean Leray Nantes Université Nantes France

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Volume 26, issue 7 (2026)