Higher crossed modules of algebras over an operad

Abstract

We study crossed modules in the context of algebras over an operad. To do so, in the first section, we adapt the methods of Janelidze by reviewing the notions of internal actions, precrossed modules and crossed modules in the operadic case. Moreover, we extract the Peiffer relations, well known in the Lie case, for precrossed modules over an arbitrary operad. We prove that our notion of crossed modules is equivalent to the one of Janelidze by proving that our crossed modules of algebras over a fixed operad are equivalent to categories internal to algebras over this fixed operad. In the second section, we study the notion of crossed modules of algebras over an operad as introduced by Leray-Riviere-Wagemann in arXiv:2411.04614 and prove it to be equivalent to the already existing notions of crossed modules. Roughly speaking, a crossed module in the sense of Leray-Rivi\`ere-Wagemann is an algebra structure on a chain complex concentrated in degrees 0 and 1. We highlight a "codescent" process which allows us to endow the degree one and zero terms of the chain complex with algebra structures such that the differential is an algebra morphism. Moreover, we prove it to be a crossed module. In fact we prove that the approach to crossed modules of Leray-Riviere-Wagemann is equivalent to the one of Janelidze. This new approach, together with the codescent process, allows us to introduce higher crossed modules in a very concise and explicit way, and we prove them to be equivalent to n-fold categories internal to algebras, for some n.