标题： 操作数上代数的高交叉模 显示英文标题

标题： Higher crossed modules of algebras over an operad

摘要： 我们研究在操作子代数背景下的交叉模。 为此，在第一部分中，我们通过回顾内部作用、预交叉模和操作子情况下的交叉模的概念，来适应Janelidze的方法。 此外，我们提取了在李情况下众所周知的Peiffer关系，适用于任意操作子上的预交叉模。 我们证明了我们的交叉模概念与Janelidze的概念等价，通过证明固定操作子上的代数交叉模与该固定操作子上的代数内部范畴等价。 在第二部分中，我们研究了Leray-Riviere-Wagemann在arXiv:2411.04614中引入的操作子上的代数交叉模的概念，并证明它与已有的交叉模概念等价。 粗略地说，按照Leray-Rivière-Wagemann的观点，交叉模是在0度和1度上集中的链复形上的代数结构。 我们强调了一个“协下降”过程，该过程允许我们为链复形的一次项和零次项赋予代数结构，使得微分成为代数同态。 此外，我们证明它是一个交叉模。 事实上，我们证明了Leray-Riviere-Wagemann对交叉模的方法与Janelidze的方法等价。 这种方法与协下降过程相结合，使我们能够以非常简洁和明确的方式引入高阶交叉模，并证明它们与某些n重内部范畴等价。 显示英文摘要

摘要： 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.