A comonadic interpretation of Baues–Ellis homology of crossed modules

IF 0.5 4区数学 Journal of Homotopy and Related Structures Pub Date : 2018-11-27 DOI:10.1007/s40062-018-0225-3

Guram Donadze, Tim Van der Linden

引用次数: 4

Abstract

We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr–Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.

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交叉模的Baues-Ellis同调的共元解释

引入并研究了阿贝尔交叉模中带系数交叉模的同调理论。讨论了这些新同调群的基本性质，并给出了一些应用。然后我们将注意力限制在积分系数的情况下。在这种情况下，我们重新获得了由Baues最初定义并由Ellis进一步发展的交叉模块的同源性。我们证明了它是Barr-Beck共一元同调的一个实例，因此我们可以利用Everaert和Gran的结果得到所有维的Hopf公式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Journal of Homotopy and Related Structures Mathematics-Geometry and Topology

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期刊介绍： Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences. Journal of Homotopy and Related Structures is intended to publish papers on Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.

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