Some Aspects of Homotopic Algebra and Non-Abelian (CO)Homology Theories

Inassaridze, Nick

doi:10.1007/s10958-016-2703-7

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2020

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“…Later, in [14], the definition was adapted to the context of Lie algebras, where it was further studied in [26,13]. Since then, several other particular instances of compatible actions have been defined or used, in various settings: see for example [17,9,8], and [22,23] where the non-abelian derived functors of ⊗ were studied.…”

Section: Introductionmentioning

confidence: 99%

Compatible actions in semi-abelian categories

Micco

Linden

2020

Homology, Homotopy and Applications

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The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday [6] and in the case of Lie algebras by Ellis [14]. In this article we extend it to the context of semi-abelian categories (that satisfy the Smith is Huq condition). We give a new construction of the Peiffer product, which specialises to the definitions known for groups and Lie algebras. We use it to prove our main result, on the connection between pairs of compatible actions and pairs of crossed modules over a common base object. We also study the Peiffer product in its own right, in terms of its universal properties, and prove its equivalence with existing definitions in specific cases.

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