Compatible actions in semi-abelian categories

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.

Keywords

semi-abelian category, pair of compatible actions, internal action, crossed module, Peiffer product, non-abelian tensor product

2010 Mathematics Subject Classification

18D35, 18E10, 20J15

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The second author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS.

Received 13 August 2019

Received revised 5 December 2019

Accepted 20 December 2019

Published 29 April 2020