Peiffer product and Peiffer commutator for internal pre-crossed modules

TL;DR

This paper extends the concepts of Peiffer product and commutator to internal pre-crossed modules in semi-abelian categories, characterizing crossed modules via trivial Peiffer commutators and exploring their coproduct structure.

Contribution

It introduces the notions of Peiffer product and commutator in a broad categorical context and characterizes crossed modules through these concepts, generalizing classical ideas.

Findings

Crossed modules characterized by trivial Peiffer commutator

Peiffer product realizes coproducts in certain categories

Applicable to various algebraic structures like groups and Lie algebras

Abstract

In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator <X,X> is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varietes, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.