Peiffer product and Peiffer commutator for internal pre-crossed modules

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.

Compatible actions of Lie algebras

“…As a consequence we get that the internal definition of the Peiffer product given in [7] coincides with the one introduced in [14].…”

“…As a final result we want to show that the coproduct in XMod L (Lie R ) can be obtained through the Peiffer product: this coproduct has already been characterized in a different way in [5] by using semi-direct products instead of the Peiffer product, but this approach generalizes the one used for XMod L (Grp) in [3]. Consequently, we also obtain that the Peiffer product defined above (and hence the one from [14]) coincides with the one defined in [7] when restricted to Lie R . Definition 3.1.…”

“…In order to see this, we can use the decomposability of the action ξ M+N M on both sides of (7) obtaining that the one on the left becomes…”

Galois theory and the categorical Peiffer commutator

“…This category RG B (C) is not pointed, but quasi-pointed [5], in the sense that it has an initial object (B, 1 B , 1 B , 1 B ), a terminal object (B × B, p 1 , p 2 , (1 B , 1 B )) and, moreover, the canonical arrow from the initial to the terminal object is a monomorphism. The category RG B (C) is known to be equivalent to the category PXMod B (C) of (internal) precrossed modules [25] over a fixed object B, also studied in [31,15].…”

Higher commutator conditions for extensions in Mal'tsev categories

“…In a recent article A. Cigoli, S. Mantovani and G. Metere have investigated a categorical notion of Peiffer commutator that is certainly related to the results presented in the present work, in the special case of semi-abelian categories. In the future it would be interesting to compare the categorical commutator conditions arising from the reflection RG(C)/B → Grpd(C)/B considered in our work with this new notion of Peiffer commutator [10].…”