# Compatible actions in semi-abelian categories

## TL;DR

This paper extends the concept of compatible actions to semi-abelian categories, introduces a new Peiffer product construction, and explores its universal properties and connections to crossed modules.

## Contribution

It generalizes compatible actions to semi-abelian categories and provides a new construction of the Peiffer product with proven universal properties.

## Key findings

- Peiffer product construction generalizes known cases
- Established connection between compatible actions and crossed modules
- Proved equivalence of Peiffer product with existing definitions

## Abstract

The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday and in the case of Lie algebras by Ellis. 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.