  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.
