Partial commutators

We introduce a notion of partial commutator in the context of a normal category with cokernels, as the commutator resulting from the recently introduced notion of partial commutativity. The partial commutator is defined by making slight modifications to the definition of the Huq commutator, and they agree in the unital context. We show that some fundamental properties of the Huq commutator can still be recovered. In particular, the partial commutator vanishes if and only if the morphisms partially commute, and in a normal category with finite colimits, it always exists and can be obtained via colimits as it is the case in absolute settings. As an application, we investigate partial commutators in the category \(\textsf{LAlg}\) of L-algebras in the sense of W. Rump.