Low-dimensional non-abelian Leibniz cohomology

We construct the zero and first non-abelian cohomologies of Leibniz algebras with coefficients in crossed modules, which differ from those of Gnedbaye and generalize the zero and first Leibniz cohomologies of Loday and Pirashvili. We also introduce the second non-abelian Leibniz cohomology and describe its relationship with extensions of Leibniz algebras by crossed modules. We obtain a nine-term exact non-abelian cohomology sequence. For Lie algebras we compare the non-abelian Leibniz and Lie cohomologies.