  The present paper, though inspired by the use of tensor hierarchies in
theoretical physics, establishes their mathematical credentials, especially as
genetically related to Lie algebra crossed modules. Gauging procedures in
supergravity rely on a pairing - the embedding tensor - between a Leibniz
algebra and a Lie algebra. Two such algebras, together with their embedding
tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed
modules are particular cases. This paper is devoted to showing that any
Lie-Leibniz triple induces a differential graded Lie algebra - its associated
tensor hierarchy - whose restriction to the category of Lie algebra crossed
modules is the canonical assignment associating to any Lie algebra crossed
module its corresponding unique 2-term differential graded Lie algebra. This
shows that Lie-Leibniz triples form natural generalizations of Lie algebra
crossed modules and that their associated tensor hierarchies can be considered
as some kind of 'lie-ization' of the former. We deem the present construction
of such tensor hierarchies clearer and more straightforward than previous
derivations. We stress that such a construction suggests the existence of
further well-defined Leibniz gauge theories.
