Let and be differential graded Lie algebras and be a morphism of graded vector spaces. Define as
for any . The morphism is called a Cartan homotopy if it satisfies the two conditions
This name has an evident geometric origin: if is the tangent sheaf of a smooth manifold and is the sheaf of complexes of differential forms, then the contraction of differential forms with vector fields is a Cartan homotopy
In this case, is the Lie derivative along the vector field , and the conditions and , together with the defining equation and with the equations and expressing the fact that is a dgla morphism, are nothing but the well-known Cartan identities involving contractions and Lie derivatives.
It is a straightforward computation to see that, if is a Cartan homotopy, then the degree zero morphism of graded vector spaces is actually a dgla morphism.