  Motivated by our attempt to recast Cartan's work on Lie pseudogroups in a
more global and modern language, we are brought back to the question of
understanding the linearization of multiplicative forms on groupoids and the
corresponding integrability problem. From this point of view, the novelty of
this paper is that we study forms with coefficients. However, the main
contribution of this paper is conceptual: the finding of the relationship
between multiplicative forms and Cartan's work, which provides a completely new
approach to integrability theorems for multiplicative forms. Back to Cartan,
the multiplicative point of view shows that, modulo Lie's functor, the Cartan
Pfaffian system (itself a multiplicative form with coefficients!) is the same
thing as the classical Spencer operator.
