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The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between K-theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes differentielles non commutatives et operations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm. Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836. Dans cet article, nous presentons la cohomologie 'a coefficients arbitraires des ensembles simpliciaux en termes de formes differentielles. A priori, le sujet n'apparalt pas comme vraiment nouveau puisque, dans le meme esprit, une presentation fut proposee dans les annees 70 par A. Grothendieck [1] et E. Miller [10], apres le travail de D. Quillen et D. Sullivan en cohomologie rationnelle. Cependant, les avantages de la presentation adoptee ici, qui utilise de maniere intensive les formes diftfrentielles non commutatives et qui prolonge dans un certain sens l'article de H. Cartan sur le sujet [1], sont multiples. Tout d'abord, en specialisant les anneaux simpliciaux de base, on retrouve a la fois l'algebre des cochaines usuelles et celle des formes differentielles (en faisant commuter les formes). Ainsi, est defini un "chapeau" commun aux formes diftfrentielles usuelles et aux cochaines (a coefficients entiers). Ceci permet de simplifier considerablement la definition de la cohomologie de Deligne-Beilinson (pour une variete COO munie d'une filtration de son complexe de de Rham: cf. [6] et le ?5). Received by the editors August 9, 1994. 1991 Mathematics Subject Classification. Primary 55N35. (?) 1995 American Mathematical Society

Formes différentielles non commutatives et cohomologie à coefficients arbitraires