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Generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds we define secondary theories and characteristic classes for smoothétale groupoids. As special cases we obtain versions of the groups of differential characters for smoothétale groupoids and for orbifolds.In the first section we will give an overview of the basic elements of de Rham and Chern-Weil theory for smoothétale groupoids following [LTX]. In the following section we define multiplicative cohomology and generalized differential characters for smoothétale groupoids and study their main properties. We will make explicit use here of the simplicial machinery as developed in [FN], but for the convenience of the reader we will recall many of the details. This was in part influenced by the work of Dupont [D1], [D2] and Dupont-Hain-Zucker [DHZ]. In the third section we introduce multiplicative bundles and versions of multiplicative K-theory for smoothétale groupoids. In the final section we then construct characteristic classes of elements in multiplicative K-theory for smoothétale groupoids with values in the groups of multiplicative cohomology and generalized differential characters.In a sequel to this article we aim to study the relationship between our generalized groups of multiplicative cohomology and smooth Deligne cohomology forétale groupoids and to construct in a unifying way secondary theories and characteristic classes for principal bundles over orbifolds, foliations and differentiable stacks. Versions of smooth Deligne cohomology for particularétale groupoids were also studied before, for example in the case of particular orbifold groupoids by Lupercio-Uribe [LU] and for general transformation groupoids by Gomi [Go]. It will be interesting to analyse if these secondary theories and characteristic classes could be also defined in the context of algebraic geometry, for example in order to extend algebraic differential characters as introduced and studied by Esnault [E1], [E2] to smooth Deligne-Mumford stacks. Real Deligne cohomology groups of proper smooth Deligne-Mumford stacks over the complex numbers C appear also in the context of arithmetic intersection theory on Deligne-Mumford stacks as introduced recently by Gillet [Gi].

Abstract. We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi's multiplicative Ktheory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.

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In this article we give a construction of Max Karoubi's multiplicative K-theory as the K-theory of an appropriate functor between two categories. We use this construction to explain why the two definitions of relative multiplicative K-theory for a compact pair of manifolds we give in the article agree. Mathematics Subject Classification (2000). 55N15, 18F05.