The simplicial extension of any functor from Sets to Sets which commutes with directed colimits respects weak equivalences. In the present paper we construct a framework which allows one to extend this result to a wide class of model categories and functors between such categories.

In this paper, we describe the sheaves of 1-homotopy groups of a simply-connected Chevalley group G; these sheaves can be identified with the sheafification of certain unstable Karoubi-Villamayor K-groups.

This paper is concerned with the theory of cup products in the Hopf cyclic cohomology of algebras and coalgebras. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving the noncommutative Weil algebra. Then we introduce the notion of higher -twisted traces and use a generalization of the Quillen and Crainic constructions (see [14] and [3]) to define the cup product. We discuss the relation of the cup product above and S-operations on cyclic cohomology. We show that the product we define can be realized as a combination of the composition product in bivariant cyclic cohomology and a map from the cyclic cohomology of coalgebras to bivariant cohomology. In the last section, we briefly discuss the relation of our constructions with that in [9]. More precisely, we propose still another construction of such pairings which can be regarded as an intermediate step between the “Crainic” pairing and that of [9]. We show that it coincides with what in [9] and as far its relation to Crainic's construction is concerned, we reduce the question to a discussuion of a certain map in cohomology (see the question at the end of section 5).

The results of the current paper were announced in [12].

For any regular noetherian scheme X and every k ≥ 1, we define a chain morphism ψk between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by ℚ. It is shown that the morphisms ψk induce in homology the Adams operations defined by Gillet and Soulé or the ones defined by Grayson.