Maltsiniotis ’ S First Conjecture For

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories such that Quillen’s K-theory of an exact category E agrees with the Ktheory of its bounded derived category D(E). Schlichting [Sch02] showed that such a K-theory for triangulated categories cannot exist. It was then natural to ask about the definition of a nice K-theory for algebraic structures interpolating between E and D(E). The best known intermediate structure is C(E), the Waldhausen category of bounded complexes in E, with quasi-isomorphisms as weak equivalences and cofibrations given by chain morphisms which are levelwise admissible monomorphisms. The derived category D(E) is the localization of C(E) with respect to weak equivalences. The Gillet-Waldhausen theorem, relating Quillen’s K-theory to Waldhausen’s K-theory, states that the homomorphisms τn : Kn(E) −→ Kn(C (E)), n ≥ 0, induced by the inclusion E ⊂ C(E) of complexes concentrated in degree 0, are isomorphisms. The category C(E) is considered to be too close to E so one would still like to find an algebraic stucture with a nice K-theory interpolating between C(E) and D(E). The notion of a triangulated derivator [Gro90, Mal07] seems to be a strong candidate. Maltsiniotis [Mal07] defined a K-theory for triangulated derivators together with natural homomorphisms ρn : Kn(E) −→ Kn(DE), n ≥ 0, 1991 Mathematics Subject Classification. 18E10, 18E30, 18F25, 19B99, 55S45.