Abstract. In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.
Abstract. In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
Traces in monoidal derivators, and homotopy colimits
Abstract. A variant of the trace in a monoidal
category is given in the setting of closed monoidal derivators,
which is applicable to endomorphisms of fiberwise dualizable
objects. Functoriality of this trace is established. As an
application, an explicit formula is deduced for the trace of the
homotopy colimit of endomorphisms over finite categories in which
all endomorphisms are invertible. This result can be seen as a
generalization of the additivity of traces in monoidal categories
with a compatible triangulation.
Comments. In the meantime the same result has
been obtained independently