Homotopy theory of dg sheaves
joint work with
Utsav
Choudhury
 Submitted (29 pages)
 arXiv:1511.02828
(pdf, source)
 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.
 Comments. After completing this note we learned that this description of the fibrant objects also appeared in doi:10.1016/j.aim.2004.07.007 and in the Ph.D. thesis of Brad Drew.
An isomorphism of motivic Galois groups
joint work with
Utsav
Choudhury
 Submitted (54 pages)
 arXiv:1410.6104
(pdf, source). Last
updated: 19 November 2015
 Abstract. Ayoub's weak Tannakian formalism applied to the Betti realization
for Voevodsky motives yields a proalgebraic group, a candidate for
the motivic Galois group. In particular, each Voevodsky motive gives
rise to a representation of this group. On the other hand Nori
motives are just representations of Nori's motivic Galois
group. In this article it is proved that these two groups are
isomorphic.
Traces in monoidal derivators, and homotopy colimits
 Adv. Math. 261 (2014), pp. 2684. 10.1016/j.aim.2014.03.029
 arXiv:1303.0153
(pdf, source). Last
updated: 16 July 2014
 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
in arXiv:1406.7854.
Traces, homotopy theory, and motivic Galois groups
The LefschetzVerdier trace formula and a generalization of a
theorem of Fujiwara
after Y. Varshavsky
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