Denis-Charles Cisinski

Dendroidal sets and simplicial operads

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by A. Joyal and J. Lurie. We also explain how this theory applies to the usual notion of operad (i.e. with a single colour) in the category of spaces.

Dendroidal sets as models for homotopy operads

The homotopy theory of infinity-operads is defined by extending Joyals homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point

Théories homotopiques dans les topos

Abstract The purpose of these notes is to give an ad hoc construction of a closed model category structure on a topos inverting an arbitrary small set of arrows. Moreover, a necessary and sufficient condition for those structures to be proper is given. As an example, the Joyal closed model category structure on the category of simplicial objects of a topos is constructed without the use of (boolean) points.

Mixed Weil cohomologies

Abstract We define, for a regular scheme S and a given field of characteristic zero K, the notion of K-linear mixed Weil cohomology on smooth S-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of G m behaves correctly), and Kunneth formula. We prove that any mixed Weil cohomology defined on smooth S-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over S to the derived category of the field K. This implies a finiteness theorem and a Poincare duality theorem for such a cohomology with respect to smooth and projective S-schemes (which can be extended to smooth S-schemes when S is the spectrum of a perfect field). This formalism also provides a convenient tool to understand the comparison of such cohomology theories.

Locally constant functors

We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives

V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.