Oliver Röndigs

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Sn-action. After taking into account only finitary functors-which may be done in two different ways-the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].

Motivic slices and coloured operads

Colored operads were introduced in the 1970s for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to colored operads and Bousfield (co)localizations are given in separate appendices.

On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P1-spectrum MGL of Voevodsky is considered as a commutative P1-ring spectrum. Setting

Slices of hermitian K–theory and Milnor’s conjecture on quadratic forms

\mathrm{MGL}^i = \bigoplus_{p-2q =i}\mathrm{MGL}^{p,q}

The Arone–Goodwillie spectral sequence for Σ∞Ωn and topological realization at odd primes

we regard the bigraded theory MGLp,q as just a graded theory. There is a unique ring morphism

Modules over motivic cohomology

\phi\colon\mathrm{MGL}^0(k)\to\mathbb{Z}

Enriched Functors and Stable Homotopy Theory

which sends the class [X]MGL of a smooth projective k-variety X to the Euler characteristic

Motives and modules over motivic cohomology

\chi(X, \mathcal{O}_X)

Motivic strict ring models for

of the structure sheaf

K

\mathcal{O}_X