Falk Beckert

Bergische Universität Wuppertal, Fachbereich C
Gaußstr. 20
42097 Wuppertal

Raum: G.15.13
Tel: (0202) 439-2528
Mail: beckert (at) math.uni-wuppertal.de
Arbeitsgruppe Topologie


Tutorium zu Lineare Algebra II
Prof. Dr. Jens Hornbostel


Abstract cubical homotopy theory (2018, arxiv , j/w Groth)
Triangulations and higher triangulations axiomatize the calculus of derived cokernels when applied to strings of composable morphisms. While there are no cubical versions of (higher) triangulations, in this paper we use coherent diagrams to develop some aspects of a rich cubical calculus. Applied to the models in the background, this enhances the typical examples of triangulated and tensor-triangulated categories. The main players are the cardinality filtration of n-cubes, the induced interpolation between cocartesian and strongly cocartesian n-cubes, and the yoga of iterated cone constructions. In the stable case, the representation theories of chunks of n-cubes are related by compatible strong stable equivalences and admit a global form of Serre duality. As sample applications, we use these Serre equivalences to express colimits in terms of limits and to relate the abstract representation theories of chunks by infinite chains of adjunctions. On a more abstract side, along the way we establish a general decomposition result for colimits, which specializes to the classical Bousfield–Kan formulas. We also include a short discussion of abstract formulas and their compatibility with morphisms, leading to the idea of universal formulas in monoidal homotopy theories.

The bivariant parasimplicial S-construction (PhD-Thesis, in preparation)
In abstract stable homotopy theory coherent strings of composable morphisms play an important role in various important constructions (for example algebraic K-theory or higher Toda brackets). In a first step we will prove a strong comparison result relating composable strings of morphisms and presheaves on cubes with support on a path from the initial to the final object. We observe that both structures are equivalent (by passing to higher analogues of mesh categories) to distinguished coherent presheaves on special classes of morphism objects in the 2-category of parasimplices. Furthemore we show that the notion of distinguished coherent presheaves generalizes well to arbitrary morphism objects in this 2-category. Understanding these presheaves in general will require a detailed analysis of the 2-category of parasimplices as well as basic results from abstract cubical homotopy theory (since subcubes of distinguished presheaves very often turn out to be bicartesian). Finally we show that the previous comparison result extends to a duality theorem on general categories of disinguished coherent presheaves.

Dualities and adjunctions of stable derivators (2013)
We analyse the structure of the 2-category of derivators and their pointed and stable variants. As a first main result, we prove that cofree derivators are dualizable. Moreover, we show that the stability of a derivator can be characterized by the existence of certain functors, which are adjoint to the homotopy Kan extensions. Furthermore, we prove that in the stable situation, a well-behaved Spanier-Whitehead-Duality exists, if we assume additionally some finiteness conditions. This will enable us to prove that certain duality functors describe the passage to adjoints in a particular sub-2-category of stable derivators.
last modified: 19.03.2018