Characterizations and Uniqueness of the stable motivic homotopy theory

• Applicant: Professor Dr. Jens Hornbostel
• Subject Area: Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 269515708

The idea of motivic homotopy theory is to apply constructions and techniques from classical homotopy theory to solve problems in algebraic and arithmetic geometry. (A list of succesful examples is provided in section 2.2.1 of the Proposal for the DFG-Priority Program.) The key object here is the stable motivic homotopy theory SH(k) for a given base field k as invented by Morel and Voevodsky in the late 90s. Important algebraic cohomology theories such as motivic cohomology, algebraic and hermitian K-theory and algebraic cobordism are representable in SH(k). The goal of this project is to obtain a better conceptual understanding of the stable motivic homotopy category SH(k) from different points of view, that is comparing various descriptions and characterizations of it, including the question of uniqueness. In particular, we wish to study descriptions of SH(k) in terms of derivators and of infinity-categories, building upon work of Ayoub and Robalo. Moreover, we will investigate if the rigidity theorem of Schwede concerning uniqueness of models for the classical stable homotopy category allows some kind of refinement to the motivic case. An underlying theme of all these problems is the question how much of the ``higher structure'' of SH(k) is already determined by its triangulated structure, that is independent of the choosen model resp. description.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1786: Homotopy Theory and Algebraic Geometry