Motivic Verona by Peter Arndt 5/6: Working with motivic spectra and a unifying abstract approach
Channel: Logic Seminar Verona
Date: 13/02/2019

Minicourse by Peter Arndt (University of Düsseldorf) on abstract and motivic homotopy theory at the University of Verona

Lecture notes: https://github.com/nakaday/Motivic_Verona

Motivic homotopy theory is a fusion of homotopy theory and algebraic geometry. In analogy to the homotopy category of topological spaces, obtained by making the unit interval contractible, one obtains a homotopy category of schemes by making the affine line contractible. This category has a rather topological flavour; one can for example talk
about suspensions, loop spaces and classifying spaces and represent cohomology theories by spectra. Apart from the ground-breaking proof of the Bloch–Kato conjecture the approach has led to a multitude of results, for example on quadratic forms, vector bundles over schemes, algebraic cycles, algebraic K-theory and computations of stable homotopy groups of spheres in topology.

The minicourse will start with an introduction to abstract homotopy theory: We will see how inverting a class of arrows in a category leads to an infinity category, and how one can work with this using several formalisms. The special case of topological spaces with the class of weak equivalences serves both as a guiding example and as a technical
foundation; thus we will simultaneously review some classical homotopy theory of topological spaces.

We will then introduce motivic homotopy theory, emphasizing the parallels to classical homotopy theory. We will use an abstract setup, due to the speaker, which also encompasses complex and non-archimedean analytic geometry and derived algebraic geometry, as well as many new geometric settings. Starting from a cartesian closed, presentable infinity category and a commutative group object therein, we will see a
representation theorem for line bundles, a Snaith type algebraic K-theory spectrum, Adams operations, rational splittings and a rational motivic Eilenberg-MacLane spectrum. We will also present some more special results for schemes and analytic spaces.

For most of the course the only prerequisites are some basic category theory and topology. In particular no knowledge of algebraic geometry is supposed until the final parts.