Abstract:

The homotopy category of spaces can be constructed as a localization of the category of topological spaces
or simplicial sets. On the other hand, in algebraic geometry one has to enlarge the category of geometric
objects under consideration (schemes) in order to obtain the analogue -- Morel-Voevodsky's A¹-homotopy category.
In particular, in algebraic geometry homotopy types are not necessarily represented by schemes.

In noncommutative geometry schemes are replaced with small stable categories and the role of cohomology theories
is taken by localizing functors on the category of small stable categories, Cat_st. The corresponding category of
motives Mot_loc representing all localizing functors was originally constructed completely analogously to the
Morel-Voevodsky's construction via enlarging Cat_st. It turns out that Mot_loc is equivalent to a localization
of Cat_st with respect to a class of morphisms we call motivic equivalences. We will discuss this claim and
explore its consequences. This talk is based on a joint work with Maxime Ramzi and Christoph Winges.