# Motivic Homotopy Theory Program

## Current events

Lectures on cross functors

Wed. 11am in Dilworth Room (first lecture Oct. 10)

For any category $C$ with fiber products and any 2-category $D$ we defined a class of cross functors from $C$ to $D$. A cross functor with values in the 2-category of categories assigns to an object of $C$ a category and to a morphism four functors between the corresponding categories together with some additional data. The main example was a cross functor on the category of algebraic varieties which sends a variety to the derived category of $l$-adic sheaves on it and such that the corresponding four functors are the standard direct and inverse images. In the first part of the lectures we introduced and studied cross functors in the general categorical context. In the second we studied properties of homotopy invariant cross functors on the category of schemes including a general duality theorem. In the last part we constructed the cross functor of motivic stable homotopy and used it to prove duality and the blow-up long exact sequences for the motivic (co-)homology theories.

## About the motivic homotopy theory

The motivic homotopy theory is the homotopy theory for algebraic varieties and, more generally, for Grothendieck's schemes which is based on the analogy between the affine line and the unit interval. Eventually, the motivic homotopy theory is expected to provide techniques which may help to solve problems in algebraic geomerty such as various "standard conjectures on algebraic cycles", Beilinson-Soule vanishing and rigidity conjectures, the Bloch-Kato conjecture etc. Unlike many other approaches tried in the past decades, the motivic homotopy theory attempts to understand the category of algebraic varieties "from the inside" i.e. without explicit use of such constructions as Betti cohomology or Hodge structures which require "external" topological or analytic tools. The first success of this theory was the proof of the Milnor Conjecture (see also the Bourbaki seminar talk La conjecture de Milnor by Bruno Kahn) relating Galois cohomology with Milnor's K-theory mod 2 and quadratic forms. There is also a good progress on the Bloch-Kato conjecture which is the generalization of the Milnor Conjecture to odd primes. For the most recent info see the following part of the Markus Rost web page. The basic constructions of the motivic homotopy theory are described in

In the current state the motivic homotopy theory has as its primary objects of study categories of three types: the unstable $A^1$-homotopy categories, the stable $A^1$-homotopy categories and the triangulated categories of motives. Each of these types consists of a family of categories depending on the base scheme and in the two later cases on the ring of coefficients.