LECTURES ON CROSS FUNCTORS


For any category C with fiber products and any 2-category D we define
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 is 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 will introduce and study cross
functors in the general categorical context. In the second we will
study properties of homotopy invariant cross functors on the category
of schemes including a general duality theorem. In the last part we
will construct the cross functor of motivic stable homotopy and use it
to prove duality and the blow-up long exact sequences for the motivic
(co-)homology theories.