## Lecture (Winter 2022/23): Six-functor formalisms

Tuesdays, 10:15 -- 12:00, Kleiner HörsaalThe idea of a six-functor formalism emerged in the study of etale cohomology. To any scheme, one can associate its derived category of l-adic sheaves, and this comes equipped with six operations: Tensor product and internal Hom; pullback and pushforward; proper pushforward and exceptional inverse image. Moreover, these satisfy some compatibilities, notably proper base change and the projection formula. Similar formalisms exist in many other contexts, for example using usual sheaves on topological spaces, or D-modules on algebraic varieties, or ...

Recently, there have been some interesting developments in the area:

-- The development of some new 6-functor formalisms;

-- An axiomatization of the notion of a 6-functor formalism, including the implicit higher-categorical coherences;

-- A relation between 6-functor formalisms and "geometric rings" (i.e. ring objects in certain geometric categories such as schemes or stacks);

-- The category of cohomological correspondences and resulting simplifications in the proofs of Poincare duality and Lefschetz fixed point theorems.

The goal of this course will be to take account of these developments, with an emphasis on the general theory of 6-functor formalisms. It is not meant to be a first course on six-functor formalisms; it will be assumed that one is familiar with at least one of the traditional six-functor formalisms.

Lecture Notes