Six-Functor Formalisms (Fall 2023)

The notion of duality appears rather ubiquitously in nature, and unravels important symmetries that are often useful to exploit. A six functor formalism often explains this symmetry, and this seminar will aim to explore the abstract framework recently developed by Mann, Liu, Lu, Zhang and many others. Scholze has recently written up notes [1] regarding these developments, and these notes will comprise the bulk of what we intend to cover. In particular, we intend to cover the abstract generality of the six functor formalism, with emphasis on higher homotopical techniques, Cohomological correspondences and Poincare-Verdier duality, and lots of important examples, including duality for coherent sheaves aka Grothendieck duality, and duality for analytic spaces. Depending on time, we may also discuss applications to nearby cycles.

Schedule

Sept 6
Caleb Ji
Organizational meeting and introduction Notes
We will give an introduction to six functor formalisms using the example of etale cohomology. Key features involve proper base change, the projection formula, and Poincare duality. Then we will describe the modern approach due to Mann in which six functor formalisms are encapsulated through the infinity category language.
Sept 13
Ivan Zelich
Abstract 6-functor formalisms
We plan to delve a little deeper in the abstract generality behind the six functor formalism, in particular describing the underlying homotopical categories that play a crucial role in the sequel. This follows Lecture 2 of Scholze's notes.
Sept 20
Caleb Ji
Derived infinity categories Notes
The purpose of this talk is to explain where derived categories and triangulated categories are situated in the modern perspective to a classical algebraic geometer. We will explain where the non-functoriality of cones in triangulated categories comes from and use this to motivate the definition of stable ∞-categories. We will then construct the derived ∞-category. Finally, we will say some words about symmetric monoidal ∞-categories and the Grothendieck construction.
Sept 27
Ivan Zelich
Symmetric monoidal infinity categories
We discuss the infinity category of infinity categories and symmetric monoidal infinity categories, being as concrete as possible.
Oct 4
Ivan Zelich
TBA
TBA