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.


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 Notes
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
Construction of 6-functor formalisms and animation Notes
We continue our discussion of symmetric monoidal infinity categories and correspondences, and use them to define 6-functor formalisms. Finally, we discuss animation.
Oct 11
Ivan Zelich
3 functor formalisms I
We discuss in some detail the construction of a 3 functor formalism.
Oct 18
Ivan Zelich
3 functor formalisms II Notes
We continue our construction of 3 functor formalisms.
Oct 25
Ivan Zelich
Nov 1
Caleb Ji
F-isocrystals and crystalline cohomology
The six functor formalism for crystalline coefficients are not well-understood. In this talk we will begin with a rapid overview of crystalling cohomology. Then we will discuss Ogus's construction of convergent F-isocrystals and his proof that they are stable under pushforward along a smooth proper morphism.
Nov 8
Caleb Ji
Grothendieck duality via homotopy theory
The original construction of the six functor formalism for coherent sheaves was complicated and intricate. In this talk, I will introduce Neeman's approach which applies very generally to compactly generated triangulated categories. The method uses Thomason's localization theorem and Brown representability.
Nov 15
Ivan Zelich
Cohomological smoothness and Poincare duality I
We discuss cohomological smoothness in the context of 3-functor formalisms. This will allow us to define a general notion of Poincare duality in 6 functor formalisms and give a framework for proving it.
Nov 29
Amal Mattoo
Dec 6
Ivan Zelich
Cohomological smoothness and Poincare duality II
We describe conditions that allow us to check cohomological smoothness, which is generally difficult. This allows us to give a framework for proving Poincare duality in a general 6 functor setup. We end with the example of locally compact Hausdorff topological spaces.