# 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.

- Organizers: Ivan Zelich, Caleb Ji
- When: Wednesdays 5:00 pm - 6:00 pm
- Where: Math 528
- References:
**[1]** Peter Scholze, Six-Functor Formalisms
- Additional notes from the seminar are not here.

#### 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