Sheaves and Cohomology

Prerequisites:

Fundamental algebra and analysis II, Category theory, Manifolds

Course description:

This course is designed for third year or fourth year undergraduate students as well as graduate students, who intend to pursue studies in fundamental mathematics. Sheaf theory plays a fundamental role in modern geometry, including algebraic geometry, complex geometry, etc. It also plays a more and more important role in many other subjects, even in math physics. In this course, we pursue a modern approach to sheaf theory via category theory, in particular, the infinity category theory. The teaching objective is to enable students to understand in depth of the notions and tools in sheaf theory. This will prepare them for further study of many areas in pure mathematics, including but not limited to, algebraic geometry, complex geometry, higher category theory and even mathematical physics. This course will employ a variety of teaching methods, including lectures, class discussions, problem-solving sessions, and independent reading.

Learning objectives:

By the end of this course, students will be able to:

1. Explain the basic definitions of sheaves, and the sheafification of a presheaf. Provide various kinds of examples, mostly importantly, in topological manifolds and algebraic geometry..

2. Understand the definition of stalks, and use them to understand the local properties of sheaves.

3. Understand the classical definition of six functors formalism in the setting of both topological manifolds and algebraic geometry.

4 Understand the basic definitions of properties of infinity categories. Understand how to transfer various notations in ordinary category theory to infinity category theory.

5 Understand the basic definitions and properties of stable infinity categories, and their relation with ordinary abelian categories and triangulated categories.

6 Explain how to use infinity category theory to reformulate the six functors formalism.

Detailed topics covered:

1 Basic definitions of presheaves and sheaves taking values in a general category, the sheafification functor.

2 The classical six functors formalism, including their tensor product, internal hom, pullback functor, pushforward functor, proper pushforward, and proper pullback, as well as their relations, including, projection formula, proper base change, Kunneth formula, Verdier duality. Their applications to cohomology theory.

3 The basic definition of infinity categories via the theory of quasi categories based on simplicial sets. The generalization of various notations in category theory, notably, limits and colimits, presentable categories, adjoint functors.

4 The basic definition and properties of stable infinity categories, the limits and colimits in stable infinity categories. The basic definition and properties of symmetric monoidal infinity categories.

5 The six functors formalism via infinity category theory.