Category theory is a wonderful tool to understand mathematical structures. Since its origins in algebraic topology in the 1940’s, it has become a unifying language for many fields of mathematics. A category encodes the relations between mathematical objects by specifying the morphisms between them.
There are nevertheless interesting mathematical phenomena (for instance in algebraic topology, homological algebra, algebraic and differential geometry, mathematical physics, etc.) which category theory does not capture well. The basic problem, which has been understood almost since the beginning of category theory, is that we often need to record finer relationships between morphisms than just equality. The basic examples of such finer relations are natural isomorphisms of functors, homotopies between continuous maps of topological spaces, chain homotopies between morphisms of chain complexes.
Infinity-category theory is a modern solution to this old problem. Roughly speaking, in an infinity-category, there is a space of morphisms between any two objects, considered up to coherent homotopy. The theory provides a framework which unifies classical category theory, homotopy theory and homological algebra, and which has already found many applications throughout mathematics.
This course will explain the basic theory of infinity-categories, following the approach based on simplicial sets and quasicategories developed by André Joyal and Jacob Lurie.
The course will be roughly divided into four parts:
1) Foundations
We will cover
- Simplicial sets
- Kan complexes and Kan fibrations
- Definitions of infinity-categories, functors, natural transformations...
- Homotopy category of a quasicategory, infinity-groupoids
- Lifting calculus, saturated classes and the small object argument
- Exponentiation of various classes of fibrations
- Slice infinity-categories
- Joyal lifting theorem and Grothendieck's homotopy hypothesis.
- Mapping spaces in infinity-categories
- Constructing infinity-categories in practice from 2-/topological/simplicial/dg-categories
2) The Grothendieck construction for infinity-categories
A major issue in infinity-category theory is that many constructions of functors are only defined up to an infinite hierarchy of compatible choices, which quickly become unmanageable. The Grothendieck construction (which is in this context a difficult theorem of Lurie) is an elegant solution to this problem: the collection of all possible choices can be organised into a (co)cartesian fibration, and this is equivalent to the original problem of constructing a functor. We will discuss various aspects of this (but not provide a complete proof).
3) Infinity-category theory
Equipped with the tools of Parts 1) and 2), we will cover (with varying levels of details) analogues of the central tools of usual category theory:
- Yoneda embedding
- (Co)limits
- Adjunctions
4) Advanced topics
We will give a brief introduction to a few topics which are central in many applications of infinity-categories:
- Presentable infinity-categories
- Stable infinity-categories and spectra
- (Symmetric) monoidal infinity categories
- Homological algebra and dg-categories from the infinity-categorical viewpoint
- verantwortliche Lehrperson: Simon Pepin-Lehalleur