Higher Category Theory

A category is something consisting of a collection of objects, a collection of morphisms, and a rule for how to compose such morphisms. This notion is ubiquitous in mathematics; any kind of objects one wishes to study (sets, vector spaces, manifolds, schemes, etc.) can usually be organized into a category. A higher category (or ∞-category) has objects and morphisms between objects, but also 2-morphisms between morphisms, 3-morphisms between 2- morphisms, etc. This notion was first invented in the context of homotopy theory, where morphisms are continuous maps, 2-morphisms are homotopies between them, and so on. Over the past 20 years, higher category theory has seen enormous development through the works of Joyal and Lurie and has rapidly found many applications to other fields, such as algebraic geometry, number theory, and representation theory. It is now a fundamental tool in modern mathematical research.

The aim of this course is to develop the basic notions of the theory of ∞-categories. A student who completes the course should then be able to dive into any of the standard references on the subject and read research papers that depend on the methods of higher category theory. Some of the topics we will cover are the following:

• Simplicial sets
• Nerves of categories
• Infinity-groupoids and ∞-categories
• Limits and colimits in ∞-categories
• Functors between ∞-categories
• Fibrations of ∞-categories

This is an advanced course, intended to introduce students to a set of methods very important to modern research. We will assume basic familiarity with the concepts of category theory, such as functors and natural transformations, limits and colimits, and adjoint functors. Some familiarity with algebraic topology will be helpful, but not strictly necessary.

Lectures: Mondays 3.15-5pm in BBG 017. First lecture is on September 20.

Material: We will rely on various sources during the course, which I'll keep track of here. Standard references are Lurie's Higher Topos Theory, the online platform Kerodon, and Cisinski's Higher Categories and Homotopical Algebra. Many sets of lecture notes are now available; let me mention those of Rezk, of Groth and of Antolín-Camarena.

Sept 20: Introduction. Simplicial sets. Examples: the nerve of a category, the singular complex of a topological space, standard simplices. The homotopy category of a simplicial set, geometric realization. Problem set 1.

Sept 27: Skeletal filtration and geometric realization. Horns, Kan complexes. Characterizing the nerve of a category. Problem set 2.

Oct 4 (online): Infinity-categories, composition, homotopies, isomorphisms, the homotopy category of an infinity-category.

Oct 11: See Oct 4.

Oct 18: Functors and functor categories. Lifting properties, weakly saturated classes. Inner anodyne maps and pushout-products. Problem set 3.

Oct 25: More on pushout-products of anodyne maps. Uniqueness of composition. Types of fibrations (trivial, Kan, left, right, inner). Problem set 4.

Nov 1: Examples of left/right fibrations. Joins and slices. Isofibrations, conservative functors. Joyal's lifting theorem. Problem set 5.

Nov 8: Joyal's lifting theorem (continued). The core of an infinity-category. The pointwise criterion for natural isomorphisms.

Nov 15: Topological categories, simplicial categories, and infinity-categories. The path category and the homotopy-coherent nerve. Problem set 6.

Nov 22: Equivalences of infinity-categories. Categorical equivalences of simplicial sets. Problem set 7.

Nov 29: Essentially surjective and fully faithful functors. Some homotopy theory of Kan complexes.

Dec 6: Characterizing equivalences of infinity-categories. Problem set 8.

Dec 13: Straightening of left fibrations. The case of an ordinary category. Problem set 9.

Dec 20: Straightening, continued.