Higher Categories for the Working Mathematician

Scientific Programme

An introduction to stable ∞-categories and spectra (Rune Haugseng)

This lecture series will give an introduction to spectra and their tensor product, focusing on categorical aspects of the ∞-category of spectra, as well as the broader context of stable ∞-categories. It will mainly be based on work of Lurie and Gepner–Groth–Nikolaus.

1. From abelian groups to additive ∞-categories. I will start by introducing the natural ∞-categorical analogues of commutative monoids and abelian groups, and then consider their connection to semiadditive and additive ∞-categories. We will also discuss the equivalence between commutative groups in spaces and infinite loop spaces.

2. Spectra and stable ∞-categories. We will motivate the ∞-category of spectra by considering several descriptions thereof: as non-connective delooping sequences, as cohomology theories, and as a natural enlargement of the stable homotopy category of finite CW-complexes. We will then define stable ∞-categories and consider spectra as an instance of stabilization.

3. The tensor product of spectra. We will first define symmetric monoidal ∞-categories and discuss some constructions of these. In particular, we will look at Day convolution, which gives us a way to define tensor products of commutative monoids and groups in spaces, as well as of spectra. We will then define algebras and modules in symmetric monoidal ∞-categories, which in particular lets us consider ring spectra and modules over them.

4. The presentable tensor product and the universal property of spectra. In the last lecture we will briefly introduce presentable ∞-categories and their tensor product. Then we will explain how presentable stable ∞-categories can be described as being precisely modules over the ∞-category of spectra, and how this gives a universal property of the symmetric monoidal ∞-category of spectra, as well as the analogous results for presentable (semi)additive ∞-categories.

(∞,1)-categories and beyond (Viktoriya Ozornova)

In this lecture series, we want to build basic language for the use of (∞,1)-categories and also (∞,n)-categories, both in other talks and in mathematical 'real life'. In a model, a lot of this was done by Joyal, Lurie, Riehl and Verity, whereas a model-independent approach is currently being developed by Cisinski, Cnossen, Nguyen and Walde.

1. Introduction to (∞,1)-categories. Why are ordinary categories not enough? What is actually the difference between (∞,1)-categories and just categories? What things go through essentially the same and which require more work? We will start addressing these questions in the first talk.

2. Examples. How to construct examples of (∞,1)-categories? We will devote this talk to several ways of doing so. To relate to previously done 'homotopy-coherent mathematics', we discuss a presentation of (∞,1)-categories given by model categories; this allows for further examples.

3. Introduction to (∞,n)-categories. After building a basic understanding of (∞,1)-categories, we want to turn our attention to changing the parameter 1 to an arbitrary natural number n. In this talk, we will discuss the axiomatization by Barwick–Schommer-Pries and some examples. If time permits, we will also look into models of (∞,n)-categories.

4. Complicial sets as a model of (∞,n)-categories. After basics of (∞,n)-categories, I will speak about a particular implementation referred to as 'complicial sets' based on work of Verity, Riehl, Rovelli and myself, and further insights by Loubaton. I will discuss advantages and drawbacks of this model.

An introduction to higher principal bundles and their applications (Christian Saemann)

This lecture series introduces structures in higher differential geometry with a focus on applications in mathematical and high-energy physics. The main goal is to define higher principal bundles with an interesting notion of connection and to discuss several applications of these structures.

1. Higher principal bundles. We begin by motivating the need for a higher notion of parallel transport, drawing on both physical and mathematical perspectives. This leads naturally to the concept of categorification, which we briefly review. After giving a concise overview of various descriptions of higher (Lie) groups, illustrated with examples, we conclude the first lecture with the definition of higher principal bundles and a number of important examples.

2. Connections on higher principal bundles — locally. By differentiating higher Lie groups, we obtain higher Lie algebras, which are closely related to homotopy Lie algebras. We recall the local description of principal bundle connections in terms of Lie-algebra–valued differential forms. Together with linearized bundle isomorphisms, these assemble into an action algebroid—the BRST-algebroid. A naive categorification of these structures encounters obstructions, and additional algebraic data ("adjustments") are required to arrive at a generally useful notion of higher connections. As an example, we study the particularly rich BRST-algebroid underlying the tensor hierarchies of gauged supergravity.

3. Connections on higher principal bundles — globally. We discuss the integration of (higher) BRST-algebroids to the corresponding BRST-groupoids. Stackification of these groupoids then yields the definition of higher principal bundles with connections. A key example is given by string structures, which are higher analogues of spin structures.

4. Applications. We explore several examples of interest in both mathematics and physics: higher monopoles and instantons, T-duality in higher geometric settings, the Penrose–Ward transform, and six-dimensional superconformal field theories. If time permits, we also review the case of Lie groupoid bundles.

(∞,n)-categories in action (Claudia Scheimbauer, Walker Stern)

In this lecture series, we will see ∞-categorical techniques in action: we will introduce 𝔼ₙ-algebras and factorization algebras in an ∞-category 𝒞 and discuss properties such as gluing and additivity statements. Time permitting, we will discuss applications to topological field theories. In the second half we discuss an application to representation theory: we discuss how one can use (lax) 𝔼ₙ-algebras in higher categories of spans to produce an 𝔼₂-structure on a higher category of representation-theoretic flavor.

1. (Scheimbauer) 𝔼ₙ-algebras in an ∞-category 𝒞 are algebras for the operad 𝔼ₙ of little disks; equivalently, they are objects in 𝒞 equipped with n compatible associative multiplications. Examples include n-fold loop spaces, (braided) monoidal categories, and A∞-algebras. Alternatively, they are a special case of a (potentially pointless) factorization algebra, a structure inspired by the observables of a quantum field theory. In this lecture, we will introduce these notions, discuss examples and first properties.

2. (Scheimbauer) In the second lecture, we will see how Kan extensions can be used in practice to prove things about factorization algebras. We will discuss gluing of factorization algebras and additivity of 𝔼ₙ-algebras and constructible factorization algebras. Time permitting, we will discuss applications to topological field theories; namely, the relevance for Morita (∞,n)-categories.

3. (Stern) We continue our discussion of 𝔼ₙ-algebras in higher categories by defining a variant of the Boardman–Vogt tensor product which allows us to consider lax 𝔼₂-algebras in symmetric monoidal (∞,n)-categories. We will further discuss under which conditions lax 𝔼₂-algebras simplify to strong 𝔼₂-algebras.

4. (Stern) A span—in algebraic geometry also called a correspondence—is a diagram of the form X ← F → Y in a higher category 𝒞. These can be viewed as the morphisms in a higher category Span(𝒞), where composition is given by pullback. We discuss (∞,n)-categories of spans, their universal properties, and how we can construct lax 𝔼₂-algebras in such categories. Time permitting, we show how 𝔼₂-monoidal categories (braided monoidal categories) in representation theory can be constructed from lax 𝔼₂-algebras in spans.

Discussion session

As part of the school there will be parallel discussion sessions on Thursday. Please suggest topics! We will search for volunteers to lead a discussion on the topics that are of interests to the most people. 

Sample suggestions: 
•⁠  ⁠Can you give examples of ... ?
•⁠  ⁠How does the proof of X work?
•⁠  ⁠Can you provide more details about that black-box that was mentioned during the minicourse?
•⁠  ⁠Can you tell more about that nice poster you showed?

You can put your suggestions in the suggestions box in the main lecture hall by Tuesday night, and we will have a ballot on Wednesday after the coffee break to choose which topics to cover.