Higher Categories for the Working Mathematician

27 Jul 2026, 08:00 → 31 Jul 2026, 12:15 Europe/Berlin

Maximum (Georg-August-Universität Göttingen, Mathematisches Institut)

Higher Categories for the Working Mathematician
A summer school of the RTG 2491

This school aims to provide an introduction to the world of higher categories and their applications, suitable for non-experts. There will be four lecture series focusing on the use of higher categories in various fields of research, ranging from homotopy theory to differential geometry and mathematical physics. It is organized as an activity of the Research Training Group Fourier Analysis and Spectral Theory (RTG 2491).

Live Stream

We will live stream the lectures via zoom. You can register here.

Please note that the lectures will not be recorded.

Program

Speakers

• Rune Haugseng (NTNU Trondheim). An introduction to stable ∞-categories and spectra
• Viktoriya Ozornova (MPIM Bonn). (∞,1)-categories and beyond
• Christian Saemann (Heriot-Watt University Edinburgh). An introduction to higher principal bundles and their applications
• Claudia Scheimbauer, Walker Stern (TU Munich). (∞,n)-categories in action

Poster Session

There will be a poster session on Tuesday.

Discussion Session

As part of the school there will be parallel discussion sessions on Thursday. We decided on the following topics. Note the room change for the last two topics. These two discussions will take place in the Nebengebäude, the building next door (see the links to a map).

• Model categories to (∞,1)-categories and homotopy (co-)limits vs infinity-(co-)limits (Maximum, chaired by Matteo)
• Explicit models of (∞, 1)- and (∞, n)-categories (Sitzungszimmer, chaired by Clémence)
• Enrichment for (∞,1) and (∞,n)-categories (HS 6, chaired by Alex)
• (Co-)cartesian fibrations (HS 5, chaired by Michael)

Conference Dinner

You can have a look at the menu here. One main dish from the list and water will be covered by us; anything further you will have to pay for yourself.

Preliminary Material

Here, you can find some preliminary material from a series of informal talks that cover helpful definitions.

Registration

If you want to participate, please fill out the registration form by May 22 at the latest. We can provide accomodation to registered participants but are not able to reimburse travel costs. Spots are limited, and we will reach out to you soon after the registration deadline.

Scientific organizers

Oscar Cosserat, Luca Dal Forno, Lars-Lennert Kerti, Kalin Krishna, Sergio Romero Alba, Stefano Ronchi, Federico Vigolo, Milena Weiershausen, Florian Wilsch

Administration

Linda Haber

 

Conference Dinner Menu

Notes on Simplicial Sets, Nerve, Sing, and Spectra

Poster

Program (ics)

Monday 27 July

1 Registration

2 Opening

Ozornova: (∞,1)-categories and beyond: Lecture 1

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.

3 Lecture

Speaker: Viktoriya Ozornova (MPIM Bonn)

12:00 Lunch break

Scheimbauer, Stern: (∞,n)-categories in action: Lecture 1

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.

4 Lecture 1

Speaker: Claudia Scheimbauer (TU Munich)

15:00 Coffee break

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 1

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.

Convener: Rune Haugseng (NTNU Trondheim)

5 Lecture

Speaker: Rune Haugseng (NTNU Trondheim)

16:30 Coffee Break

Ozornova: (∞,1)-categories and beyond: Lecture 2

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.

Tuesday 28 July

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 2

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.

10:30 Coffee break

Scheimbauer, Stern: (∞,n)-categories in action: Lecture 2

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.

12:00 Lunch break

Saemann: An introduction to higher principal bundles and their applications: Lecture 1

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.

15:00 Coffee break

Poster session

16:30 Coffee break

Ozornova: (∞,1)-categories and beyond: Lecture 3

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.

Wednesday 29 July

Saemann: An introduction to higher principal bundles and their applications: Lecture 2

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.

10:30 Coffee break

Discussion session: Preparation

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 3

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.

12:10 Lunch and free afternoon

Thursday 30 July

Ozornova: (∞,1)-categories and beyond: Lecture 4

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.

10:30 Coffee break

Saemann: An introduction to higher principal bundles and their applications: Lecture 3

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.

12:00 Lunch break

Scheimbauer, Stern: (∞,n)-categories in action: Lecture 3

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.

15:00 Coffee break

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 4

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.

16:30 Coffee break

Discussion session

6 Conference Dinner

Friday 31 July

Saemann: An introduction to higher principal bundles and their applications: Lecture 4

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.

10:30 Coffee break

Scheimbauer, Stern: (∞,n)-categories in action: Lecture 4

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.

7 Closing