Higher Categories and Their Applications

John Baez

January 9-13, 2007

From January 9th to 13th, 2007, Peter May, Eugenia Cheng and I ran a workshop at the Fields Institute in Toronto, entitled Higher Categories and Their Applications. This was part of the Fields Institute program on Geometric Applications of Homotopy Theory, organized by Rick Jardine, Gunnar Carlsson and Dan Christensen.

What follows is a taste of what people said and did: abstracts of their talks, links to their talks (or related work), and photos. You can even hear a bunch of the talks using Real Audio.

You can also read a description of this conference in week245 of my column, This Week's Finds.

Tuesday January 9 - "Low Dimensions"

Dror Bar-Natan

Tom Fiore and Dan Christensen

9:30-10:30 — Tom Leinster: A Survey of the Theory of Bicategories

Abstract: A mature version of the coherence theorem for bicategories should not be limited to 'every weak 2-category is equivalent to a strict one': it shouldalso say something about the functors etc. between 2-categories. With this in mind, I will discuss the various ways to collect together (strict or weak) 2-categories to form a single structure, and what coherence does and does not say about how these structures are related. For instance, if Str2Cat denotes the 3-category consisting of strict 2-categories, strict 2-functors etc., and similarly Wk2Cat (everything weak), then the inclusion Str2Cat → Wk2Cat is not an equivalence. This is a precise expression of the view (long advocated by Bénabou) that the most important aspect of the theory of bicategories is not that they themselves are weak, but that the maps between them are weak.

Although this talk will consist of elementary observations, I will assume knowledge of basic bicategory theory: see for instance the references below.

• Francis Borceux, Handbook of Categorical Algebra 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press, 1994.
• Ross Street, Categorical structures, in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 1, North-Holland, 1996.
• Tom Leinster, Basic bicategories.

Tom Leinster

10:30-11:00 — Coffee

From left to right: Joshua Nichols-Barrer, André Joyal, Igor Bakovic and Urs Schreiber

At right: Steve Awodey

From left to right: Tom Leinster (hidden), Richard Garner, Alissa Crans, Nick Gurski and Jeffrey Morton

11:00-12:00 — Steve Lack: A Survey of the Theory of Gray-Categories

I'll discuss the theory of Gray-categories, including the connections with braided monoidal categories and with homotopy 3-types, the relationship between Gray tensor products and other tensor products of 2-categories, Gray-enriched limits, and pseudomonads in Gray-categories.

Steve Lack

12:00-3:00 — Lunch

3:00 — Russian seminar with Nick Gurski: Tricategories

I will discuss tricategories from the ground up assuming no prior knowledge of their technical aspects. I will show how the definition can be obtained as a naive categorification of the definition of bicategory, and I will discuss the coherence theory for tricategories with the focus being on how to use this theory to make tricategories more manageable.

Nick Gurski

Note: A "Russian seminar" is run by a moderator who asks the speaker questions and also urges the audience to ask as many questions as they want. It runs for an indefinite amount of time, basically until everyone gets tired and hungry. This tradition goes back at least to Gel'fand, and has been continued by Drin'feld at Chicago University.

7:00-8:00 — Reception

8:00 — Informal talk by Mike Shulman: Model Categories for Beginners

This talk will begin with some fuzzy thoughts about 'sameness' and 'homotopy', what a 'homotopy category' is and what it's good for (like comparing different models for homotopy types, or for higher categories), and the tools people use to get a handle on it. Then we'll work up to model categories, which are a neat compact package that incorporates all the information about equivalences, homotopy, and useful tools for reasoning about them. We'll also briefly discuss Quillen adjunctions and Quillen equivalences, which are two of the ways model categories mae it easier to compare different models of the 'same' objects. The goal is to convey enough of the language of model categories that everyone can follow later talks in which it is used.

Michael Shulman

Wednesday January 10 - "n-Categories and Homotopy Theory"

9:30-10:30 — John Baez: The Homotopy Hypothesis

Crudely speaking, the Homotopy Hypothesis says that n-groupoids are the same as homotopy n-types — nice spaces whose homotopy groups above the nth vanish for every basepoint. We summarize the evidence for this hypothesis. Naively, one might imagine this hypothesis allows us to reduce the problem of computing homotopy groups to a purely algebraic problem. While true in principle, in practice information flows the other way: established techniques of homotopy theory can be used to study coherence laws for n-groupoids, and a bit more speculatively, n-categories in general.

The modelling of homotopy types provides an important link between higher category theory and homotopy theory. In this talk we compare two models of connected n-types: cat^n-1-groups and Tamsamani's weak n-groupoids (with one object). The first model arose in homotopy theory, generalizing earlier work of Whitehead on crossed modules. The second arose in higher category theory. As a result of this comparison we identify a subcategory of Tamsamani's weak n-groupoids whose objects are 'less weak' than the ones used by Tamsamani to model connected n-types. Thus objects of this subcategory are 'semistrict'. We show that every Tamsamani weak n-groupoid representing a connected n-type is in a suitable sense equivalent to a semistrict one. A large part of the talk will be devoted to the case n = 3, where we will also make a connection with Gray groupoids. We will then illustrate the main ideas involved in the argument for higher n.

3:00 — Russian Seminar with Eugenia Cheng: Batanin ω-Groupoids and the Homotopy Hypothesis

I will discuss work of Batanin, Berger and Cisinski towards proving a version of the Homotopy Hypothesis for Batanin's theory of weak ω-groupoids. Building on the work of Berger, Cisinski has proved part of Batanin's hypothesis that Batanin weak ω-groupoids model homotopy types; specifically, he proves that the homotopy category of CW-complexes can be embedded in the homotopy category of Batanin's weak ω-groupoids. This talk will be expository. In particularly we will not go into technical model category theoretical details. A general idea of model categories and homotopy categories will be sufficient, as covered in Mike Shulman's talk on Tuesday.

Eugenia Cheng

8:00 — Peter May: Applications of Bicategories to Algebraic Topology

How does duality theory extend from symmetric monoidal categories to bicategories? What does that have to do with parametrized homotopy theory? It turns out that there are two very different kinds of duality theory in parametrized homotopy, homology, and cohomology theory, and it is virtually impossible to understand them without first understanding duality in bicategories. The theory here (joint with Johann Sigurdsson) sheds new light even on classical Poincaré duality.

Duality theory in symmetric monoidal categories leads to traces of maps. What are traces in bicategories? What do they have to do with fixed point theory? It turns out (work of Kate Ponto) that to understand the converse of the Lefschetz fixed point theory, one should understand Reidemeister traces as traces in a bicategory.

I'll give an informal introduction to these ideas.

Peter May

Thursday January 11 - "Quasicategories"

9:30-10:30 — Michael Shulman: Introduction to Quasicategories

Quasicategories are one way to define ∞-categories in which all cells of dimension above 1 are invertible, also known as '(∞,1)-categories'. Following the homotopy-hypothesis intuition, we can also describe them as `categories up to coherent homotopy'; this is how they originally arose in the work of Boardman and Vogt. Of the known definitions for (∞,1)-categories, quasicategories are one of the simplest and most tractable. For instance, they support good theories of limits and colimits, functor categories, fibrations, and adjunctions. This talk will be a basic introduction to the theory of quasicategories, with an emphasis on intuition and why you should care. We'll do some things explicitly, to get a feel for how things work, and then say a bit about the model structure for quasicategories and related tools that help manage the unavoidable combinatorial complexity. Later talks will expand on various aspects of the theory of quasicategories.

The model category structure for quasicategories can be regarded as providing a model for homotopy theories or a model for (∞,1)-categories, depending on ones point of view. In this talk we wil describe various Quillen equivalences connecting this model structure with several others: the model structure on the category of simplicial categories, two Segal category model structures on the category of Segal precategories, and the complete Segal space model structure on the category of simplicial spaces.

Julie Bergner

12:00-2:30 — Lunch

2:30-3:30 — Joshua Nichols-Barrer: Fibred Quasicategories and Stacks

We discuss three distinct models for the quasicategories of left/right/op-Cartesian/Cartesian fibrations over a simplicial set S: space-valued functors, the smplicial nerve construction of Joyal and Lurie, and the opFib/S and Fib/S constructions ('fibered quasicategories') of the speaer's thesis, all of which give equivalent quasicategories. We examine at least one form of the Yoneda lemma and look at descent in these different pictures, arriving at a few appropriate quasicategories of stacks on a site. We'll talk a little about how moduli problems arise in geometry and how they fit naturally into the fibred quasicategory picture. If there is time, we'll talk a bit about Lurie's Giraud-type characterization theorem for quasicategories of stacks in Kan complexes (what he calls ∞-topoi, and what might more precisely be termed (∞,1)-topoi).

4:00-5:00 — André Joyal: From Quasicategories to Higher Categories

Quasicategories were introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures. They belong as much to the world of homotopy theorists as to the world of category theorists. It turns out that category theory can be wholly extended to quasicategories. The extension is nontrivial and it opens new avenues in every domain where category theory plays an important role: homotopical algebra, algebraic geometry, etc. We call the resulting extension of algebra "quasi-algebra". Quasi-algebra is a new methodology for handling homotopy invariant algebraic structures and coherence problems in homotopy theory. It can be used for handling coherence problems in higher category theory as well. As an example, we propose a notion of "n-quasicategory" using quasi-algebra. For this we must use the notion of reduced category, a set theorietic version of the notion of complete Segal space of Rezk. We conjecture that the notion of n-quasicategory is equivalent to the notion of Segal n-category introduced by Hirschowitz and Simpson.

André Joyal

7:30-8:15 — Kathryn Hess: Parallel Transport in Bundles of Bicategories

Kathryn Hess

8:15-9:00 — Dorette Pronk: Bicategories and Tricategories of Fractions

Dorette Pronk

Friday January 12 - "Higher Gauge Theory"

9:30-10:30 — Alissa Crans: A Survey of Higher Lie Theory

In preparation for the day's talks, we focus on categorifying Lie theory, concentrating on Lie 2-groups and Lie 2-algebras. These are categorified versions of Lie groups and Lie algebras, where we have replaced the associative law and Jacobi identity, respectively, by natural isomorphisms called the "associator" and the "Jacobiator". We will consider alternative proposals for what categorified Lie theory should look like and discuss the advantages and limitations of these different choices.

11:00-12:00 — Danny Stevenson: Lie 2-Algebras and Higher Gauge Theory

There has been recent interest in generalisations of principal bundles, in which the structure group is replaced by a 2-group, which is a certain kind of groupoid. Understanding the geometry of these generalised principal bundles, known as "2-bundles" or "nonabelian gerbes", forms part of the subject of higher gauge theory.

In the case of ordinary principal bundles, Atiyah gave an elegant formulation of the notion of a connection in terms of a splitting of an extension of Lie algebras. Just as the usual cohomology of groups allows one to classify central extensions, Schreier theory allows one to classify arbitrary extensions of groups. Atiyah's approach to connections and curvature for bundles can be nicely understood using a Lie algebra version of Schreier theory. After reviewing the basic concepts of Lie 2-algebras, we shows that Schreier theory for Lie 2-algebras clarifies the theory of connections and curvature for 2-bundles.

12:00-3:00 — Lunch

3:00 — Russian seminar with Urs Schreiber: Parallel Transport in Low Dimensions

A vector bundle with connection can be conceived as a suitable parallel transport 1-functor from paths in base space to vector spaces. We categorify this and discuss various examples of locally trivializable 2-transport, some of them relevant for the description of charged 2-particles in formal high energy physics. We also point out how some surface transport really has to be conceived as 3-transport and we discuss the Chern-Simons 3-transport.

Listen to Urs' talk

Urs Schreiber

Simon Willerton

8:15-9:00 — Igor Bakovic: Bigroupoid Principal 2-Bundles

We describe bigroupoid principal 2-bundles, and we give a complete classification of these objects in terms of a newly defined nonabelian cohomology. We also show that bigroupoid principal 2-bundles give Glenn's simplicial 2-torsors after the application of the Duskin nerve functor for bicategories.

Saturday January 13 - "QFT and TQFT"

From left to right: André Joyal, Tom Leinster, Steve Lack, John Baez

9:30-10:30 — Aaron Lauda: Frobenius Algebras, Quantum Topology and Higher Categories

After recalling the well-known construction of 2d topological quantum field theories (TQFTs) from commutative Frobenius algebras, we show how this fits into a bigger picture involving open-closed TQFTs and the Fukuma-Hosono-Kawai state sum model. Some of these ideas can already be categorified to study higher dimensions. If time permits, we will also sketch how this machinery can be used in a construction of Khovanov homology, not just for links but also for tangles.

10:30-11:00 — Coffee

11:00-12:00 — Urs Schreiber: Quantum 2-States: Sections of 2-Vector Bundles

The quantization of the charged point particle relates two 1-functors into vector spaces: the parallel transport on target space is turned by quantization into propagation on parameter space. We categorify this and discuss how the states of an open string can be thought of as the sections of a 2-vector bundle. Further details can be found here:

• Urs Schreiber, On 2-dimensional QFT: from arrows to disks.
• Urs Schreiber, D-branes from tin cans: arrow theory of disks.
• Urs Schreiber, D-branes from tin cans, II.
• Urs Schreiber, D-branes from tin cans III: hom of homs.

From left to right: Nils Baas, Bruce Bartlett, André Joyal, Sergey Arkhipov and Urs Schreiber

From left to right: Kathryn Hess, Simon Willerton, Steve Lack, Simona Paoli, Tom Leinster

And later that evening....

From left to right: Tom Leinster, Julie Bergner, Simona Paoli and John Baez

You can see a lot more photos here. For example:

Tom Leinster

Toby Bartels and Aaron Lauda

Dan Christensen has some more photos. The photo above from the morning coffee hour of January 13th is one of these. Julie Bergner also has some pictures on her website, like this:

Julie Bergner, Simona Paoli and Tom Fiore (from front to back)

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