## December 14, 2006

### Higher Categories at the Fields Institute

#### Posted by John Baez

Excitement is building throughout Toronto as news of this workshop spreads:

Crowds are lining up in front of the Fields Institute, trying to get seats early, wondering when the show will start. To prevent rioting and looting, I have decided to post a tentative schedule including abstracts of some of the talks.

#### Tuesday January 9 - “Low Dimensions”

9:00-9:30 — Coffee and Registration, Welcoming Remarks

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

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.
• Tom Leinster, Basic bicategories, 1998.
• Ross Street, Categorical structures, in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 1, North-Holland, 1996.

10:30-11:00 — Coffee

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

To include:

• relationship between Gray tensor products and others
• Gray-weighted limits
• Gray-groupoids and homotopy types
• braided monoidal categories

12:00-3:00 — Lunch

3:00 — Russian seminar with Nick Gurski: Tricategories

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.

#### Wednesday January 10 - “n-Categories and Homotopy Theory”

9:00-9:30 — Coffee

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.

11:00-12:00 — Simona Paoli: Semistrict Tamsamani n-groupoids and connected n-types

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: ${\mathrm{cat}}^{n-1}$-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.

12:00-3:00 — Lunch

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.

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.

#### Thursday January 11 - “Quasicategories”

9:00-9:30 — Coffee

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.

11:00-12:00 — Julie Bergner: Model Categories Quillen Equivalent to the Quasicategory Model Structure

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.

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).

3:30-4:00 — Coffee

4:00 — Russian Seminar with André Joyal: Applications of Quasicategories to Higher Categories

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

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

#### Friday January 12 - “Higher Gauge Theory”

9:00-9:30 — Coffee

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.

10:30-11:00 — Coffee

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.

7:30-8:15 — Simon Willerton: Hopf Monads

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

#### Saturday January 13 - “QFT and TQFT”

9:00-9:30 — Coffee

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-Hosano-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: On 2-Dimensional QFT: from Arrows to Disks

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 2-dimensional quantum field theory describing the charged 2-particle relates two 2-functors with values in 2-vector spaces.

Our goal is to explain and illuminate this way aspects of the categorical description of rational conformal field theory by the FRS theorem; like the relation of boundary fields (living on “D-branes”) to internal modules and of bulk fields to internal bimodules.

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. Some of the evening talks may also be run as Russian seminars.

Posted at December 14, 2006 10:19 PM UTC

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### Re: Higher Categories at the Fields Institute

Perhaps we might set up a list of questions we’d love to have answers for to pose to the experts gathering for this workshop.

You mentioned Nils Baas might help us with Baas-Sullivan theory.

Then there are a couple of questions about strictifying 2-functors.

And, of course, we’d like solutions to the Millennium prizes.

Posted by: David Corfield on December 16, 2006 8:20 AM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

Tom Leinster’s talk should answer some of our questions about strictifying 2-functors. I just put up a more detailed abstract of his talk - see above (the very first talk).

Posted by: John Baez on December 16, 2006 9:12 PM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

I completely forgot to ask Nils Baas for help with Baas–Sullivan theory.

As far as I can tell, all my hunches about strictification were confirmed.

It’s not on your list above, but I I got a lot of help from Tom with understanding the Euler characteristic of a category. I just explained some of that in week244. But, there’s more to say — see this.

Posted by: John Baez on February 4, 2007 10:19 PM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

Titles for all the talks are now available!

For example, André Joyal will be talking about ‘Applications of quasicategories to higher categories’.

Posted by: John Baez on December 21, 2006 4:41 PM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

You can see the slides of my talk here:

Posted by: John Baez on January 3, 2007 2:01 AM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

What does ‘We probably have adjoint 2-functors’ (p. 8) mean?

Posted by: David Corfield on January 3, 2007 8:52 AM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

It means these 2-functors are probably adjoint, but I haven’t proved it or seen a proof.

How could this have been unclear? If you explain, I can try to say it more clearly.

Or, maybe I’ll buckle down and prove it before my talk.

Posted by: John Baez on January 3, 2007 7:30 PM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

I just wasn’t sure whether it was:

• I know this hasn’t been proved but expect it to be true.
• I don’t know whether this has been proved but expect that it has.
• I don’t know whether this has been proved but expect that it could be.
Posted by: David Corfield on January 3, 2007 9:35 PM | Permalink | Reply to this

### Re: Higher Categories at the Fields Institute

Okay, thanks. I don’t think I want to pick one those 3 alternatives in my notes: as Niels Bohr said, you should never write more clearly than you think.

It should be a routine matter to check whether these functors between 2-categories are weak adjoints: the ‘fundamental groupoid’ functor

${\Pi }_{1}:1\mathrm{Type}\to 1\mathrm{Gpd}$

from homotopy 1-types to 1-groupoids, and the ‘Eilenberg–Mac Lane space’ or ‘geometric realization of the nerve’ functor

$\mid -\mid :1\mathrm{Gpd}\to 1\mathrm{Type}$

from 1-groupoids back to homotopy 1-types.

I think it’s true. It would be appalling if nobody had settled the question yet, but I don’t know a reference. I asked on the category theory mailing list but didn’t get any helpful replies. So far I’ve been too busy/lazy to check that the triangulators:

satisfy the swallowtail coherence laws:

Posted by: John Baez on January 4, 2007 12:09 AM | Permalink | Reply to this
Read the post Higher Categories and their Applications
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