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October 18, 2006

Australian Category Theory

Posted by John Baez

The prowess of Australians in higher category theory is renowned worldwide. They’re way ahead of us. If you don’t know what I’m talking about, read this:

For example, check out the fun story on page 4 where an Australian student, asked if he knew the definition of a topological space, correctly but incomprehensibly replied:

Yes, it is a relational β-module!

So, it behooves the rest of us to watch what they’re up to down under. For this reason, I plan to occasionally post comments to this article, announcing talks at the Australian Category Seminar. I won’t be obsessively systematic, so I apologize in advance for everyone’s talks that I forget to announce.

You can easily spot new comments by looking under “Recent Comments” on the right side of this blog. We engage in a lot of protracted discussions, so that’s a handy feature.

  • Dorette Pronk, Bicategories of fractions II, Wednesday October 11, 2006.

    Abstract: In 1967, Gabriel and Zisman introduced the conditions needed on a class of arrows to admit a calculus of fractions. If one has a category CC with a class WW of weak equivalences that satisfy these conditions, the arrows in the homotopy category of fractions C[W −1]C[W^{-1}] can be described as spans where the left leg is in WW. This category is the universal category obtained from CC by inverting the arrows in WW.

    In this talk I will discuss the bicategorical generalization of this: I will discuss the conditions needed for a bicalculus of fractions and give an explicit description of the 2-cells in C[W −1]C[W^{-1}], the free bicategory obtained by turning the arrows in W into equivalences. This talk is based on my paper in Compositio Mathematica from 1996, but I will discuss some minor improvements on that paper as well as ways to generalize this further to monoidal bicategories of fractions, and eventually, tricategories of fractions.
Posted at October 18, 2006 10:08 PM UTC

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12 Comments & 3 Trackbacks

Re: Australian Category Theory

Kea has fascinating you-are-there posts on maths in Australia.

Posted by: Louise on October 19, 2006 2:22 AM | Permalink | Reply to this

Re: Australian Category Theory

Here are some more talks:

  • Ross Street, Kan extensions in basic group representation theory. Wednesday November 1, 2006.

  • Dorette Pronk, Conformal field theory as a nuclear functor. (Joint work with Prakash Panangaden and Rick Blute.) Wednesday November 1, 2006.

    Abstract: Segal’s definition of a conformal field theory discusses “functors” for which the domain is not a category. There is an associative notion of composition for arrows, but there are no identities. Segal claim that this is not important, but we claim that one should not just freely add identities. Segal’s original category can be seen as a nuclear ideal in a larger category and the functors one wants to consider as conformal field theories are nuclear functors. We will present one example of such a functor, due to Neretin.

The issue being tackled here is technical but important. Following Segal, we would like to think of a conformal field theory as a symmetric monoidal functor Z:2Cob ℂ→VectZ : 2Cob_{\mathbb{C}} \to Vect where the morphisms in 2Cob ℂ2Cob_{\mathbb{C}} are 2d cobordisms equipped with conformal structure. However, there’s an obstacle to realizing this. If the morphisms in 2Cob ℂ2Cob_{\mathbb{C}} are 2d cobordisms equipped with conformal structure, there are no identity morphisms! The identity for a circle, for example, would need to be an “infinitely short” cylinder - something like [0,t]×S 1[0,t] \times S^1 with the obvious conformal structure, but letting t→0t \to 0.

One can follow Segal and formally throw in identity morphisms, but the problem persists. The linear operator associated to the cylinder [0,t]×S 1[0,t] \times S^1 should be exp(−tH) \exp(-t H) where HH is the Hamiltonian, an operator on the Hilbert space for the circle. So, the partition function of the torus formed by gluing the two ends of this cylinder together should be tr(exp(−tH)) \tr(\exp(-t H)) But, exp(−tH)\exp(-t H) reduces to the identity operator at t=0t = 0, and trace of the identity operator is infinite - so we cannot compute the partition function in this case!

It seems like a niggling small issue. But, the challenge is to find a formalism that smoothly deals with it, by accepting it. This is apparently what Blute, Panangadgen and Pronk are working on.

The same issue is lurking in Stolz and Teichner’s paper on elliptic cohomology. The issue can already be seen in Example 2.1.4. on page 11 - what happens when t=0t = 0? And, it raises its ugly head again in Definition 4.1.1. on page 48 - what are the identity morphisms in this category 𝒞(X)\mathcal{C}(X)? I don’t know if it’s clearly resolved here, though I remember talking to Stephan Stolz about it, and he had some nice ideas about it, again involving “ideals” in categories.

Posted by: John Baez on October 31, 2006 3:03 AM | Permalink | Reply to this

Re: Australian Category Theory

Kea reports from both Pronk’s and Street’s talks. Street has two related papers written with Panchadcharam online (93 and 94 here).

A week earlier Paolo Bertozzini had spoken about Horizontal categorification of Gel’fand theory and categorical non-commutative geometry:

We present a version of “Gel’fand duality theorem” for commutative full C*-categories and we discuss its relevance in a wider research project aiming at the development of categorical methods in the context of A.Connes’ non-commutative geometry.

Does anyone know what’s special about horizontal categorification? Is there also vertical categorification? Maybe the horizontal version just means spreading constructions from one level of the n-category ladder to adjacent fields without any climbing.

Posted by: David Corfield on November 2, 2006 9:02 AM | Permalink | Reply to this

Re: Australian Category Theory

I’ve never heard about “horizontal categorification”, but I agree with you: it sounds like what I call “many-objectification” or “oidization”. For example, the process of replacing groups by groupoids, rings by ringoids, algebras by algebroids, and so on.

Posted by: John Baez on November 3, 2006 4:53 PM | Permalink | Reply to this

Functorial QFT

Dorette Pronk, Conformal field theory as a nuclear functor. (Joint work with Prakash Panangaden and Rick Blute.)

I am having trouble finding any notes on this on the web. Is there anything available?

Segal’s definition of a conformal field theory discusses “functors” for which the domain is not a category. There is an associative notion of composition for arrows, but there are no identities.

Just for my own convenience, I would like to link this statement to the following observation:

Some people using the functorial QFT formalism in practice (for actually computing correlators, for instance) use a formulation which does in fact not mention functors and cobordism categories (in this context) at all, but which still captures the crucial property one wants to have, equivalently known as “factorization”, “sewing”, “locality” – the property that going from A to C over B is the same as first going from A to B and then from B to C.

For instance the axiomatization of 2-dimensional conformal field theory in math.CT/0512076 goes as follows:

Let 2Cob S2\mathrm{Cob}_S be the set of 2-dimensional cobordisms with given extra structure SS and with boundaries labeled as incoming or outgoing.

Define an operation cut\mathrm{cut} which to any such cobordism with suitably embedded circle associates the cobordism obtained by cutting along that circle and labelling the two new resulting boundary components as incoming and outgoing, respectively.

Instead of thinking of this directly as a category and considering functors to Vect\mathrm{Vect} on it, now consider assignments that send every boundary component to a vector space and every cobordisms to a linear map between these (just as a functor would) such that the factorization axiom is satisfied by this assignment:

the linear map associated to a given cobordisms must be the obvious trace over the linear map associated to any cobordism obtained from the given one by cutting along any circle.

This is called axiom C1 on p. 3 of the above paper.

If we had cobordisms with no extension in our cobordisms category, this axiom would simply follow from functoriality: it simply describes functoriality under composition with the cobordisms with no incoming and two outgoing boudary component, or two incoming and no outgoing, and no extension in between.

Posted by: urs on December 20, 2006 11:46 AM | Permalink | Reply to this

Re: Functorial QFT

To add identities to the quoted version

of Segal’s theory, one need only consider surfaces which are all boundary/empty interior/no bulk; in particular, S 1S^1

as a surface of infinite thinness. Apparently the problem is more with the representation. At a recent Sullivan seminar, Segal gave a specific realization of a relevant topological vector space HH as a representation of the relevant operad.

Crucial for his presentation: let the surface have a Riemannian metric.

To a point xx on the surface, the corresponding H xH_x is achieved as follows:

take a disk of radius ϵ\epsilon and let H x,ϵH_{x,\epsilon} be L 2 L^2 or your own favorite space of founctions on S 1S^1 regarded as the boundary of that disk

if we compare ϵ 10.\epsilon_1 0.

Posted by: jim stasheff on December 20, 2006 3:16 PM | Permalink | Reply to this

Re: Functorial QFT

To add identities to the quoted version of Segal’s theory, one need only consider surfaces which are all boundary/empty interior/no bulk; in particular, S 1S^1

I think that’s what John addressed above:

If we talk about conformal cobordisms (whose boundaries will need to be equipped with conformal “collars” that allows gluing) and the obvious composition of these, then none of these will act as an identity morphism.

Still, we can turn them into a true category by formally throwing in identity morphisms.

But, when we do that, as you say

the problem is more with the representation

Apparently Segal addresses this issue:

Segal gave a specific realization of a relevant topological vector space HH as a representation of the relevant operad.

Thanks for the information!

Your description ends with

[…] if we compare ϵ 10\epsilon_1 0

I am not sure I understand, yet. Are you saying that, for every ϵ\epsilon (describing a small annulus) we get a topological vector space H x,ϵH_{x,\epsilon}, and that we want to find something like the limit of that as ϵ\epsilon tensds to 0?

Posted by: urs on December 20, 2006 3:27 PM | Permalink | Reply to this

Re: Functorial QFT

I wrote:

Some people using the functorial QFT formalism in practice (for actually computing correlators, for instance) use a formulation which does in fact not mention functors and cobordism categories (in this context) at all, but which still captures the crucial property one wants to have, equivalently known as “factorization”, “sewing”, “locality” – the property that going from A to C over B is the same as first going from A to B and then from B to C.

For instance the axiomatization of 2-dimensional conformal field theory in math.CT/0512076 goes as follows:

[…]

These authors now have a new and refined formulation of this concept, presented in their latest: hep-th/0612306.

The way it is formulated there, it has some smell of multicategory to it.

Somebody should go over these definitions and clearly identify the category-theoretic notion that is implcitily invoked.

Roughly, the new definition now goes like this:

A category WShWSh is defined, whose objects are surfaces with boundaries, each component of which labelled as incoming or outgoing, and various further structures not relevant to the main idea.

Morphisms are

A) homemorphisms between these surfaces (respecting all extra structure)

B) sewing of surfaces.

A sewing is a choice of pairs of outgoing boundary components with matching ingoing boundary components, followed by gluing according to this identitfication.

Clearly, we should be able to think of the collection of objects here as (composable collections of) morphisms in a multicategory. Sewing morphisms are the processes describing composition in these collections.

Somebody who knows the theory of multicategories in more detail than I do should go over section 3.1 in the above paper and extract the corresponding description.

Posted by: urs on January 3, 2007 6:15 PM | Permalink | Reply to this

Re: Australian Category Theory

More Australian Category theory this week at the The Morgan-Phoa Mathematics Workshop.

Ross Street spoke yesterday on Quantum categories, Frobenius algebras and weak Hopf algebras in braided monoidal categories.

Michael Batanin speaks tomorrow on Deligne’s conjecture: an interplay between algebra, geometry and higher category theory.

Posted by: David Corfield on November 29, 2006 11:28 AM | Permalink | Reply to this

Re: Australian Category Theory

Ross Street spoke yesterday on Quantum categories, Frobenius algebras and weak Hopf algebras in braided monoidal categories.

Interesting. That’s relevant for 2-dimensional conformal field theory #.

Does anyone know if there is a paper available, corresponding to this talk?

Posted by: urs on November 29, 2006 3:42 PM | Permalink | Reply to this

Re: Australian Category Theory

Posted by: John Baez on November 30, 2006 1:39 AM | Permalink | Reply to this

Re: Australian Category Theory

Kea has blogged about this Workshop.

Posted by: David Corfield on December 3, 2006 6:24 PM | Permalink | Reply to this
Read the post FFRS on Uniqueness of Conformal Field Theory
Weblog: The n-Category Café
Excerpt: A strengthening of the FFRS theorem on 2-dimensional rational conformal field theory.
Tracked: January 3, 2007 7:37 PM
Read the post Spaceoids
Weblog: The n-Category Café
Excerpt: Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul on categorified spaces and the many-object version of C-star algebras.
Tracked: January 17, 2008 7:48 PM
Read the post Generalized Multicategories
Weblog: The n-Category Café
Excerpt: A brief introduction to generalized multicategories, in honor of a new draft of a paper about them.
Tracked: January 4, 2010 2:21 AM

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