## January 19, 2007

### Traces in Ottawa

#### Posted by John Baez

If you’re interested in logic, category theory, and diagrams, you’ll like this workshop:

Here’s the conference announcement:

The abstract theory of traces has had a fundamental impact on a variety of fields within mathematics. These range from functional analysis and noncommutative geometry to topology and knot theory, and more recently to logic and theoretical computer science. The theory of traced monoidal categories, due to Joyal, Street and Verity, is an attempt to unify various notions of trace that occur in these diverse branches of mathematics. More recent developments include several theories of partial traces in monoidal categories.

The Logic and Foundations of Computing Group at the University of Ottawa, with funding from the Fields Institute, is proud to host a workshop to explore these topics. The purpose of this workshop is to bring together researchers in these fields to look for common developments, models, and applications of trace theory. Among the applications are various notions of parametrized traces arising in operator algebras, in the theory of feedback and recursion in theoretical computer science, in braid closure in knot theory, and in dynamics of proofs as expressed by Linear Logic and the Geometry of Interaction.

Some invited speakers include:

• Samson Abramsky (Oxford)
• Robin Cockett (Calgary)
• Andre Joyal (UQAM)
• Louis Kauffman (Illinois)
• Mathias Neufang (Carleton)
• Timothy Porter (Bangor)

We will be announcing further speakers shortly. This is intended to be a workshop, with student participation in mind, including introductory lectures. We will have some funding for student travel and accommodation. Students interested in receiving financial aid should contact the organizers by January 30th.

Anyone interested in attending or contributing a talk should contact us by the same date.

You may know about the trace of square matrix, which is a number: just the sum of its diagonal entries. In index notation:

$Tr(f) = f^i_i$

where the Einstein summation convention says we sum over the repeated inde $i$.

In quantum physics we often do something more general: we take the ‘partial trace’ of a density matrix describing the state of a two-part system, to get the state of one of the parts. Mathematically, the partial trace of a linear operator

$f: A \otimes U \to B \otimes U$

is an operator

$Tr^U_{A,B}(f): A \to B$

given by

$Tr^U_{A,B}(f)^a_b = f^{ai}_{bi}$

where $i$ runs over a basis of the vector space $U$.

In terms of diagrams, the partial trace looks like this:

Time in this diagram goes up, as in traditional Feynman diagrams. If we’re talking about linear operators, the loop stands for the summation over the repeated index $i$. But these diagrammatic methods are much more general: we can define partial traces in a large class of monoidal categories, including all compact symmetric monoidal categories — but also others! For example, in logic and computer science, partial traces are used to describe ‘fixed points’.

The picture above is from a talk by Paul-André Melliès, which explains some of the uses of traces in logic. Click on it and learn more!

Posted at January 19, 2007 4:19 AM UTC

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### Re: Traces in Ottawa

I have mentioned/asked the following before, but here in the context of general traces it might be the right point to mention/ask it again:

In what should probably be called “globular extended 2-dimensional quantum field theory” we need partial traces in braided monoidal categories to do things like this:

The dashed boxes are supposed to indicate a 2-morphism in a 2-category. The full lines inside the boxes are supposed to denote the Poincaré-dual “string diagram” notation of that same globular 2-morphism.

Being a 2-globe, the 2-morphism itself always has the topology of a disk.

But I might want to think of this 2-morphism of having the topology of a trinion: a sphere with three disks cut out.

This can be thought of as obtained from the shown disk-like bigon by identifying all outer parallel dashed lines.

What does such a kind of identification mean operationally? If we compare with John’s diagram for any partial trace above, we see that it should mean that we somehow join the dual “string tangles” that run perpendicular to the edges that are to be identified.

In other words: that we take particl traces at the ingoing and outgoing edges that are to be identified.

In order to do that, though, as the picture makes clear, we need to braid past some other strands.

The diagram above depicts the right result.

Is there a general theory of generalized traces in a general class of categories that would be applicable to this context and reproduce the above idea?

I’d be grateful for any pointers to the literature.

Posted by: urs on January 19, 2007 7:30 AM | Permalink | Reply to this

### Re: Traces in Ottawa

I suspect all the partial traces you need are those that exist in what some people call a ‘balanced’ braided monoidal category.

To warm up to this notion, start with what people call a ‘compact’ braided monoidal category. This is one where every object has a dual.

In case anyone out there forgets what that means:

We say an object $x$ in a monoidal category has a left dual if there’s an object $x^*$ equipped with morphisms called the counit

$e: x^* \otimes x \to 1$

drawn as

                 x*     x
\    /
\  /
\/


and unit

$i: 1 \to x \otimes x^*$

drawn as

                   /\
/  \
/    \
x*     x


satisfying the zig-zag identities, which say

                   x       x
/\       |       |
/  \      |       |
/    \     |       |
|      \    /  =    |
|       \  /        |
|        \/         |
x                   x


and

       x*                  x*
|       /\          |
|      /  \         |
|     /    \        |
\    /      |  =    |
\  /       |       |
\/        |       |
x*      x*


We can also define right duals in a similar way — above, we say $x$ is a right dual of $x^*$.

If they exist, left and right duals are unique up to canonical isomorphism.

In a braided monoidal category, an object has a left dual iff it has a right dual, thanks to the braiding. So, we say a braided monoidal category is compact if every object has a left (or right) dual.

In a compact braided monoidal category, we can make sense of partial traces, using string diagrams. However, to manipulate them nicely, we want a little extra structure called the ‘balancing’, which describes a 360° twist in one of our strings. Then everything we could possibly want to do with partial traces by manipulating string diagrams in 3d space will work nicely.

• Christian Kassel, Quantum Groups, Springer Verlag, 1994.

The category of framed oriented tangles in 3 dimensions is the free balanced monoidal category on one object. Categories of representations of quantum groups give other examples.

In applications to physics, you may want duals for morphisms as well as objects, to discuss ‘unitarity’ and ‘self-adjointness’. Then you want something better than a balanced braided monoidal category: you want a ‘braided monoidal category with duals’.

For more, try this.

Posted by: John Baez on January 20, 2007 6:13 PM | Permalink | Reply to this

### Re: Traces in Ottawa

I suspect all the partial traces you need are those that exist in what some people call a ‘balanced’ braided monoidal category.

Okay, yes. In the applications that I have in mind, the category will be “ribbon”, which is balanced and compact, as far as I can see.

And indeed, when one draws the above diagram more carefully, remembering that all the dual lines are really ribbons, one sees the “twist” appear here and there.

I am imagining that, more generally, there should be something called “extended globular QFT”. It would differ from other notions of extended QFT in that it does not use sophisticated domain $n$-categories (like here) that take track of all the global structure of an “open/closed extended $n$-cobordism”. Instead, it would use only globular $n$-cobordisms probing the global structure locally, and that global structure would then be re-introduced by means of higher traces.

I am wondering if I would have to completely invent this sort of structure, or if I could use any existing work. In particular existing work on higher traces, maybe.

Maybe I am looking for a statement that relates $n$-tangles to higher traces, or something like that.

Posted by: urs on January 20, 2007 9:55 PM | Permalink | Reply to this

### Re: Traces in Ottawa

Urs wrote:

Maybe I am looking for a statement that relates n-tangles to higher traces, or something like that.

I’m biased, but personally I hope you’re looking for the Tangle Hypothesis: $n$-tangles in codimension $k$ are the free $k$-tuply monoidal $n$-category with duals on one object.

Posted by: John Baez on January 21, 2007 6:55 AM | Permalink | Reply to this

### Re: Traces in Ottawa

Then you want something better than a balanced braided monoidal category: you want a ‘braided monoidal category with duals’.

Just to be clear … a braided monoidal category is automatically balanced if it has all duals. Yes?

Posted by: Toby Bartels on January 21, 2007 4:42 AM | Permalink | Reply to this

### Re: Traces in Ottawa

Toby wrote:

Just to be clear … a braided monoidal category is automatically balanced if it has all duals. Yes?

Yes — if it has duals for all objects and morphisms.

Posted by: John Baez on January 21, 2007 6:49 AM | Permalink | Reply to this
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