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Concrete Categories October 26, 2009

Posted by David Speyer in Algebraic Topology, Category Theory.
12 comments

In many introductions to category theory, you first learn the notion of a concrete category: A concrete category is a collection of sets, called the objects of the category and, for each pair (X,Y) of objects, a subset of the maps X \to Y. (There are, of course, axioms that these things must obey.) In a concrete category, the objects are sets, and the morphisms are maps that obey certain conditions. So the category of groups is concrete: a map of groups is just a map of the underlying sets such that multiplication is preserved. So are the category of vector spaces, topologicial spaces, smooth manifolds and most of the other most intuitive examples of categories.

Using terminology from a discussion at MO, I’ll call a category concretizable if it is isomorphic to a concrete category. For example, \mathrm{Set}^{op} can be concretized by the functor which sends a set X to the set 2^X of subsets of X, and sends a map of sets f:X \to Y to the preimage map f^{-1}: 2^Y \to 2^X.

At one point, I learned of a result of Freyd: The category of topological spaces, with maps up to homotopy, is not concretizable. I thought this was an amazing reflection of how subtle homotopy is. But now I think this result is sort of a cheat. As I’ll explain in this post, if you are the sort of person who ignores details of set theory, then you might as well treat all categories as concrete. My view now is that specific concretizations are very interesting; but the question of whether a category has a concretization is not. I’ll also say a few words about small concretizations, and Freyd’s proof.

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A hunka hunka burnin’ knot homology September 24, 2009

Posted by Ben Webster in Category Theory, category O, combinatorics, homological algebra, link homology, low-dimensional topology, quantum groups, representation theory.
19 comments

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). (more…)

Quaternions and Tensor Categories July 30, 2009

Posted by David Speyer in Algebraic Geometry, Category Theory, things I don't understand.
27 comments

As you can tell from the title of this post, I am trying to drag John Baez over to our blog.

Let Q be the ring of quaternions, i.e., \mathbb{R} \langle i,j,k \rangle with the standard relations. Let Q-mod be the category of left Q-modules. This has an obvious tensor structure (including duals), inherited from the category of \mathbb{R} vector spaces. Actually, that structure doesn’t quite work; I’ll leave to you good folks to work out what I should have said.

Let q=a+bi+cj+dk be a quaternion. Anyone who works with quaternions knows that there are two notions of trace. The naive trace, 4a, is the trace of multiplication by a on any irreducible Q-module, using the obvious tensor structure. But there is a better notion, the reduced trace, which is equal to 2a. Similarly, there is a naive norm, (a^2+b^2+c^2+d^2)^2, and there is a reduced norm a^2+b^2+c^2+d^2.

This all makes me think that there is a subtle tensor category structure on Q-mod, other than the obvious one, for which these are the trace and norm in the categorical sense. Can someone spell out the details for me?

By the way, a note about why I am asking. I am reading Milne’s excellent notes on motives, and I therefore want to understand the notion of a non-neutral Tannakian category (page 10). As I understand it, this notion allows us to evade some of the standard problems in defining characteristic p cohomology; one of which is the issue above about traces in quaternion algebras.

Working equivariantly for the action of a monoidal category May 22, 2009

Posted by Ben Webster in Category Theory.
15 comments

I recently got an email question from Sergey Arkhipov with a question, which I couldn’t answer to my own satisfaction, so I thought I would throw it open to the peanut gallery.

One construction I’ve used a lot in my recent work is the equivariant derived category for the action of a group G on a space X (in basically whatever category you like). This is basically the poor man’s way of understanding sheaves on the quotient stack of that space by the group.

But, of course, one could forget that there was ever a space there, and just remember that you have a category of sheaves on X, which the group G acts on. So, questions:

  1. Is there a construction of the equivariant derived category which makes no reference to the space and just uses the category of sheaves?
  2. If there a generalization of this construction where the action of G can be replaced by one of an arbitrary monoidal category?

The first question is in that class of things I’m sure I could do myself if I forced myself to sit down and do it: the answer is something like replacing the category with the category of locally constant sheaves on BG valued in your category. The second, I’m less sure about.

The Witt group, or the cohomology of the periodic table of n-categories March 30, 2009

Posted by Noah Snyder in Category Theory, conferences, homological algebra, quantum groups, talks.
10 comments

A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!

In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.

First recall the Baez-Dolan periodic table.

Periodic Table

Periodic Table

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SF&PA: Subfactors = finite dimensional simple algebras March 23, 2009

Posted by Noah Snyder in Category Theory, representation theory, subfactors.
2 comments

Since my next post on Scott’s talk concerns the construction of a new subfactor, I wanted to give another attempt at explaining what a subfactor is. In particular, a subfactor is just a finite-dimensional simple algebra over C!

Now, I know what you’re thinking, doesn’t Artin-Wedderburn say that finite dimensional algebras over C are just matrix algebras? Yes, but those are just the finite dimensional algebras in the category of vector spaces! What if you had some other C-linear tensor category and a finite dimensional simple algebra object in that category?

Let me start with an example (very closely related to Scott Carnahan’s pirate post).
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Coincidences of tensor categories March 22, 2009

Posted by Noah Snyder in Category Theory, quantum groups, small examples, talks.
5 comments

This week Scott and I were at a wonderful conference on Modular Categories at Indiana University. I find that I generally enjoy conferences on more specific subjects, especially in algebra. Otherwise you run the danger of every talk starting by defining some algebra you’ve never heard of (and won’t hear of again the rest of the conference) and then spend a while proving some properties of this random algebra that you still don’t know why you care about (let alone why you should learn about its projective modules). With more specific conferences if you don’t quite get something the first time you have a good change of seeing it again and it slowly sinking in. The organizers (Michael Larsen, Eric Rowell, and Zhenghan Wang) did an excellent job putting together and interesting, topically coherent, and fun conference. I was also pleasantly surprised by Bloomington, which turned out to actually be kind of cute. I have several posts I’d like to give on other people’s talks, in particular there were several talks (by Davydov, Mueger, and Ostrik) on the “Witt group” which involves the simplest case of a kind of cohomology of the periodic table of n-categories and thus should appeal to all of you over at the n-category theory cafe. But I think I’ll start out with our talks (which Scott and I prepared jointly based on our joint work with (Emily Peters)^2 and Stephen Bigelow).

The first of these talks (click for beamer slides) was on coincidences of small tensor categories. The strangest thing about this talk was that I was introduced as a “celebrity math blogger.”

Please note that in the slides I’ve completely elided the distinctions between a quantum group, its category of representations, and (when q is a root of unity) its semisimplified category of representations (where you quotient out by the kernel of the inner product as in David’s post).

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A warmup: GL_t for t not an integer February 25, 2009

Posted by Noah Snyder in Category Theory.
17 comments

This post is meant as a warm-up to my planned follow-up to David’s post.  You don’t have to have read his post to understand this one, but there are a few technical details at the end where I’ll refer you to the end of his post.  Most of what I learned here I learned from reading this expository paper by Ostrik which I read in preparation for some talks I gave my second year of grad school.

If you like to draw pictures, how do you think about the representation theory of groups?  Well, you use an oriented strand for some basic or fundamental representation V of a group, you orient the strand the other way for the dual representation, you use disjoint union for tensor product.  Now you can try to draw pictures for maps between tensor products V\otimes V \otimes V^* \rightarrow V^* \otimes V^* say.  Stacking these pictures is composition, and disjoint union is tensor product.  This should be pretty familiar to you if you’ve read the archives for “this week in mathematical physics.”

Since we’re looking at the category of representations of a group we have a bonus bit of information: this tensor category is symmetric.  There’s a canonical map V\otimes W \rightarrow W \otimes V which satisfies the relations of the symmetric group.  In pictures this can be drawn using a crossing.  (Warning: this is not a crossing in 3-dimensional space, you need to either think of your pictures as being in 4-dimensions or not embedded at all.)

Ok, so what is \mathrm{GL}_t?  It should be a linear symmetric tensor category with a representation V that has no properties other than having dimension t.  What does it mean to have dimension t in picture language?  It means that a closed loop should have the value t.  So here’s our proposed category \mathrm{GL}_t:

  • Objects are collections of oriented points on a line.
  • Morphisms are linear combinations of oriented strands (unembedded or in dimension greater than 4 so that the crossings satisfy the relations of the symmetric group) whose boundaries match the objects that they’re mapping between.
  • Composition is stacking of diagrams with the relation that a closed loop can be removed for a multiplicative factor of t.
  • Tensor product is disjoint union.

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Bleg: testing algebraic integrality by computer. October 13, 2008

Posted by Noah Snyder in Category Theory, Number theory, blegs, knot atlas, quantum algebra, subfactors, things I don't understand.
8 comments

Update 2: we’ve found a nice answer to our question. Maybe it will appear in the comments soon. –Scott M

Scott, Emily, and I have an ongoing project optimistically called “The Atlas of subfactors.” In the long run we’re hoping to have a site like Dror Bar-Natan and Scott’s Knot atlas with information about subfactors of small index and small fusion categories. In the short run we’re trying to automate known tests for eliminating possible fusion graphs for subfactors.

Right now we’re running into a computational bottleneck: given a number that is a ratio of two algebraic integers how can you quickly test whether it is an algebraic integer? Mathematica’s function AlgebraicIntegerQ is horribly slow, and we’re not sure if that’s because it’s poorly implemented or whether the problem is difficult. So, anyone have a good suggestion? After the jump I’ll explain what this question has to do with tensor categories (and hence subfactors which correspond to bi-oidal categories as I’ve discussed before).

To whet your appetite, here’s an example. Is a/b, where

a=-293 \lambda^{11}+4624 \lambda^9-23668 \lambda^7+50302 \lambda^5-44616\lambda^3+14017 \lambda

b=131\lambda^{10} - 2033 \lambda^8 + 9974\lambda^6-18951\lambda^4+12233 \lambda^2-1475

and where \lambda is the largest real root of

1 - 58 x^2 + 175 x^4 - 186 x^6 + 84 x^8 - 16 x^{10} + x^{12},

an algebraic integer? Mathematica running on Scott’s computer (using the builtin function AlgebraicIntegerQ) takes more than 5 minutes to decide that it is.

Update: Thanks to David Savitt for pointing out that both this example and an earlier one are answered instantly by MAGMA. Blegging is already working. But what’s the trick? Is it something we can teach Mathematica quickly? –Scott M

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Group rings arrr commutative September 18, 2008

Posted by Scott Carnahan in Category Theory, hopf algebras, quantum algebra, quantum groups, representation theory.
10 comments

If you are familiar with group rings, you might think that the title of this post is false. If G is a nonabelian group, multiplying the basis elements g and h in \mathbb{Z}G can yield gh \neq hg, so we have a problem. In general, if you have a problem that you can’t solve, you should cheat and change it to a solvable one (According to my advisor, this strategy is due to Alexander the Great). Today, we will change the definition of commutative to make things work.

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