Motive-ating the Weil Conjecture Proof

This post concludes a series of posts I’ve been writing on the attempt to prove the Weil Conjectures through the Standard Conjectures. (Parts 1, 2, 3, 4, 5.) In this post, I want to explain the idea of the category of motives. In the modern formulation of algebraic topology, cohomology theories are functors from some category of spaces to the category of abelian groups. The category of motives is meant to be a universal category through which any such functor should factor, when the source space is the category of algebraic varieties. At least in the early days of the subject, the gold test of this theory was the question of whether the Weil Conjectures could be proved entirely in this universal setting. Nowadays, this question is still open, but the use of motives has grown. To my limited understanding, this growth has two reasons: among number theorists, it has become clear that motivic language is an excellent way to formulate results on Galois representation theory; among birational geometers and string theorists, many applications have been found for motivic integration. There will be a bunch of category theory in this post, which I hope will make it more attractive to the tensor category crowd.

I am much less comfortable with this topic than the other posts in this series; my understanding doesn’t go much further than Milne’s survey article. So I’m going to make this post a pretty short introduction to the main ideas. That will be the end of my expository posts; I also want to write one more post raising some questions about motives that seem natural to me.

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Concrete Categories

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

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). Continue reading

Quaternions and Tensor Categories

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

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

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

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

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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