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July 30, 2010

Azimuth

Posted by John Baez

I’d like to invite all of you to visit my new blog:

It’s about various topics that don’t fit neatly into the nn-Category Café. So far, that mainly means talking about ecological issues, and live-blogging on talks about quantum technologies.

Posted at 5:40 AM UTC | Permalink | Followups (18)

July 28, 2010

Zeta Functions: Dedekind Versus Hasse-Weil

Posted by John Baez

I’m a bit confused. It serves me right: I’ve been trying to learn too much math from Wikipedia articles.

I think I understand the Dedekind zeta function of a number field. You can write it as a product over prime ideals in the ring of algebraic integers for that field… and you can group that product to make it into a product over ordinary primes. This is called the ‘Euler factorization’ of the zeta function.

I also have some rough understanding of the Hasse–Weil zeta function of an algebraic variety defined over an algebraic number field. It too has an Euler factorizatoin. For each ordinary prime we get a factor called a local zeta function, and for the primes that are unramified there’s an easy formula for this local zeta function. But the Wikipedia article mumbles rather portentuously when it comes to the ramified primes, emitting smoke and lightning but not (as far as I can tell) a precise formula…

Posted at 7:01 AM UTC | Permalink | Followups (18)

July 24, 2010

Ternary Factorization Systems

Posted by Mike Shulman

We’ve been having a bit of discussion on the nForum about ternary and higher-ary factorization systems in categories and higher categories. A kk-ary factorization system is supposed to give a way of factoring any morphism into a composite of kk morphisms of specified types. Ordinary “factorization systems” are binary factorization systems, which factor every morphism into a binary composite; for instance, the factorization of a set function into a surjection followed by an inclusion.

The first place we noticed these higher-ary factorization systems is in the “Postnikov decomposition” of higher categories. The basic example is that any functor f:ABf\colon A\to B factors as Aim 2(f)im 1(f)B A \to im_2(f) \to im_1(f) \to B where im 1(f)Bim_1(f)\to B is full and faithful, im 2(f)im 1(f)im_2(f) \to im_1(f) is essentially surjective and faithful, and Aim 2(f)A\to im_2(f) is essentially surjective and full. Similarly, we expect a functor between nn-categories to have a natural (n+2)(n+2)-ary factorization. This is called the “layer-cake” view of cohomology as John described here (while I took notes, and somehow ended up with my name on the paper too). However, higher-ary factorization systems (and in particular, ternary ones) also turn up in some other surprising places. But surprisingly, they don’t ever seem to have been precisely defined by anyone, and there are lots of unanswered questions!

Posted at 8:59 AM UTC | Permalink | Followups (28)

July 22, 2010

Commutative Separable Algebras II

Posted by John Baez

Thanks to everyone who helped me with my questions about commutative separable algebras! I have taken my new-found wisdom, meager as it may be, and placed it here:

But now I have two more questions — one precise, one more open-ended.

Posted at 6:07 AM UTC | Permalink | Followups (6)

July 16, 2010

Commutative Separable Algebras

Posted by John Baez

Now that I’m in Singapore, I’ll probably be thinking more about technology and the environment, and less about pure math. But I still want to keep working with Jim Dolan. It’s just too fun to quit.

For the last few years, we’ve been trying to learn algebraic geometry and number theory and recast certain portions of these subjects in terms more friendly to (nn-)category theorists. Not by adding fancy new layers on top of the existing work: rather, by making the basics even simpler.

This project is slowly picking up speed. But I don’t really want to talk about it now. It’s not ready for prime time yet. I mention it mainly to explain why I’m asking an elementary — I hope! — question about commutative separable algebras.

Posted at 1:02 AM UTC | Permalink | Followups (16)

July 14, 2010

What is a Theory?

Posted by Mike Shulman

Over at the nForum John just asked me an interesting question, which I think deserves a wider discussion. It’s about the relationship between logical theories and their walking models, a.k.a. “syntactic categories.” In particular, which is more deserving of the name “theory”? I’m going to present my point of view, but I’d love to hear others as well.

Posted at 5:34 AM UTC | Permalink | Followups (41)

July 12, 2010

The Dold–Kan Theorem: Two Questions

Posted by Tom Leinster

The Dold–Kan Theorem states that the category of simplicial abelian groups is equivalent, in a particular way, to the category of chain complexes. I was never particularly captivated by it until André Joyal explained to me that it can be viewed as a categorification of the fundamental theorem of Newton’s finite difference calculus. He spoke about this again at Category Theory 2010 in Genova last month, but there are no notes available online and that’s not what this post is about—although his talk is what reignited my interest. So that’ll just have to be a teaser.

The purpose of this post is to ask two questions. First I’ll explain why I’m asking them. Chain complexes (of abelian groups) can be regarded as strict \infty-categories in Ab\mathbf{Ab}, the category of abelian groups. So the Dold–Kan Theorem states that in Ab\mathbf{Ab}, strict \infty-categories are the same as simplicial objects. I’m sure other people have contemplated this: hence the following questions.

Posted at 9:28 PM UTC | Permalink | Followups (53)

July 8, 2010

Grinding to a Halt?

Posted by David Corfield

There’s a perception about that

The nn-Category Café is grinding to a halt.

Lieven Le Bruyn attributes this decline to John’s shift in focus away from pure mathematics to more pressing environmental concerns. But, as I remarked there, I think it’s also largely due to energy that used to be expended at the Café now going into the nnLab and nnForum. Discussions at nnForum are often just notifications of changes at nnLab, but sometimes they blow up into the kind of thing we used to do here.

I’d be interested to know from our clientele their thoughts on this shift. Are there any among you who would liked to have read and even contributed to plethysm, Schur functors, the long-running path category vs cobordisms for bundles or some other nnForum discussion, but who can’t see themselves seeking out what might interest them there?

I’d think it a great shame if we never hear again from some of the excellent people who have dropped by here over the years simply because of the change of venue.

Posted at 12:03 PM UTC | Permalink | Followups (69)

July 5, 2010

Day on Higher Category Theory in Utrecht

Posted by Urs Schreiber

A small workshop on higher category theory takes place

Tuesday, July 13, 2010, beginning 10am, in Utrecht.

For details see the webpage Day on Higher Category Theory - July 2010.

The talks are

If you want to participate or have any questions, please drop me a note, here or by email.

Posted at 4:57 PM UTC | Permalink | Followups (2)

July 1, 2010

Homological Algebra Puzzle

Posted by John Baez

Whenever I have time to talk with James Dolan, he likes to pose puzzles — partially to test out ideas he just had, partially to teach me stuff, and partially just to watch me squirm. I’m always happiest when I find a mistake in his solution of the puzzle, but the next best thing is to solve it.

Here’s the puzzle he threw at me tonight while we were eating pizza. It’s part of a much bigger story, but I’ll rip it out of context and throw it at you:

Puzzle: What is the free finitely cocomplete linear category on an epi?

First, remember what Tom said:

Someone taught me how to spot a fake knot theorist. (You can never be too careful.) Simply ask them to draw a trefoil. If they hesitate in the slightest, they’re faking it. Actually, this will only catch really bad fakes, but in some weak sense it’s a shibboleth, a mark of identity, or at the very least, something that any professional can do.

The point I want to make is that for category theory, the ability to throw around phrases of the form

the free such-and-such category containing a such-and-such

is something like the ability to draw a trefoil without hesitation. Of course, this comparison isn’t entirely serious: one of these skills is much more meaningful than the other. But there is something serious here. While there are plenty of people who are fluent in quite sophisticated category- and topos-speak, it seems to me that outside the smallish group for whom category theory is a central research interest, not many people are comfortable with phrases of this form. And that’s a shame.

So, while the essence of this puzzle turns out to be homological algebra, being able to think about it without instantly fainting is a test of whether you’re a category theorist.

Of course, it also helps to know what the words mean here:

“What is the free finitely cocomplete linear category on an epi?”

Here ‘epi’ is a cute nickname for epimorphism — and soon to come we’ll see ‘mono’, which is a cute nickname for monomorphism. Also: we are secretly fixing a field kk, so a linear category is a category enriched over VectVect, the category of vector spaces over kk. Finally, by ‘finitely cocomplete’ we mean our category has finite colimits. For linear categories, this means it has cokernels and finite coproducts. If you don’t know much about finitely cocomplete categories, you can read an article about finitely complete categories and then turn around all the arrows! Or just stand on your head while reading it.

Okay: try the puzzle. Our attempted solution — not guaranteed to be correct — is below the fold. Even if it’s correct, we could certainly use some help filling in the details.

Posted at 6:30 AM UTC | Permalink | Followups (6)