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June 19, 2007

Cohomology and Computation (Week 27)

Posted by John Baez

In the last of this year’s classes on Cohomology and Computation, we sketched a few of the simplest consequences of the bar construction:

  • Week 27 (June 7) - Cohomology of algebraic gadgets. The bar construction "puffs up" any algebraic gadget, replacing equations by edges, syzygies by triangles and so on, with the result being a simplicial object with one contractible component for each element of the original gadget. Examples: Ext and Tor, group cohomology and homology, Lie algebra cohomology and homology. How Ext and Tor arise from the adjoint functors between the category of abelian groups and the category of modules of a ring. Free resolutions. Group cohomology as a special case of Ext. Group cohomology as the cohomology of the the classifying space BG=EG/G.

Last week’s notes are here.

We never reached what I was really aiming for, namely an automatic weakening procedure that turns λ-calculi into “simplicial λ-calculi”, in which proofs become 1-simplices, and so on. But, this should be easy for anyone who followed the whole year’s course. So, I leave it as an exercise for the reader.

I just saw Derek Wise graduate — in fact, I just “hooded” him. I thank him for taking wonderful notes on these seminars for the last 4 years. He’s going on to a postdoc at U. C. Davis.

I’m now taking a week-long vacation at a secret location far from the internet.

Have a fun summer! Next year’s seminar will be on groupoidification.

Posted at June 19, 2007 6:32 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1327



11 Comments & 1 Trackback

Read the post Cohomology and Computation (Week 26)
Weblog: The n-Category Café
Excerpt: An example of the bar construction: puffing up a point to the free contractible G-space EG, important in group cohomology.
Tracked: June 19, 2007 6:49 AM

Re: Cohomology and Computation (Week 27)

Congratulations to Derek!!

Posted by: urs on June 19, 2007 6:51 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Yes, well done Derek.

Perhaps you can be allowed a sabbatical back in Riverside to pass on your note-taking skills for the Quantum Gravity seminar.

Posted by: David Corfield on June 19, 2007 9:25 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Woohoo!

Congrats! Too bad I never made it out the Riverside given I’m so close. It would have been fun to discuss your lattice EM stuff over: (pick one) tea, wine, beer :)

Posted by: Eric on June 19, 2007 3:24 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Well done Derek! Who will take notes now…? We are sunk. It won’t be the same with someone else taking notes.

Posted by: Bruce Bartlett on June 19, 2007 11:26 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Congrats, Derek!

Vacancy: mathematical note-taker. The candidate should have a mathematical degree, a good record of attendance and punctuality, and legible handwriting …

Posted by: Tim Silverman on June 19, 2007 5:27 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Congratulations to Derek! John’s use of the term ‘hooded’ seems sinister!

I was interested in the notes on the bar construction as I have been going through very similar material in a grad course at Ottawa. People might note that although very good for Abelian cohomology the chain complex bar resolution is not optimised for non-Abelian stuff. The simplicial approach given in John’s seminar/lectures is (see Breen on Bitorsors for a taste of the results.) There are half way houses such as the standard crossed resolution beloved of Ronnie Brown and myself for low dimensional non-Abelian cohomology, and there is great fun to be had with passage between this and the chain complex viewpoint. (This leads to discussions of Fox derivatives, Alexander matrices and Jacobians, (see for instance the very nice paper by Loday on homotopical syzygies.) That stuff is fun, doable and relevant to a lot of cohomology. It points out the sort of thing that may be lost if one `linearises’ a theory too early on in a calculation. Of course, it also links in with Knot theory, but this audience will not really need reminding of that.

I have been typing up notes on my lectures and have added sections, beyond where I will get to, on torsors, bitorsors, etc. These are informal and have been done to help me understand that stuff (long overdue!) If people are interested I can make preliminary versions available.

Posted by: Tim Porter on June 26, 2007 12:39 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Tim Porter wrote:

Congratulations to Derek! John’s use of the term ‘hooded’ seems sinister!

Yeah, that’s one reason I included a link explaining this sort of ‘hood’. Don’t worry, it’s not the sort used Ku Klux Klan members. Rather, it’s a colorful cape-like thing draped over the shoulders, which hangs down in back. It’s pretty common in Britain — what do you call it over there?

Another reason for my squeamishness is that I’d never heard the verb ‘hood’ used transitively until I was asked to participate in this ritual. I’ve hoodwinked some people, but never simply hooded them.

Since no one explained these things to me as a child, I find them all quite confusing:

In most American schools, the color of the velvet outside of the hood is distinctive of the disciplines - or as closely related as possible - to which the degree earned pertains (see the table of degrees below). For instance, one who has earned a master’s degree in public administration focusing on education would wear velvet trim of “light blue” to signify education rather than “peacock blue” to denote a general public administration concentration. The width of the velvet increases from 2 inches to three inches and finally to five inches for the bachelor’s, master’s and doctorate degrees, respectively. The silk inside lining shows the colors of the school from which the wearer is a graduate. A number of other items, cords or sashes, may be also seen worn, representing various academic achievements. The length of the hood will vary with the level of academic achievement as well: bachelor’s wear a 3 foot length, master’s a 3.5 foot length, and doctors a 4 foot length.

Anyway, it’s great to have one of my higher-dimensional algebra mentors join our blog discussions! I still rememeber meeting you in Bangor, and your explanation of the virtues of simplicial methods, back when I was gung-ho on globular.

I have been typing up notes on my lectures and have added sections, beyond where I will get to, on torsors, bitorsors, etc. These are informal and have been done to help me understand that stuff (long overdue!) If people are interested I can make preliminary versions available.

Sure! Just add a link to a comment here — that’s probably the simplest thing. We like informal expositions and samizdat material.

Posted by: John Baez on June 26, 2007 4:30 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

One reason Tim may not have reacted to
hooding as being familiar, many (all?) British degrees do not have hoods but may be even more colorful:

http://good-times.webshots.com/photo/1304028017062118478dbUKKS

Oxon DPhil is bicolored.

For history:

http://www.umich.edu/news/index.html?Releases/2000/Apr00/r042000b

Posted by: jim stasheff on June 27, 2007 2:11 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

Hmm, near the top of that second link it says Edward III granted Oxford University a charter in 1214. 1214 sounds about right for the university, but since Edward III was fighting the hundred years war in the mid-14th century, it was unlikely to have been him who granted the charter! The king at that time was John (he signed Magna Carta the following year, so it was obviously a good time for starting long-lasting traditions).

However, the Wikipedia article on Oxford University doesn’t mention 1214. The closest it comes is recognition as a corporation in 1231, which would have been under Henry III. This accords with what I previously understood, that the university emerged rather gradually and haphazardly, with records of the process being rather scanty. So I suspect the article linked to is not entirely to be relied upon …

Posted by: Tim Silverman on June 27, 2007 1:26 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

This is more reliable. The false statement seems to be a conflation of:

By 1201, the University was headed by a magister scolarum Oxonie, on whom the title of Chancellor was conferred in 1214, and in 1231 the masters were recognized as a universitas or corporation,

and

In 1355, Edward III paid tribute to the University for its invaluable contribution to learning; he also commented on the services rendered to the state by distinguished Oxford graduates.

Posted by: David Corfield on June 27, 2007 2:05 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 27)

I’d be very interested to read the notes on torsors, bitorsors, etc.

Posted by: Bruce Bartlett on June 29, 2007 2:13 PM | Permalink | Reply to this

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