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April 20, 2007

Cohomology and Computation (Week 21)

Posted by John Baez

This time in our course on Cohomology and Quantization we explained why mathematicians like to turn algebraic gadgets and topological spaces into simplicial sets — and how this actually works, in the case of topological spaces:

  • Week 21 (Apr. 19) - Simplicial sets and cohomology. Two sources of simplicial sets: topology and algebra. The topologist’s category of simplices, Δ top\Delta_{top}. How a topological space XX gives a simplicial set called its ‘singular simplicial set’ SXS X. How this gives a functor S:TopSimpSetS: Top \to SimpSet.

Last week’s notes are here; next week’s notes are here.

There won’t be any class on Tuesday the 24th or Thursday the 26th. I’ll be in France! I’m giving a talk at this workshop:

The other people attending are: Howard Barnum, Katherine Brading, Jeffrey Bub, Brigitte Falkenburg, Michael Friedman, Lucien Hardy, Patricia Kauark-Leite, Marc Lachieze-Rey, Hermann Nicolai, Paolo Parrini, Jean Petitot, Oliver Pooley, Thomas Ryckman, Matteo Smerlak, Rob Spekkens, Paul Teller, and Christopher Timpson.

My talk will be related to some earlier themes of this class. You can see it here:

It’s a lot like the talk I gave at the Perimeter Institute last spring, but with less physics, more philosophy, nothing on higher categories, and a bit on spans and cospans.

On the way back I’ll spend a day in Nice talking to Eugenia Cheng.

Posted at April 20, 2007 8:23 PM UTC

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Re: Cohomology and Computation (Week 21)

The construction of a simplicial set for groups, rings, Lie algebras, etc. – does that have a name? I’ve always suspected something like that existed, but I don’t know what to look for.

Posted by: Walt on April 22, 2007 5:10 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 21)

It’s sometimes called the bar construction. Eilenberg and Mac Lane first used that phrase for a special case, but it applies very generally: you get simplicial sets from any algebraic gadget described by a monad.

People usually turn these simplicial sets into simplicial groups, then turn those into chain complexes, and then take their cohomology. This is sometimes called ‘monad cohomology’ — or, since monads are also called triples, ‘triple cohomology’. This subsumes group cohomology, Lie algebra cohomology, Hochschild cohomology, etc..

This might be a good place to start:

  • J.W. Duskin, Simplicial methods and the interpretation of “monad” cohomology, Mem. Amer. Math. Soc., 3 (1975).

Or, take this course! Monad cohomology is the main thing I’ll be talking about.

Posted by: John Baez on April 22, 2007 5:04 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 21)

I tracked down a description of the monad construction, and after reading your introductory material it seemed pretty easy to understand. Without that, my reaction would have been “huh?”. (And I know that, because I did the experiment. Years ago, I read the description of the construction, and my reaction was “huh?”.)

Posted by: Walt on April 24, 2007 6:51 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 21)

Great! If I had an infinite lifespan, I’d definitely spend one day a year working on a book called Scary Concepts in Mathematics, which would try to transform “Huh?” to “Cool!” for as many scary concepts as possible. Someday I might even try it without the benefit of immortality.

Posted by: John Baez on April 24, 2007 8:29 PM | Permalink | Reply to this

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