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May 11, 2007

Cohomology and Computation (Week 23)

Posted by John Baez

In this week’s seminar on Cohomology and Computation, we finally came within sight of the promised land. We began sketching how to get simplicial sets from quite arbitrary algebraic gadgets, using the mind-bogglingly beautiful ‘bar construction’:

  • Week 23 (May 10) - Simplicial sets from algebraic gadgets. Algebraic gadgets and adjoint functors. The unit and counit of an adjunction: the unit ‘includes the generators’, while the counit ‘evaluates formal expressions’. The canonical presentation of an algebraic gadget. Simplicial objects from adjunctions: the bar construction. 1-simplices as proofs.

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

To understand adjunctions, it’s really good to keep concrete examples firmly in mind, like the adjunction between

L:SetAbGpL: Set \to AbGp

(the functor sending each set to the free abelian group on that set) and

R:AbGpSetR: AbGp \to Set

(the functor sending each abelian group to its underlying set).

One of the students was quite impressed by the example of LRLR\mathbb{Z}, the group of formal integer linear combinations of integers.

But, this was just a warmup for understanding abelian groups like LRLRLRLR\mathbb{Z}, which consists of formal integer linear combinations of formal integer linear combinations of integers, and LRLRLRLRLRLR\mathbb{Z}, which consists of formal integer linear combinations of formal integer linear combinations of formal integer linear combinations of integers!

Applying the bar construction to \mathbb{Z}, we take all the above groups and lump them into a simplicial abelian group, which has:

  • formal integer linear combinations of integers as vertices,
  • formal integer linear combinations of formal integer linear combinations of integers as edges,
  • formal integer linear combinations of formal integer linear combinations of formal integer linear combinations of integers as triangles,

and so on! The (1)(-1)-dimensional simplices are just the integers themselves.

Amazing what the human mind will dream up — and eventually understand — and eventually find a necessary tool for further thought!

Posted at May 11, 2007 5:22 AM UTC

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15 Comments & 2 Trackbacks

Re: Cohomology and Computation (Week 22)

Is it possible to discuss simplicial sets, homotopies, topologies, … related to formal theories? I can see one direct way: if our theory defines some category of (algebraic) models, and for each model we can build corresponding simplicial set, then I think it’s possible to do that for whole category or some interesting “small” fragment of this category… But maybe someone could suggest more direct way for calculating topological parameters of formal theories (using n-categories for example).

If we can (theoretically) do such things then we can (also theoretically) analyze all the humans knowledge with this qualitative approach, calculating topological invariants for each theory and comparing it?

Posted by: osman on May 11, 2007 2:36 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Ousman wrote:

Is it possible to discuss simplicial sets, homotopies, topologies,… related to formal theories?

Yes, that’s exactly where I’m heading — though I don’t know how far I’ll get before the class ends.

Last quarter I focused on formal theories of a very simple sort, using a limited language called the ‘λ\lambda-calculus’. Theories in this langauge can be regarded as ‘presentations’ of cartesian closed categories. So, such theories are a lot like group presentations — or presentations of monoids, or other algebraic gadget.

In class, I sketched how any monoid presentation gives a monoidal category in which the morphisms are proofs that two products of generators give the same element in the monoid. Then I sketched how you could build a cartesian closed 2-category from a theory in the λ\lambda-calculus. In this 2-category the 2-morphisms are computations — or if you prefer, ‘proofs’ that two expressions are equal.

There’s nothing special about the particular examples of monoids, or theories in the λ\lambda-calculus. These are just illustrations of a very general idea. The idea is:

A presentation of an algebraic structure allows us to automatically categorify it, adding new morphisms which correspond to proofs, or computations.

But, one really wants to go further, and consider ‘homotopies between proofs’ — proofs that two proofs are essentially the same. One also wants to study ‘homotopies between homotopies between proofs’, and so on, ad infinitum.

The slick way to do this is a generalization of the bar construction. The bar construction uses the ‘canonical presentation’ of an algebraic gadget, where we use all possible generators and all possible relations.

Today we sketched the bar construction. Then, starting from the canonical presentation of an abelian group, we applied the bar construction to build a simplicial abelian group in which paths are proofs that two products of generators are equal. Paths of paths are ‘homotopies between proofs’, and so on.

The goal is to do the same thing for theories in the λ\lambda-calculus, or other formal theories. In the case of the λ\lambda-calculus, the result should be something like a ‘simplicial cartesian closed category’ where paths are proofs, paths-of-paths are homotopies between proofs, and so on.

The ultimate goal would be to set up a ‘homotopy theory of proofs’ where we can actually compute stuff.

So far this has only been done in some simple cases, which I’ll try to discuss before class ends!

Posted by: John Baez on May 11, 2007 5:16 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Thank you, now I understand your idea better. But I don’t understand why physicist John Baez is the only thinker (at least in the web) who develops gnoseology of mathematics. When I try to understand why some particular notion is so important in mathematics - Google always points to John Baez. So I just added your websites to my bookmarks.

Posted by: osman on May 12, 2007 7:31 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

I think it’s partly because very few people write very much mathematics aimed at a general audience, especially semitechnical treatments of heavy ideas. Even among the recent spate of blaths the vast majority are not solely mathematical, work on only toy problems, are more concerned with mathematics education, or update very sporadically.

John_B’s writings clearly tackle substantive mathematics, and have done so for so long that TWF has become the referent for many people on the web. Anyone who wants to refer to “the walking adjunction” will link to TWF, and so it rises in search engine rankings. Then more people see it, more people read other pages on it, and more people link to those other pages, and the cycle goes around.

Posted by: John Armstrong on May 12, 2007 4:48 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Off the web, though, I have noticed a recent spate of fairly heavy-duty but popular books about mathematics, e.g. Fearless Symmetry (about Galois representations and Fermat’s Last Theorem), Symmetry and the Monster (about the classification of the finite simple groups, especially the sporadics), several books about the Riemann Hypothesis (The Music of the Primes and others). I don’t know whether this is really new or whether I’ve just noticed it, but it’s interesting to see this sort of book on the popular science shelves.

Posted by: Tim Silverman on May 12, 2007 5:22 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Well Osman did specifically say, ” Google always points to John Baez”, so I restricted to the web.

As for those books, they’re really pretty hit-or-miss in my opinion. For instance, I read “Symmetry and the Monster” with the intent to review it on my own weblog, but I really can’t say much for it. Especially annoying is the author’s insistence on trying to wave his hands at real mathematics, and then to pull the punch by introducing supposedly-suggestive names for everything. If you have to define a term for the audience anyway, why give them “atom of symmetry” rather than “simple group”?

What runs through much of the popular mathematics literature is this assumption that the audience is both generally curious about mathematical subjects and terrified of actual mathematics. What John_B manages to do in TWF is to give a sense of the reasons behind the structures unburdened by the technical baggage, and he does so without euphemizing the subject to the point of euthanizing it.

Posted by: John Armstrong on May 13, 2007 6:03 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

When I ask “why this notion is so important?”, I need exact epistemological answer, not popular. Why “derivative” is so important? Popular answer: this is “speed” in physics and differetiable functions “ARE GOOD” (I hate this answer) or something… Current epistemological answer: hmm, I don’t know, try combinatorial species, free modules, categorified QO… I need an answer in terms of homomorphisms, which is central in mathematics from my point of view. And John tries to give us such answers, knowingly or not.

Posted by: osman on May 13, 2007 7:31 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Ousman wrote:

I don’t understand why physicist John Baez is the only thinker (at least in the web) who develops gnoseology of mathematics.

I’m not really a physicist — I teach in a math department. I call myself a ‘mathematical physicist’, but (curiously enough) this is the name for a mathematician who works on problems of physics. I mainly use this term as an excuse to think about as many things as possible. Even gnosiology!

One thing I don’t know about gnosiology is whether it’s the same as gnoseology and gnosology. I guess gnosology is quite different, while gnoseology is just an alternate spelling, used mainly by Eastern Europeans.

Posted by: John Baez on May 12, 2007 4:23 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

And there’s also someone on the web called … let me see now … <scratches head> … David Corfield? He seems to have a more than passing interest in the gnosiology of mathematics … :-)

Posted by: Tim Silverman on May 12, 2007 5:05 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

I mean no irony and I sincerely think you are genius, since that day when I read about categorification of quantum oscillator. I’m just stupid Russian programmer. Sorry for my English language :D

Let’s say epistemology. :)

Posted by: osman on May 12, 2007 6:03 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Don’t worry about the English — broken English is the international language of science. I hadn’t known you were Russian. I just noticed that many references to ‘gnoseology’ on the web were written by Eastern Europeans! A major exception is the Columbia Encyclopedia article, which redirects one to an article on ‘epistemology’.

So yeah, I guess you meant ‘epistemology’. But, it’s always nice to learn a new word!

It’s also nice being called a genius … but I should emphasize that James Dolan and I came up with that categorified harmonic oscillator stuff together. So, I’m at best a half-genius.

Which reminds me of the guy who tried to hire a ‘wit’ to entertain the audience at his restaurant. His assistant came back and said ‘I couldn’t find a wit, but I found two half-wits….’

Posted by: John Baez on May 12, 2007 7:06 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

The assistant was drawn and quartered.

Given the “zoo” of higher-order Lie alebra analogues, I have to ask if there can be continuous symmetries of symmetries of proofs, perhaps q-deformed proofs?

Posted by: Jonathan Vos Post on May 13, 2007 3:33 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Which reminds me of the guy who tried to hire a ‘wit’ to entertain the audience at his restaurant. His assistant came back and said ‘I couldn’t find a wit, but I found two half-wits….’

There’s a George Carlin bit along those lines, but it’s the sort of thing Zuckerman would laugh at in private while chastizing Vogan for telling in public, so not really the subject matter of this forum…

Posted by: John Armstrong on May 13, 2007 6:06 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

just a off-topic remark.. this should be week 23

Posted by: ericv on May 12, 2007 10:58 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 22)

Whoops. Thanks.

Posted by: John Baez on May 12, 2007 4:50 PM | Permalink | Reply to this
Read the post On the Bar Construction
Weblog: The n-Category Café
Excerpt: Todd Trimble on the bar construction.
Tracked: June 1, 2007 11:24 AM
Read the post Cohomology and Computation (Week 24)
Weblog: The n-Category Café
Excerpt: What makes the bar construction tick?
Tracked: June 7, 2007 5:03 PM

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