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March 9, 2016

Category Theory Seminar Notes

Posted by John Baez

Here are some students’ notes from my Fall 2015 seminar on category theory. The goal was not to introduce technical concepts from category theory—I started that in the next quarter. Rather, I tried to explain how category theory unifies mathematics and makes it easier to learn. We began with a study of duality, and then got into a bit of Galois theory and Klein geometry:

If you discover any errors in the notes please email me, and I’ll add them to the list of errors.

You can get all 10 weeks of notes in a single file here. Take a look at them and choose your favorite student!

Or, you can look at individual topics:

  • Lecture 2 (Oct. 8) - How to see topology as dual to a special kind of commutative algebra. C*-algebras, the commutative C*-algebra of continuous functions on a compact Hausdorf space, and the spectrum of a commutative C*-algebra. The Gelfand-Naimark theorem: the category of compact Hausdorff space as opposite of the category of commutative C*-algebras.
  • Lecture 8 (Nov. 16) - Klein geometry. How Euclidean plane geometry, spherical geometry and hyperbolic geometry are associated to different symmetry groups GG, with the ‘space of points’ and also the ‘space of lines’ being a homogeneous GG-space in each case. Projective geometry, and how duality lets us switch the concept of point and line in projective geometry. Klein’s general framework where a ‘geometry’ is just a group GG and a ‘type of figures’ is just a homogeneous GG-space. How to classify homogeneous GG-spaces in terms of subgroups of GG.
  • Lecture 9 (Nov. 23) - Klein geometry. A category GRelG \mathrm{Rel} with GG-sets as objects and GG-invariant relations. The example of projective plane geometry: if G=PGL(3,)G = PGL(3,\mathbb{R}), the set YY of ‘flags’ (point-line pairs, where the point lies on the line) is a homogeneous space, and there are 6 ‘atomic’ invariant relations between flags. Enriched categories. The category GRelG \mathrm{Rel} is enriched over complete atomic Boolean algebras.
  • Lecture 10 (Nov. 30) - Klein geometry. Enriched categories and internal monoids. The example of projective plane geometry: if G=PGL(3,)G = PGL(3,\mathbb{R}) and CABA\mathrm{CABA} is the monoidal category of complete atomic Boolean algebras, GRelG \mathrm{Rel} is a CABA-enriched category. Taking YY to be the set of flags, hom(Y,Y) \mathrm{hom}(Y,Y) is a monoid in CABA. We can work out the multiplication table for the atoms in this CABA, and the result is closely related to the 3-strand braid group and the symmetric group S 3S_3.
Posted at March 9, 2016 6:17 PM UTC

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Re: Category Theory Notes

Thank you for posting their notes. Would you also please concatenate each student’s notes into a single pdf? This way everyone could download three files instead of 27 files + errata.

Cheers, Kevin

Posted by: Kevin on March 10, 2016 10:25 AM | Permalink | Reply to this

Re: Category Theory Notes

Okay: I made PDFs of the complete notes of Jordan Tousignant, Christina Osborne, Jason Erbele, and Samuel Britton—Samuel gave me a big PDF and I haven’t broken it down into weeks.

It’s interesting how different student’s notes are, even the good ones. Some record the jokes, some don’t, etc. Jason’s starts with a joke.

Posted by: John Baez on March 10, 2016 5:33 PM | Permalink | Reply to this

Re: Category Theory Seminar Notes

The first three PDF’s are missing. The first weekly is there, though.

Posted by: Perry Wagle on March 11, 2016 12:37 AM | Permalink | Reply to this

Re: Category Theory Seminar Notes

Sorry, for the first few PDFs I forgot to stick in the

stuff that would make the URL point to my website. It should be fixed now!

Posted by: John Baez on March 11, 2016 2:01 AM | Permalink | Reply to this

Re: Category Theory Seminar Notes

Thank you!

Posted by: Kevin on March 11, 2016 6:09 PM | Permalink | Reply to this

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