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Dear nLab folks,
An advanced undergrad with some familiarity with category theory and algebraic topology is interested in doing some additional reading this semester, on their own. I was looking for some suggestions. I seem to remember there were some “discussion notes” that John Baez had, where they read like a conversation, because they were actually a back and forth between him and a student, but I can’t seem to find them. Other suggestions are welcome as well. Thanks in advance.
Best,
Dave
Throw the students into the thick of it. I really liked Angelo Vistoli’s notes on fibered categories, stacks, and descent, and I moved onto Lurie’s HTT from there way back when.
I would not suggest HTT for a first course though. Joyal’s unreleased book on quasicategories is a much friendlier treatment of the basics because it actually bothers to go through the simplicial combinatorics and arguments. Personally I did not like Categories for the Working Mathematician. I found it extremely dry.
I think that John Baez’s Oz thing is too much of a baby treatment of the stuff. The best way to learn this stuff is to jump in headfirst and then recurse backwards into references when you don’t understand the detail of a proof.
Why not Emily’s CT textbook? Or even her first book?
Thanks for the suggestions so far. He’s already gone through a lot of Emily Riehl’s book, as I used it in the course on category theory I taught that he audited, and he’s taken algebraic topology with me out of May’s book.. Vistoli’s notes are quite nice though, that’s a good idea.
Thanks for the Baez links David.
Any other recommendations would be great as well.
If you’re aiming for a more category-theoretic direction, I like Steve Lack’s “2-categories companion”, and Kelly’s book on enriched category theory is also good.
In a rather different direction, there is of course the HoTT book too… (-:O
David, you are welcome to have a copy of the notes I started for a course in Ottawa some years ago (known as the Crossed Menagerie) There is a short version on the Lab but a lot of other stuff has been added since that was put there in 2010. The notes try to use both a modern model category type of approach (although the reader is often assumed to know the meaning of the terms from that language) as well as the more combinatorial descriptions using cocycles, etc. There is a lot of simplicial stuff in them as well also some of the crossed complex theory that is not easy to find in the literature. I leave lots of hints for explorations away from the notes, and initiations to other approaches as I felt that a student has to be able to use several different mathematical languages to swim well in these waters. They thus tackle some classical topics as well as stacks, etc.
Has your student worked through Urs’s courses? You can see them listed at the start of Introduction to Stable Homotopy Theory.
@Mike: Thanks, I’ll pass that on. I actually haven’t read those myself, so I should have a look. I think HoTT might be a bit out of reach at the moment though.
@Tim, sorry for the delay. I’d love a copy of your notes. (Even for myself). I’ll send you an email.
@DC: Thanks for the link. I think he’d need to take the algebraic topology sequel I’m teaching next semester before tackling stable homotopy though, but I’ll be sure to point him (and the rest of my class) towards Urs’ notes.
@Tim: Can’t find your email. Mine is davidcarchedi@gmail.com
@David: remember the notes change… but more frequently than my e-mail. I will send you the notes, and thus my e-mail!
(Edit: some hours later: I have sent you the notes.)
@David: the notes I pointed to in #8 concern more than stable homotopy. Note the two background files and the prelude.
@Tim may I have an updated version of the notes, too?
Of course, they will be speeding their way to Oz shortly. (There are not that much more added since the earlier version you know of but …. )
Thanks!
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