This will be an experimental class which focuses less on computation and more on themes surrounding abstraction and category theory. Our starting point will be what we might think sets and functions are intuitively, before abstracting away desirable features, placing an emphasis on the morphisms (functions). Along the way, we will look at examples of ideas like universal properties, adjunctions, and functorial semantics which have proved to be powerful techniques both as organizational tools and devices that aid in exploring new topics. The eventual goal is to develop a picture of categorical structures, like elementary toposes and modules over monads.
Building this categorical machinery will also provide us with the language and tools to have a discussion on topics like questions of identity/equality versus equivalence, how certain invariants we assign to a collection of objects like topological spaces or representations end up having certain algebraic structures, what it means to have a collection of "spaces" or "algebras", and dualities between them.
Keeping in mind that this is a decal, we won't be assigning any mandatory homework (as in problem sets to hand in). The minimum is to have a question to ask for each reading, or a worked out problem if you don't have a question.
To help move the discussion along, please have a question prepared from each reading. We have a better chance of preparing a proper answer to a question if the question is sent in at least a day in advance (via email).
A PDF of the syllabus is here. The syllabus might contains some course logistics that we have not transferred to this webpage yet, so be sure to read the syllabus! The schedule is tentative (so please bookmark this page and check back once in a while weekly).
We will try to keep the prerequisites to a minimum. It would be nice to have some familiarity with what is usually covered in Math 55. The bare minimum should probably be at least some exposure to matrices and proof by induction. Again, this course is experimental.
The textbooks that we will actually be using are Conceptual Mathematics (CM) by Schanuel and Lawvere, and Introduction to Categories and Categorical Logic (ICCL) by Abramsky and Tzevelekos. You can obtain CM either from the library or a PDF through the library proxy. ICCL is available as a PDF here.
We'll also try to post some accompanying notes in the schedule section summarizing what happened on that day.
Most books on category theory will provide a decent number of examples as they go along, but here are some books that really load up on the examples.
Categories in Context by Riehl is very recent (published in 2016; there's an even more recent book by Grandis publsihed in 2018) and contains a huge collection of examples. The chapter on Kan extensions is mostly from Riehl's earlier book Categorical Homotopy Theory which may be of interest to some of you. Depending on how the class is feeling, we could have a look at the chapters on the Bar construction towards the end once we get to comonads. We had to delay starting for a while due to various issues so there won't be time for miscellaneous topics.
Abstract and Concrete Categories by Adámek, Herrlich, and Strecker has lots of examples and also covers factorization systems and algebraic and topological categories. Beware of some outdated terminology though (e.g. their notion of quasicategory does not refer to a model of an -category).
Sets for Mathematics by Lawvere and Rosebrugh shares quite a bit with CM, including discussions on a structural set theory, but moves forward at a faster pace and goes more in-depth.
Generic Figures and their Gluings by Reyes, Reyes, and Zolfaghari is meant to be a sequel to CM, and focuses heavily on the study of presheaf categories.
Mathematical Physics by Geroch introduces the sort of structures we might see later, like groups, fields, algebras over a field, topological spaces, and organizes them using some category theory. The actual category theoretic content is pretty lacking though.
Algebra, 3rd edition by Mac Lane and Birkhoff is an algebra textbook, but they integrate in an introduction to category theory in addition to using category theory terminology.
Topology and Groupoids by Ronald Brown may be of interest. Aside from the usage of some categorical terminology, it spends a lot of time talking about the fundamental groupoid. In particular, it gives a version of Seifert–van Kampen which doesn't require that the domain category be closed under finite intersections, and so allows us to compute .
Books on algebraic topology, homological algebra, and homotopy theory give plenty of ways to see "category theory at work", but these will go far beyond anything we'll cover. However, at some point you might be interested in Category Theory at Work edited by Herrlich and Porst (especially the article titled Concrete Dualities by Porst and Tholen).There are a wealth of resources that you can find just by searching. But here are a few you might find helpful:
The Catsters is a YouTube channel with plenty of videos explaining concepts we'll eventually see.
There's a lot of fascinating writing you can find on John Baez's website, but perhaps the most relevant to us is this page. There is also "the tale of -categories" which starts here.
Peter Smith has a reading list for people interested in philosophy here.
Some articles from Lab may be helpful. Certain entries are meant to be introductory, while other entries are fairly technical.
For convenience, the readings are posted here in addition to being in the syllabus. Most of them are due by the day they are next to, with the exceptions noted.
02/13: Article 1 of CM (due after our first session)
02/20: Article 2 of CM, pages 1 to 11 of When is one thing equal to another thing? by Mazur.
02/27: Section 1.1 of ICCL
03/06: Session 10 of CM
03/13: Sections 1 to 6 of Article 3, Session 13 of CM
03/20: Sections 7 to 12 of Article 3
04/03: Nothing, although section 2 of this page is helpful.
04/10: Article 4 of CM
04/17: Section 1.5.1 of ICCL
04/24: Section 1.5.2 of ICCL
05/01: Mac Lane's coYoneda lemma would probably be closer than what's written on the syllabus if not just examples of dualizing objects, but that isn't necessary.
02/13: We will start meeting on the 13th! No readings are due, but please think about what you'd like to get out of this decal and what you're willing to put in. It would be great if you could introduce yourself on this day, since I'd like to get an idea of who is taking the class. Dennis won't be joining us this day since he will still be traveling. Notes
02/20: Notes
02/27: A few links
03/06: Notes
03/13: Notes
03/20: Notes on presheaves and sheaves
04/03: New page for miscellaneous links and references here! Notes
04/10: Some notes ended up being part of the instiki page.
04/17: Notes are at the instiki page again.
04/24:
05/01: We were originally going to talk about the co-Yoneda lemma and (co)end calculus but it's more likely that we'll wrap up Isbell duality and either sketch some ideas about internal languages/syntactic categories or models of (weak) higher categories and some other topics e.g. periodic table, stabilization, tangles, factorization systems, etc. We'll be spending our last meetings on examples of Isbell duality or examples for which Isbell duality serves as a template for; if we have time we'll get to other kinds of dualities (syntax-semantics, recursion-corecursion, formal-concrete, etc.) since so far we had mostly just talked about dual categories in broad terms.