Category Theory Course

I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced.

The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.

Lecture 1: What is pure mathematics all about? The importance of free structures.

Lecture 2: The natural numbers as a free structure. Adjoint functors.

Lecture 3: Adjoint functors in terms of unit and counit.

Lecture 4: 2-Categories. Adjunctions.

Lecture 5: 2-Categories and string diagrams. Composing adjunctions.

Lecture 6: The ‘main spine’ of mathematics. Getting a monad from an adjunction.

Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

Lecture 8: The walking monad, the augmented simplex category and the simplex category.

Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

Lecture 8: The walking monad, the
augmented simplex category and the simplex category.

Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

Lecture 13: The adjunction between G-sets and sets.

Lecture 14: The bar construction for groups.

Lecture 15: The simplicial set \mathbb{E}G obtained by applying the bar construction to the one-point G-set, its geometric realization EG = |\mathbb{E}G|, and the free simplicial abelian group \mathbb{Z}[\mathbb{E}G].

Lecture 16: The chain complex C(G) coming from the simplicial abelian group \mathbb{Z}[\mathbb{E}G], its homology, and the definition of group cohomology H^n(G,A) with coefficients in a G-module.

Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group G by an abelian group A gives an action of G on A and a 2-cocycle c \colon G^2 \to A.

Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

Lecture 19: Group cohomology. The chain complex for the cohomology of G with coefficients in A, starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of G by A in terms of H^2(G,A).

Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. \mathbb{E}G as the nerve of the translation groupoid G/\!/G. BG = EG/G as the walking space with fundamental group G.

24 Responses to Category Theory Course

  1. Toby Bartels says:

    What’s the reduced functionality? It looks normal to me.

  2. Amusingly, that example on the first page on lecture one about fd vector spaces having skeleton the standard R^ns is one that Mochizuki (and Go Yamashita, acting as a proxy) claim shouldn’t do! See eg the bottom of page 2 in this FAQ by Yamashita http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20IUTeich.pdf namely the dialogue in A4. Odd…

    • Todd Trimble says:

      I’m somewhat sympathetic to the sentiment that working with a skeleton can be occasionally confusing. Mainly because it can cause one to “see” things which are not actually there! One of my favorite examples is the conceptual distinction between linear orderings of the set \{1, 2, \ldots, n\} and permutations thereon. Because it’s hard not to notice the usual ordering there, it’s very tempting to conflate the two — an urge which goes away when one works not with this skeleton of finite sets, but finite sets more generally, where the distinction becomes totally clear. I gather that Mochizuki (or Yamashita) is driving at something similar.

      • I agree that blind reduction to the skeleton is not the way to do things, but I have taught first-year linear algebra a number of times, and our course uses exclusively the skeleton :-). Not to mention in physics, where everything is R^3 or R^4, and one just makes sure the not-standard basis is explicit.

  3. John Baez says:

    Three new episodes!

    Lecture 8: The walking monad, the augmented simplex category and the simplex category.

    Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

    Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

  4. Samuel Vidal says:

    Thank you so much for sharing. I’m trying to understand the statement that a general morphism in Hom(C, C) looks like a wiggly diagram between (UF)^m and (UF)^n (https://johncarlosbaez.files.wordpress.com/2018/10/qg-fall2018_7.pdf p.3). Are m and n always finite ? If so, this is quite impressive to me. If I understand correctly this means in particular, that given any adjunction between two functors U and F, then any endofunctor is described by such a ‘resolution’. For example something as simple as the “maybe monad” can serve to produce such a resolution for any endofunctors of sets ?

    • John Baez says:

      Samuel wrote:

      Are m and n always finite?

      Yes. We can compose a functor with itself a bunch of times, but only a finite number of times.

      If I understand correctly this means in particular, that given any adjunction between two functors U and F, then any endofunctor is described by such a ‘resolution’.

      Not any endofunctor! Only those that are powers of UF: C → C (these are certain endofunctors on C) or FU: D → D (these are certain endofunctors on D).

      Loosely speaking, these are the endofunctors that “must exist by virtue of there being an adjunction between C and D”.

      If so, this is quite impressive to me.

      There’s nothing impressive going on here: we’re only getting what we pay for.

  5. John Baez says:

    More lectures notes from Christian Williams!

    Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

    Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

    Lecture 13: The adjunction between G-sets and sets.

    The main point here: the bar construction shatters an algebraic structure and then reassembles it, replacing equations by edges, equations-between-equations by triangles, ad infinitum.

  6. John Baez says:

    More lecture notes!

    Lecture 14: The bar construction for groups.

    Lecture 15: The simplicial set \mathbb{E}G obtained by applying the bar construction to the one-point G-set, its geometric realization EG = |\mathbb{E}G|, and the free simplicial abelian group \mathbb{Z}[\mathbb{E}G].

    Lecture 16: The chain complex C(G) coming from the simplicial abelian group \mathbb{Z}[\mathbb{E}G], its homology, and the definition of group cohomology H^n(G,A) with coefficients in a G-module.

    Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group G by an abelian group A gives an action of G on A and a 2-cocycle c \colon G^2 \to A.

    Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

    Lecture 19: Group cohomology. The chain complex for the cohomology of G with coefficients in A, starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of G by A in terms of H^2(G,A).

    • For lecture 15, shouldn’t that be BG? EG is what you get from the bar construction for the action of G on itself, no?

      • John Baez says:

        It might depend on what you mean by the ‘bar construction’, but in this class the bar construction applied to an algebra X of a monad T on a category C gives a simplicial T-algebra \overline{X} whose underlying simplicial C-object is equipped with a deformation retraction to X.

        In particular, applied to the trivial G-set, this construction gives the universal contractible simplicial G-set, which is EG.

        Another way to put it is that the bar construction is taking a G-set and creating a simplicial G-set that’s a cofibrant replacement of that. We’re also using it to take a \mathbb{Z}[G]-module and create a chain complex that’s a projective resolution of that.

        I haven’t quite come out and said most of this jargon, but that’s what’s going on.

      • Todd Trimble says:

        There’s a two-sided bar construction, denoted B(Y, M, X), which works for any triple consisting of a monad M: C \to C in a bicategory together with a left M-module X: B \to C and a right M-module Y: C \to D. In the situation John was describing, the total space of the classifying bundle would be EG = B(G, G, 1) and the base space would be BG = B(1, G, 1); the functor B(-, G, 1) applied to the map G \to 1 induces the projection EG \to BG.

        • John Baez says:

          Yes, this is why I said “it might depend on what you mean by the bar construction”. So far in class I’m only talking about the version where Y = 1_C for some category C and B is the terminal category: I’m starting with a monad on a category and and algebra of that monad.

          I’m debating whether to talk more about the the general abstract bar construction, or do more examples of the bar construction, or talk more about simplicial sets and topological spaces as models of \infty-groupoids. I think the last will be the most exciting to the students.

        • Todd Trimble says:

          I was just responding to David’s question for readers out there who might be wondering: what about BG? But you probably mean Y = M, because M will not act on the identity functor.

        • John Baez says:

          You’re right of course.

  7. James Smith says:

    The links for lectures 14ff appear to be broken, at least for me.

  8. John Baez says:

    Until December 13th I will be on jury duty every day except Friday, so the rest of this course there will be just one class per week — and none next week, when Friday is part of Thanksgiving vacation. However, some students are willing to sit through two hours of this stuff on Friday, so today’s class notes are extra long:

    Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. \mathbb{E}G as the nerve of the translation groupoid G/\!/G. BG = EG/G as the walking space with fundamental group G.

You can use Markdown or HTML in your comments. You can also use LaTeX, like this: $latex E = m c^2 $. The word 'latex' comes right after the first dollar sign, with a space after it.

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.