## 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 11: Comonads from adjunctions. The walking comonad. The bar construction.

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 $G.$

Lecture 21: Homotopification and higher algebra. Internalizing concepts in categories with finite products. Pushing forward internalized structures using functors that preserve finite products. Why the ‘discrete category on a set’ functor $\mathrm{Disc} \colon \mathrm{Set} \to \mathrm{Cat},$ the ‘nerve of a category’ functor $\mathrm{N} \colon \mathrm{Cat} \to \mathrm{Set}^{\Delta^{\mathrm{op}}},$ and the ‘geometric realization of a simplicial set’ functor $|\cdot| \colon \mathrm{Set}^{\Delta^{\mathrm{op}}} \to \mathrm{Top}$ preserve products.

Lecture 22: Monoidal categories. Strict monoidal categories as monoids in $\mathrm{Cat}$ or one-object 2-categories. The periodic table of strict $n$-categories. General ‘weak’ monoidal categories.

Lecture 23: 2-Groups. The periodic table of weak $n$-categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with $G$ as the group of objects and $A$ as the abelian group of automorphisms of the unit object in terms of $H^3(G,A).$ The Eckmann–Hilton argument.

### 30 Responses to Category Theory Course

1. Toby Bartels says:

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

• John Baez says:

I can’t save any new files on the website. I’m trying to get that fixed.

2. Amusingly, that example on the first page on lecture one about fd vector spaces having skeleton the standard $R^n$s 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.

• Todd Trimble says:

Hitting each of those links results in a Not Found message. I suppose a continuation of the earlier problems with the UCR server.

• John Baez says:

No, just an extra zero in “qg-fall20018”. Thanks for catching that!

A large chunk of this course is a slowed-down, diluted version of your notes on the bar construction. The ideas involved make a nice introduction to category theory for students who already know the basics.

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 $G.$

9. I hope you will write that promised short book on category theory. I hope it’ll be aimed at a general audience.
Margaret Wertheim

• John Baez says:

Thanks, but I think I need to spend about 5 or 10 years explaining math at the level math grad students can understand, better than people have done before, while I’m still up to it. I’ve spent a long time building up my understanding, and I feel some duty to share it. There’s an interesting debate to be had about what’s more useful: explaining things in a less detailed way to a larger audience, or in more detail to a smaller audience. If I have time I’d like to do both. But I think the more detailed approach requires a sharper, younger mind, so I want to do that first.

10. John Baez says:

Okay, the course is done! Here are the remaining lectures:

Lecture 21: Homotopification and higher algebra. Internalizing concepts in categories with finite products. Pushing forward internalized structures using functors that preserve finite products. Why the ‘discrete category on a set’ functor $\mathrm{Disc} \colon \mathrm{Set} \to \mathrm{Cat},$ the ‘nerve of a category’ functor $\mathrm{N} \colon \mathrm{Cat} \to \mathrm{Set}^{\Delta^{\mathrm{op}}},$ and the ‘geometric realization of a simplicial set’ functor $|\cdot| \colon \mathrm{Set}^{\Delta^{\mathrm{op}}} \to \mathrm{Top}$ preserve products.

Lecture 22: Monoidal categories. Strict monoidal categories as monoids in $\mathrm{Cat}$ or one-object 2-categories. The periodic table of strict $n$-categories. General ‘weak’ monoidal categories.

Lecture 23: 2-Groups. The periodic table of weak $n$-categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with $G$ as the group of objects and $A$ as the abelian group of automorphisms of the unit object in terms of $H^3(G,A).$ The Eckmann–Hilton argument.

11. Hendrik Boom says:

Thank you for these notes. I’m planning to work through them, filling in details as I need them. If I succeed, I’ll end up with the set of notes that I would have liked to start with. If not, well, I’ll still have learned something enlightening.

When I studied algebraic topology in 1967-8, the prof really showed us homology of topological spaces. And dismissed cohomology by saying, it’s just the dual. A few years ago I tried to imagine what mappings and such would constitute such a dual and concluded that the prof would have been right if he had left out the word “just”. There seems to be a lot there.

• Toby Bartels says:

Heh. If your class only covered homology on closed oriented manifolds, then it would be fair to say that cohomology is ‘just’ the dual. But in general, cohomology is even more interesting.

• John Baez says:

In the modern outlook cohomology seems a bit more fundamental than homology. For example the nth cohomology of a space $X$ with coefficients in an abelian group $A$ is the set of homotopy classes of maps from $X$ to a space $K(A,n)$ called the nth Eilenberg–Mac Lane space of $A.$ The nth homology also has a description in terms of Eilenberg–Mac Lane spaces, but I’ll still maintain that cohomology is simpler. The approach using Eilenberg–Mac Lane spaces may seem “fancy” at first, but a lot of modern work on cohomology (like “generalized” cohomology theories) is based on this approach. Here’s an overview:

• John Baez, Classifying spaces made easy.

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