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Ruby Khondaker (she/her) (Aug 05 2025 at 10:40):Didn't want to necro the old #community: discussion > teaching CT topic, so thought I'd make a new one. I wanted to draw from the experience people here have had, and ask - what sorts of difficulties have people found from teaching category theory?
My guess is that these fall into 2 general types:
Of course, I'd also be interested in ways people have managed to successfully address these issues!
Adrian Clough (Aug 05 2025 at 11:23):I am a big fan of the following 3-step process followed (implicitly) in Aluffi's Algebra: Chapter 0.
Step 1 addresses your first point: Abstraction is about avoiding doing the same thing over and over again afresh. Most students are happy to learn a + b = b + a and not 1 + 2 = 2 + 1, 2 + 3 = 3 + 2, etc. Similarly, I have found when I introduce point 1 above most students are happy to learn that quite technical and often unmotivated constructions like the free product of groups follow a general pattern.
Once one has completed Step 1, one should be sufficiently motivated to proceed to Step 2 -- thus addressing your second point -- because one recognises the utility of being able to say things like preserves finite coproducts.
After this, in my experience, people either become hooked, and want to learn about AFTs, free cocompletions, Grothendieck constructions... or simply recognise the utility of categories in certain situations and move on to things they find more interesting.
John Baez (Aug 05 2025 at 11:27):I don't find teaching category theory particularly harder than other subjects: for example teaching the full modern version of the fundamental theorem of calculus seems harder, since it involves subtle concepts that the students haven't mastered yet.
However, teaching mathematics in general is extremely hard, which is why most professors are so bad at it, and only the most talented students survive. I wrote some tips on teaching here:
For example, rather few teachers notice that teaching is akin to acting, and learn the necessary acting skills to keep the students focused and eager to hear what comes next
Adrian Clough (Aug 05 2025 at 11:31):Also, just to get a pet peeve of my chest. One of the worst things to do -- this turned off many of my peers from category theory -- is to make off-hand remarks when teaching Algebra, say, about how "of course, all this could be explained much more succinctly using category theory, but that would be far too advanced for this class", making category theory seem much more foreboding that it should be.
Adrian Clough (Aug 05 2025 at 11:32):But maybe this was specific to my education :man_shrugging:
John Baez (Aug 05 2025 at 11:37):Regarding Aluffi, I start by talking about "categories of mathematical objects", which he calls "categories as containers", but I balance this with a lot of talk about "categories as mathematical objects", like how a group or a poset or a set with an equivalence relation is a kind of category. I even talk a lot about the most important categories with 3, or 2, or 1, or 0 objects, and I draw these on the board. This prevents people from thinking categories are just containers, "just a framework," with their objects being the object of interest.
If all the categories you know contain infinitely many objects, or worse a proper class of objects, you're more likely to think of category theory as abstract nonsense.
Adrian Clough (Aug 05 2025 at 11:42):John Baez said:
Regarding Aluffi, I start by talking about "categories of mathematical objects", which he calls "categories as containers", [...]
Oh, categories as containers, is a formulation I came up with. Unless, by "he" you mean "me" (just to avoid having people search for "the part about containers" in Chapter 0) :slight_smile:
John Baez (Aug 05 2025 at 11:52):Okay, sorry. "Categories as containers" is a very good term for that attitude toward categories. I only got interested in categories when physicists started using categories in other ways.
Ruby Khondaker (she/her) (Aug 05 2025 at 11:54):John Baez said:
Regarding Aluffi, I start by talking about "categories of mathematical objects", which he calls "categories as containers", but I balance this with a lot of talk about "categories as mathematical objects", like how a group or a poset or a set with an equivalence relation is a kind of category. I even talk a lot about the most important categories with 3, or 2, or 1, or 0 objects, and I draw these on the board. This prevents people from thinking categories are just containers, "just a framework," with their objects being the object of interest.
If all the categories you know contain infinitely many objects, or worse a proper class of objects, you're more likely to think of category theory as abstract nonsense.
This is quite intriguing to me - what categories with small finite numbers of objects do you usually use as examples?
John Baez (Aug 05 2025 at 12:12):I always draw and explain the initial category, the walking object, and the walking morphism. Then I point out that these are the finite ordinals (= natural numbers) 0, 1, and 2, and I draw 3, and I mention that the finite ordinals, viewed as categories, are fundamental to modern homotopy theory (without getting into any detail - that might come much later). Then I draw the walking span and the walking cospan, since those are the categories that come up in the definition of pushout and pullback, respectively. Then I draw (or more precisely the one-object groupoid ) and mention that any group can be seen as a one-object groupoid. If I had time I might draw a more interesting lattice like the power set of , which looks like a cube. (This could come later when mentioning that a lattice is a poset with finite products and finite coproducts: it's nice to draw some greatest lower bounds and least upper bounds, to give a more visceral feeling for products and coproducts.)
Alex Kreitzberg (Aug 05 2025 at 13:54):One example that left a strong impression on me as a beginner, I actually learned from Baez.
Finding the left adjoint (or lower adjoint) of the inclusion function .
This example was easy to understand (though hard to solve for me) and only got more illustrative over time as I thought about it.
Categorical properties corresponded to "mundane" properties (Being a "Functor" is a sort of monotonicity condition)
The input and output sets play an essential role in the solution.
There is soon enough context to discuss the adjoint functor theorem, which might excite mathematicians who are otherwise intimidated by adjoint Functors, but believe they're important.
And even here, getting to play with galois connections clearly conveys adjoint Functors are a sort of generalization of invertiblity, what Baez then called "the closest you can get to invertible". Which is easy to get excited about.
I find it very satisfying that there's so much to learn from just the inclusion function. Superficially it looks like "everything wrong with the categorical mindset", that we're making up objects just to waste our time converting between them. But if you get past this false impression, immediately you get rewarded with a magic trick.
It's a pretty little example of how you can learn new stuff by seriously contemplating and recontemplating simple things.
John Baez (Aug 05 2025 at 14:01):Thanks! Anyone curious about how I explained this sort of puzzle in a course can read Lecture 4 - Lecture 7 in my applied category theory course. I think the details of how one presents these puzzles matter.
Paolo Perrone (Aug 06 2025 at 17:30):In category theory (and other abstract areas of science) there is something which is often overlooked, but which I find extremely, extremely helpful when teaching:
Category theory is often about giving names and context to patterns that you have already noticed.
The best way that I found to teach category theory is to bring the students, usually through examples, to a state where the structure or phenomenon that we are studying is "already in their head, waiting for a name".
For example, I tend to motivate the definition of a category as follows:
When we compose two functions, we get again a function. But we cannot compose any two functions, the domain of the second one has to be the codomain of the first one. The same happens with matrices and their multiplication. Think of other similar examples, for example from the following, if you are familiar with them: Markov kernels and their compositions, continuous functions, continuous curves in a space, relations, group homomorphisms. If you had to formalize the common pattern that all these follow, what structure would you define?
When I ask this, most people come up themselves with the definition of a category. (Or almost, some people drop the identities, etc.)
You can do this also with functors, monads, monoidal categories, and so on.
Of course, the more advanced the topics become, the harder it is to find many examples. That's part of the challenge.
Morgan Rogers (he/him) (Aug 08 2025 at 15:22):I'll be teaching an intro to CT in Nesin Matematik Köyü next week, so this was excellent timing for this discussion!