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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!
Matteo Capucci (he/him) (Sep 02 2025 at 16:05):Morgan Rogers (he/him) said:
Nesin Matematik Köyü
I didn't know of this place :O what a cool concept!
Morgan Rogers (he/him) (Sep 02 2025 at 17:08):It was so cool that I'm hoping to recreate it in some form in the future.
David Michael Roberts (Sep 03 2025 at 02:00):https://www.youtube.com/watch?v=4Gykqp0grE0
Jacques Carette (Sep 05 2025 at 14:01):Paolo Perrone said:
Category theory is often about giving names and context to patterns that you have already noticed.
This! Very much this!
Texts that remember that aspect of category theory throughout are, to me, wonderful. Texts that slowly forget this point of view as things advance get less and less interesting.
Valeria de Paiva (Sep 06 2025 at 20:12):Thanks @Morgan Rogers (he/him) and @David Michael Roberts for letting me know about this! so cool!
Posina Venkata Rayudu (Sep 16 2025 at 00:37):Adrian Clough said:
I am a big fan of the following 3-step process followed (implicitly) in Aluffi's Algebra: Chapter 0.
- Categories as organising containers -- Categories assemble "things of the same type", e.g., sets, groups, rings, etc. Aluffi defines some basic universal constructions such as objects freely generated from a set, products and coproducts. One can then ask what these are in each new category one encounters.
- Functors: passing from one container to another -- It becomes natural to figure out which of the structures considered in point 1 are preserved by which functors, which leads one to consider statements such as the one that right (left) adjoints preserve (co)limits.
- Categories as a theory -- Here one starts constructing new categories and functors with properties one has by this point recognised as useful. E.g., AFTs provide adjoints.
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.
If I may, Cantorian abstract sets may be thought of as containers, but categories as containers may be obscuring much of what categories are: qualitative turn (cf. categorical mistake is comparing qualitatively different apples and oranges; of course, as fruits we can compare apples and oranges to our hearts' content). Also, given that our ordinary conscious experience is categorical (e.g., cats, cars, people, buildings, trees, sounds et al. are all perceived as objects of corresponding categories), category theory is a welcome ever-proper alignment of reason with experience. Since every object of a category partakes in the essence(s) characteristic of the category, morphisms between objects are necessarily structure / essence-preserving. It might be of some use to highlight that category is not a (French ;) structural turn; categories are geometric objectifcation of concepts abstracted from particulars, thereby serving as theories, functorial interpetations of theories, and as backgrounds for these models. Encouraged by the fact that a listing of properties of functions with respect to composition readily leads to category theory, we can start with even more elementary place-value notation for numbers, which when displayed as functions with geometric domains (places) and algebraic codomains (values), and readily recognize categories (not to mention the self-foundation of domain geometry subsuming the algebra of codomain values as its subcategory). Unfortunately, functions in the then set theory didn't have codomains, while type theory didn't have domains; hence the delayed discovery of categories. Also, functoriality, without discounting the indispensability of functors in going across limited universes of discourses / categories, seems to be about compatibility with composition. It is also not without value to introduce category theory in terms of trending things: for example, a morphism between models (in the background of sets) of a theory consisting of m component objects and n component structural maps consists of m functions satisfying n equations, which can inform why divide-and-rule works: with compounding identities as component structural maps, it's difficult to change the status quo of the society. In closing, unlike physics, with its shortest path, math (category theory) appears to have a thing for crooked paths, possibly a reflection of its makers (cf. Isaiah Berlin). P.S. [variable] algebra as an advance beyond [constant] arithmetic is understandably attractive, but even elementary arithmetic, say, 17 + 8 = 25 is loaded with content and waiting for humanity to contemplate (not only problems, but also the solutions that work so wonderfully). Your 5 mBTC ;)
fosco (Sep 16 2025 at 05:31):What's the origin of the pdf on "delayed discovery of categories"?
David Michael Roberts (Sep 16 2025 at 05:55):@fosco it almost looks like a Lawvere document
Patrick Nicodemus (Sep 16 2025 at 05:55):looks like the default font in LibreOffice, Liberation Serif? No, the 5 is wrong. Hm.
Late 19th century mathematics would surely have recognized that a typical map has a
a definite domain A and a definite codomain B that is, in general, distinct from its image E
This point is addessed in the first few pages of https://pages.physics.ua.edu/faculty/fabi/CT/The%20Uses%20and%20Abuses%20of%20the%20History%20of%20Topos%20Theory.pdf
David Michael Roberts (Sep 16 2025 at 06:03):Compare the arrow on page 5 of https://ncatlab.org/nlab/files/Lawvere-OpenProblems.pdf with the arrow in 1a) in the 'Delayed discovery...' document.
David Michael Roberts (Sep 16 2025 at 06:05):BTW there are three versions of Lawvere's 'Open problems ...' document: the one on the nLab I linked to, the 2016 update (https://github.com/mattearnshaw/lawvere/blob/master/pdfs/2009-open-problems-in-topos-theory.pdf) and the Lawvere Archives version, which is typeset in a modern way (https://lawverearchives.com/wp-content/uploads/2024/12/2009.PP_.openproblemsintopostheory.pdf)
David Michael Roberts (Sep 16 2025 at 06:06):But then I can't categorically (heh) state that Lawvere wrote that. But it feels like the kind of thing he would write.
Patrick Nicodemus (Sep 16 2025 at 06:06):David Michael Roberts said:
But then I can't categorically (heh) state that Lawvere wrote that. But it feels like the kind of thing he would write.
You nailed it, his name is in the document metadata.
Patrick Nicodemus (Sep 16 2025 at 06:09):the font is Cambria, in case anyone was curious.
Patrick Nicodemus (Sep 16 2025 at 06:09):I also discovered a Google easter egg: if you Google "<font name> font", it will return the results in that font.
fosco (Sep 16 2025 at 06:31):thanks! The other thing I would like to know is what the author (at this point, probably Bill) meant with "there were several parallel developments
in Italy and Portugal prior to 1945, in which the idea (that
homomorphisms of algebras, continuous maps of
spaces, operators between Banach spaces, et cetera,
form a significant kind of algebraic structure) was being
crystallized"
Posina Venkata Rayudu (Sep 16 2025 at 06:33):fosco said:
What's the origin of the pdf on "delayed discovery of categories"?
@fosco I am terribly sorry about that! It's in response to a question I had while teaching Conceptual Mathematics at The Salk Institute.
Posina Venkata Rayudu (Sep 16 2025 at 06:38):David Michael Roberts said:
fosco it almost looks like a Lawvere document
@David Michael Roberts WOW, speak of Disturbingly Delightful (it's a chocolate & wine place in Little Italy; enjoy if you happen to be in San Diego :), what's the decision structure logic you deployed to conclude that's Professor F. William Lawvere; amazing!
fosco (Sep 16 2025 at 06:39):This is an interesting (and timely, for me) discussion, partly because I recently have been drafting a book on category theory together with many members of the ItaCa group, and also because every year since 2020 I struggle (the struggle being mainly due with my inexperience) with the task of teaching Category Theory to people who do not know mathematics.
In particular, working on the book, I have put a lot of effort in trying to convey an idea that I haven't found explicitly spelled out in any other reference, but that I believe we all agree upon, at least in its general form:
This is a category: .
But a category is also a "universe in which to do mathematics" --some would say it's a system of types.
But a category is also a simultaneous generalization of a partial order, and a monoid, meaning that every monoid is a category (bounding the size of the class of objects) and every poset is a category (bounding the size of the class of morphisms). One can fruitfully do category theory only embracing all three points of view at the same time, i.e. thinking categories as (combinatorial) shapes, as foundational universes, and as mathematical structures of their own specific kind.
Posina Venkata Rayudu (Sep 16 2025 at 06:42):fosco said:
thanks! The other thing I would like to know is what the author (at this point, probably Bill) meant with "there were several parallel developments
in Italy and Portugal prior to 1945, in which the idea (that
homomorphisms of algebras, continuous maps of
spaces, operators between Banach spaces, et cetera,
form a significant kind of algebraic structure) was being
crystallized"
@fosco here's what Professor F. William Lawvere often alludes to.
David Michael Roberts (Sep 16 2025 at 06:45):Posina Venkata Rayudu said:
David Michael Roberts WOW, speak of Disturbingly Delightful, what's the decision structure logic you deployed to conclude that's Professor F. William Lawvere; amazing!
The formatting of the document (very atypical), the subject matter (which aligns with other things Lawvere has written about historical precursors to his ideas), and what I have learned of your interests (and approach to writing about mathematics and referencing Lawvere's work).
David Michael Roberts (Sep 16 2025 at 06:47):BTW, thank you for making Lawvere's emails in response to your questions public.
Posina Venkata Rayudu (Sep 16 2025 at 06:58):David Michael Roberts said:
Posina Venkata Rayudu said:
David Michael Roberts WOW, speak of Disturbingly Delightful, what's the decision structure logic you deployed to conclude that's Professor F. William Lawvere; amazing!
The formatting of the document (very atypical), the subject matter (which aligns with other things Lawvere has written about historical precursors to his ideas), and what I have learned of your interests (and approach to writing about mathematics and referencing Lawvere's work).
@David Michael Roberts even Professsor F. William Lawvere wrote to me quite a few times that referencing his work alone is not correct (i remember him saying (via email only) that Tarski or Halmos came very close to his functorial semantics; unfortunately (or fortunately, given Tarski's penchant for paradoxes ;) for them category theory was no more than a pair of sets, while Professor F. William Lawvere recognized the difference: binary elementhood vs. ternary composition), but the ideas occupying my mind since 1997 (the year CM was published) are like border security forces dessicating any and every idea trying to sneak in ;) Here's the take of Professor Ronnie Brown and that of Professor Michael Barr, which I could readily find and thought you and @fosco might find interesting.
Posina Venkata Rayudu (Sep 16 2025 at 07:05):Posina Venkata Rayudu said:
fosco said:
thanks! The other thing I would like to know is what the author (at this point, probably Bill) meant with "there were several parallel developments
in Italy and Portugal prior to 1945, in which the idea (that
homomorphisms of algebras, continuous maps of
spaces, operators between Banach spaces, et cetera,
form a significant kind of algebraic structure) was being
crystallized"fosco here's what Professor F. William Lawvere often alludes to.
@fosco here's the reading of Professor Andree Ehresmann / Charles Ehresmann.
@fosco Professor F. William Lawvere discusses the work of Portuguese mathematician J. Sebastiao e Silva on functional analysis and in the context of covariant cohension.
Posina Venkata Rayudu (Sep 16 2025 at 07:40):fosco said:
This is an interesting (and timely, for me) discussion, partly because I recently have been drafting a book on category theory together with many members of the ItaCa group, and also because every year since 2020 I struggle (the struggle being mainly due with my inexperience) with the task of teaching Category Theory to people who do not know mathematics.
In particular, working on the book, I have put a lot of effort in trying to convey an idea that I haven't found explicitly spelled out in any other reference, but that I believe we all agree upon, at least in its general form:
This is a category: .
But a category is also a "universe in which to do mathematics" --some would say it's a system of types.
But a category is also a simultaneous generalization of a partial order, and a monoid, meaning that every monoid is a category (bounding the size of the class of objects) and every poset is a category (bounding the size of the class of morphisms). One can fruitfully do category theory only embracing all three points of view at the same time, i.e. thinking categories as (combinatorial) shapes, as foundational universes, and as mathematical structures of their own specific kind.
@fosco the following, in light of the above, might be of some interest to you: It is not correct to say that a given category C 'is' a category of structures, because there may be many such representations (a similar subtlety with the 'is' arises in linear algebra: it is not correct to say that a vector 'is' a list of numbers because a given vector space has many co-ordinatizations (isomorphisms with suitable R^n). Simplistically stated, it's not unlike writing a number n as more than one exponential (n = a^b = c^d = ..., albeit not in the case of all numbers). Professor F. William Lawvere explicitly states (i'll look it up) that a category C can be represented as B^T or D^V or ... depending on what is taken as a theory and what is taken as a background for interpreting the theory to obtain models of the theory. On a not too unrelated note: Consider the pure math modeling in terms of diagrams in the category of Cantorian abstract sets. This structure is essentially the only expression of whatever "inner essence" the dots may have. From that point of view, a category is special case of truncated simplicial set, i.e. using finite totally sets as (contravariant) diagram schemes and as "figure shapes" in any given category. Two of the maps between 3 and 2 correspond to the domain and codomain structure of any category, whereas one map from 2 to 3 corresponds to the composition operation in any category. Then the map from 2 to 1 corresponds to the "inclusion" from the set of objects to that of the maps. This inclusion should be understood in the same spirit as any inclusion map in category theory.
@David Michael Roberts knows who ;) There's more on this in Functorial Semantics of Algebraic Theories, p. 12.
Posina Venkata Rayudu (Sep 16 2025 at 08:29):David Michael Roberts said:
BTW, thank you for making Lawvere's emails in response to your questions public.
@David Michael Roberts (i can't take them with me to bardo ;)
On a serious note, I just remembered that I owe you all an apology. Please accept my sincere apologies for the mistake in my letter to the Notices of the AMS. I am attaching my corrigendum published in the September issue of the Notices.
Once again I am terribly sorry for my mistake!
Thanking you,
Yours respectfully,
posina
Corrigendum_Posina.pdf
David Michael Roberts (Sep 16 2025 at 08:37):@Posina Venkata Rayudu can I ask if the Lawvere Archive has a copy of the 'Delayed discovery...' document? Was it something he just sent to you, or a general manuscript you happen to have a copy of?
fosco (Sep 16 2025 at 08:47):Posina Venkata Rayudu said:
fosco the following, in light of the above, might be of some interest to you: [...]
It is a very useful remark and I totally agree; I made this analogy with bases and tuples of scalars when teaching, many times, especially when I have to explain what's a Lawvere theory.
How should I attribute this passage, in case I want to quote it?
Posina Venkata Rayudu (Sep 16 2025 at 09:36):David Michael Roberts said:
Posina Venkata Rayudu can I ask if the Lawvere Archive has a copy of the 'Delayed discovery...' document? Was it something he just sent to you, or a general manuscript you happen to have a copy of?
@David Michael Roberts The Lawvere Archives doesn't have a copy. When I started teaching Conceptual Mathematics (CM), I requested Professor F. William Lawvere, along with many serious category theoreticians to be on the CM mailing list, along with the students, so that they can correct any mistakes I may make in my lecture notes I used to email a week in advance of my lecture. All my questions and Professor F. William Lawvere et al. answers posted to the CM mailing list are archived at The Salk Institute (i'll see if i can find the website address / login et al). I still vividly experience the gratitude I felt when many of the who's who of category theory answered my nagging questions dating back to some 0.25 century in a very helpful manner. One of these days, if the religion of peace doesn't put life / humanity to rest in peace, I'll collect them all and organize and send it to Danilo Lawvere. (I shared Professor F. William Lawvere & Stephen Schanuel Objective Number Theory material I had, which will appear on the archives sometime soon given Danilo Lawvere's workload.) I think the Lawvere family, especially Madam Fatima Lawvere, along with their grandchildren are already working very hard to make public all of Professor F. William Lawvere's unpublished work, the volume of which is beyond anyone's imagination if one only counts his published papers.
P.S. Cambridge University Press (CUP) is very eager to publish a book on Professor F. William Lawvere's seminal insights and expansive visions (such as): a mathematical theory of the transformation of time and space into each other in a way more profound than that reflected in the notion of velocity. I don't remember what and when I wrote to CUP, if I did, but it looks like I have been marked on their mailing calender. I think Gilbert Strang is heading the math unit at CUP, but I don't think he's into category theory (although I felt a kinship in reading him say in his zillionth edition Linear Algebra textbook: formulas have a place, but not the first place :) If anyone of you are interested in working on CUP interest in Professor F. William Lawvere's brilliance, I'd be very happy! (I'll try to find any emails i wrote to CUP and their invitations to submit a book proposal.)
Posina Venkata Rayudu (Sep 16 2025 at 10:47):fosco said:
Posina Venkata Rayudu said:
fosco the following, in light of the above, might be of some interest to you: [...]
It is a very useful remark and I totally agree; I made this analogy with bases and tuples of scalars when teaching, many times, especially when I have to explain what's a Lawvere theory.
How should I attribute this passage, in case I want to quote it?
@fosco
I'm looking up for a published paper in which Professor F. William Lawvere makes the same point; until then you can go with what seems sensible and reasonable to you.