I have recently come across Conceptual Mathematics: A First Introduction to Categories by Lawvere & Schanuel. It is a gentle introduction to Category Theory and strikes me as a potential alternative text for a bridging course to abstract mathematics. Has anyone here used it as such and what were the pros and cons you experienced in doing so?

Related: Where can I find resources for creating a mathematics “bridge course”?

Edit: Another text I've encountered is David Spivak's Category Theory for Scientists. It is also a gentle introduction to Category Theory, but with an emphasis on modeling scientific ideas.


3 Answers 3


I find Conceptual Mathematics creative, illuminating, and thought-provoking. Subobject classifiers for high school students! However, I've never taught from it and I don't think it's well-suited to the goals of a typical bridging course.

If the goal of the course is teaching good proof techniques, I would focus on things like:

  • Do you use "Let x" to begin a proof of $\forall x$ or $\exists x$?
  • If you want to prove $P\rightarrow Q$, does it make sense to investigate $P\ \&\ \neg Q$ or $Q\ \&\ \neg P$?
  • What is the difference between $\forall x\exists y$ and $\exists y\forall x$?

The students in Lawvere and Schanuel's dialogues remind me of the students in Proofs And Refutations, by Imre Lakatos -- nominally naive, actually not likely to be tripped up by any of the above questions -- and therefore more mathematically sophisticated than most students that would be taking a bridging course. I'd stick with the suggestions from the other question.

  • $\begingroup$ I was thinking along the lines of a somewhat unconventional bridging course in which the focus would be more on gently learning about abstraction than on proof techniques. However, the opportunity cost of not focusing on proof techniques might be too high. $\endgroup$
    – J W
    Commented May 18, 2014 at 17:17

In case you have not yet seen it, I thought I would draw your attention to (what is currently) the most recent issue of the American Mathematical Monthly, and, in particular, the article:

Leinster, T. Rethinking Set Theory. The American Mathematical Monthly, 121(5), pp. 403-415.

An arXiv version can be found here.

The abstract says:

Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as 'the' axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead? The message of this article is that this can be done, in a simple, practical way (due to Lawvere). The resulting axioms are ten thoroughly mundane statements about sets.

In particular, the ten axioms (stated informally) are:

enter image description here

Leinster makes it a point to dispel a few misconceptions about Lawvere's presentation, namely, (1) that an underlying goal is to replace set theory with category theory; (2) that the axiomatization requires greater mathematical maturity than other systems (e.g., ZFC); and (3) that there might be some inherent circularity by mentioning both 'categories' and 'sets'.

(Respectively, he comments: this is literally set theory; it is not that complex; and the word 'category' is not used.)

The primary motivation for this paper is that most working mathematicians use ZFC set theory without really paying attention to the axioms. As the author somewhat humorously remarks:

enter image description here

The article does mention some pros and cons of using the text to teach "axiomatic set theory," but perhaps they could transfer to a bridge course:

enter image description here

The citations above are:

(5) F. W. Lawvere, R. Rosebrugh, Sets for Mathematics. Cambridge University Press, Cambridge, 2003.

(18) T. Trimble, ETCS: building joins and coproducts (2008). Retrieved online from http://ncatlab.org/nlab/show/Trimble+on+ETCS+III.

Still: If you are interested in trying this text for a bridging course, then maybe using Leinster's presentation would be of help. (Or reading it over, making it available for students, etc.)

(As a final, offhand comment about bridging courses: You can find more on the history of such courses at my MO response here.)

  • $\begingroup$ That's very interesting. Just to clarify, are you suggesting Lawvere & Schanuel or Lawvere & Rosebrugh as the text for a course? $\endgroup$
    – J W
    Commented May 19, 2014 at 16:57
  • $\begingroup$ @JW I have taught with neither, so am only pointing to possibly helpful supplementary materials if you decide to give it a shot. (The last time I was teaching for a set-theory-like proof-course we used the first half of Wilder's classic text: archive.org/details/IntroductionToTheFoundationsOfMathematics.) $\endgroup$ Commented May 19, 2014 at 22:36
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    $\begingroup$ This is a very valuable and informative answer (+1). I do want to point out though that one is certainly not learning "ZFC set theory" in any transitions course I have ever seen. One is barely learning set theory at all but rather learning how to do some manipulations with sets. $\endgroup$ Commented May 19, 2014 at 23:57
  • $\begingroup$ When I taught transitions, I pointed out as an aside that one should in theory probably define "ordered pair" and mentioned one possible way to do so. This entire consideration turned out to be too "formalized" for most students, almost to the point that I regret mentioning it. But maybe other transitions courses are different. $\endgroup$ Commented May 19, 2014 at 23:58
  • $\begingroup$ @PeteL.Clark Good point: I have also not seen ZFC covered in a transition course, and I don't think it would be wise to do so (outside of certain exceptional circumstances). The nice idea about the approach outlined here is that the axioms look very digestable - at least in their informal presentation. (You might need to make additional comments, e.g., the eighth axiom can be thought of as mapping an element to $0$ if it's excluded from the subset, or to $1$ if it's included in the subset; this also suggests a quick combinatoric proof of the cardinality of the power set of a finite set)... $\endgroup$ Commented May 20, 2014 at 0:16

My instinctive reaction is that a "category error" is being made here (in the philosophical sense, not the mathematical sense of category). Namely, category theory is an abstraction of (standard, undergraduate level) abstract algebra, which is itself an abstraction of the sort of very concrete mathematical manipulations most students have seen up to that point. In most undergraduate curricula I am familiar with, the sort of transition course you describe comes just before abstract algebra and gives students needed familiarity with (i) reading and writing proofs, (ii) very basic mathematical logic, and (iii) experience with the next level of abstraction in mathematics (i.e., rather arbitrary sets, functions and relations).

I have taught the above bridge course twice at the University of Georgia. Admittedly there is a class of undergraduates who do not take this course, so it is somehow the opposite of an honors course. Nevertheless I think the students there are representative of the sort of math majors one meets in many American universities. In this course I spend more than two weeks on mathematical induction, and the abstraction of induction as a statement about subsets of the natural numbers is very challenging for the students. Proving things like: if $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are functions, then: if $g \circ f$ is injective, $f$ is injective; if $g \circ f$ is surjective, $g$ is surjective; $g \circ f$ surjective need not imply $f$ surjective, and so forth -- is something that my students can learn eventually but certainly need to be taught. Each time I taught the course I ended up doing very little with cardinalities of infinite sets: this material occurs at the very end of the syllabus, and by the time I got there I felt it would be more worthwhile for the majority of the students if I spent the last part of the course reinforcing the basic techniques learned, rather than tacking on a new concept that (unfortunately, to be sure) does not get used much in the rest of their undergraduate career.

I should admit that I do not own the book of Lawvere and Schanuel. I looked at some of it on amazon just now, and it does look to be quite carefully written and unusually friendly. At a preliminary glance it looks plausible and even intriguing to use this text for some other undergraduate course. However, to use it for a transitions course would involve increasing the level of abstraction in such a course and therefore seems to be less appropriate for (at least the standard versions of) that course than for other courses. Using this text would involve abandoning most of the traditional content of a transitions course and, for the clientele to which the traditional content is pitched, that would be a loss.

I don't want to be too discouraging though: a sufficiently talented, knowledgeable and enthusiastic instructor can make almost anything work. In fact my first undergraduate introduction to abstract algebra began with five weeks of category theory. Before we learned about groups, we learned about monoids and the free monoid functor (called the "James construction": I have not gone back to try to track down its provenance). Before we studied monoids we studied sets and mapping from the perspective of universal mapping properties, e.g. we learned the Quotient Principle for when one map factors through another (in the category of sets). The latter at least turned out to be extremely useful. Overall the course at the time looked eccentric, and doing something more traditional would probably have worked even better, but it did work, because the instructor -- the still-present, great Arunas Liulevicius -- had so much insight, enthusiasm and charm. It also worked because the students were very talented and enthusiastic: this was the "honors algebra course" at the University of Chicago. So you can make things work that sound like they shouldn't, sometimes.

  • 3
    $\begingroup$ Until I encountered Lawvere & Schanual, I would have automatically agreed that category theory is best learned after (or just possibly concurrently with) courses such as abstract algebra and topology. Now, I wonder if there are benefits to introducing it earlier, counterintuitive as that may seem. Perhaps this should not be a replacement for a more conventional bridge course, but, as you mention, some other undergraduate course. $\endgroup$
    – J W
    Commented May 19, 2014 at 5:23
  • 2
    $\begingroup$ I actually think it makes a great deal of sense to talk about quotients in the category of sets before introducing them in group theory. $\endgroup$ Commented Oct 30, 2014 at 20:47

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