# Fundamental Theorem of Category Theory appropriate for undergraduates?

I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics build up to a “Fundamental Theorem” that is a somewhat deep result in the subject, is accessible to undergraduate level students in the time available, and serves as a kind of “capstone” which brings together the techniques and theorems studied in the course. For example, an introduction to abstract algebra and group theory may have the fundamental theorem of finite abelian groups, and an introduction to analysis may have the fundamental theorem of calculus. Is there an analogous result in Category theory that meets these criteria? Does category theory have too many prerequisites for reaching such a result? What would be a good result to conclude a short introduction to category theory with?

• Probably the Yoneda lemma is a good candidate. – Alec Rhea Oct 4 '18 at 1:53
• Special Adjoint Functor Theorem. – Oskar Oct 4 '18 at 1:55
• The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one. – Todd Trimble Oct 4 '18 at 2:15
• I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application – David Roberts Oct 4 '18 at 3:42

The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate.

Another example (that I might select, as a homotopy theorist) would be Giraud's theorem.

A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".

As mentioned above I believe the Yoneda lemma is a good candidate.

It requires an understanding of categories, functors, exponentials, natural transformations and their interplay -- how to parametrize natural transformations between the contravariant representable functors of a category $$\mathcal{C}$$ in terms of arrows in $$\mathcal{C}$$ and vice-verse.

As mentioned by Todd Trimble above the proof is short and sweet, probably easier than most standard capstone proofs, but I agree that it is 'deep' -- it elucidates something fundamental about categories themselves rather than how category theory can be applied elsewhere.

It does miss out on machinery like limits/colimits which are mentioned in many adjoint functor theorems; perhaps that 'right adjoints preserve limits' could be an additional 'capstone' theorem in the opposite direction, slightly easier for an undergraduate class than the standard adjoint functor theorems.

• The reason I did not suggest the Yoneda lemma is that, if I taught a course on category theory, it would probably come up rather earlier than the end of the semester. Of course, you could always have a first/second/third fundamental theorem of category theory, and make this the first. – David White Oct 4 '18 at 12:56
• @DavidWhite After reviewing a few category theory texts I would tend to agree with you -- I studied out of Awodey who put it back on p.162, but even there adjoints come right afterwards and serve as the high point of the book. – Alec Rhea Oct 4 '18 at 19:24