Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
@DanielSchäppi Yeah, TTT certainly misspoke there, unless by "graph" Barr and Wells meant reflexive graph (which I doubt they did). In particular, as you seem to recognize, the arrow functor to sets can't be monadic because it's not faithful (since it cannot detect behavior at isolated points of a graph).
@TimothyChow That's not bad! Actually, if Uncle John has the patience and isn't just being polite, then one could offer one of the standard (and easy!) proofs of the irrationality of $e$ as an example of this magnification process.
Yes, good point. And in the sense I was using, "topological" really refers to properties of the (forgetful) functor, much as "monadic" really refers not to a category but to a functor.
@TimothyChow Yes, and (the YouTuber) Mathologer has made a very fine video, about 30 minutes long, which explains this proof. It takes rather more than one sentence to explain.
@WillieWong In case it wasn't clear: that wasn't my question, it was mathwonk's. If you look at the edit history, it should be clear what's going on. :-)
No apology is necessary! And it's not striking a nerve, so don't worry. I share your frustration about the way "old prejudices die hard"; I think we're just disagreeing about the aptness of your answer to this question, but in the final analysis, that's no big deal.
This will be my last comment. The actual word was "branches". Whether this can mean "fields" or "subfields" or something else, who's to say. Maybe tparker can say. But we then disagree over whether category theory and set theory are correct answers, i.e., enjoy the same epistemic status as other branches like, say, algebraic geometry, according to consensus. They do, as I have argued. As opposed to History of Mathematics or Mathematics Education, mentioned by Terence. These are not deductive sciences in the same way; a consensus view is these are not mathematics proper.
