[thatmathguy]

I fully disagree, learning category theory (& friends) in undergrad does more harm than good.

An undergrad curriculum is expository in nature, the main goals being diversity of topics and developing a maturity.

Maturity is the biggest prerequisite to approaching cat theory imo. Similarly, without examples and non-examples from various fields, cat theory will feel like esperanto for its own sake. Given this, cat theory should not fall into undergrad territory, but will be present in any grad program.

Categories should show up naturally in Algebra/Algebraic topology, and not much elsewhere (in undergrad). Saying 'category of vector spaces over k' in a (linear) algebra course is effectively a waste of breath due to how little additional insight it provides.

[spekcular]

They don't show up much elsewhere even in research-level mathematics. Unless you're doing something related to algebraic geometry or algebraic topology, there's a 95% chance they're a waste of time. The stuff in the linked article is relevant mainly to homotopy theorists.

The importance of category theory in mathematics seems wildly overestimated by HN, judging by the number of stories about it.

[zozbot234]

> The stuff in the linked article is relevant mainly to homotopy theorists.

HoTT makes homotopy theory relevant to the foundations of mathematics. So every mathematician encounters this stuff in some way. It clarifies what it means for mathematical structures to be isomorphic, and what exactly one is doing when treating isomorphisms by analogy with equality, which is often dismissed as an "abuse of notation" but is something that practically everyone does in math.

[AnimalMuppet]

> HoTT makes homotopy theory relevant to the foundations of mathematics.

OK.

> So every mathematician encounters this stuff in some way.

I doubt that. First, not every mathematician cares all that much about foundations. If you're using differential equations in mathematical biology, how much do you actually care about foundations? And second, even if you do care about foundations, HoTT isn't the only possible foundation, nor is it the most common one. You could care about foundations and base those foundations on ZFC without giving a rip about HoTT.

So... I don't buy it. (Unless by "encounters in some way" you mean "hears it in hallway conversations" or "skims journal articles about it now and then".)

[spekcular]

HoTT is not really relevant to the study of the foundations of mathematics, in the sense that it does not help solve the questions considered important or interesting by the foundations/metamathematics community. The best they can say is basically that they came up with a theory that is mutually interpretable with ZFC, and in their opinion more aesthetically pleasing, which sort of misses the point. The purpose of ZFC isn't to be aesthetically pleasing or "capture how mathematicians actually think" or anything like that, it's to facilitate proving metamathematical theorems.

Also, people understood isomorphisms quite well decades before HoTT was invented.

I hear these talking points repeated often enough on HN that it makes me think I should write a centralized debunking of them and just link it every time a HoTT story hits the front page.

[benrbray]

As I mentioned elsewhere, my biggest complaint about my undergrad math education is that the topics felt entirely disjoint. We could take courses in any order, and it effectively felt like we started over from scratch each semester, rather than allowing the material to compound over four years. If courses are offered in a more rigid order, I think that the gradual introduction of categorical language could go a long way. It does not need to be emphasized, but it should not be neglected.

Textbook authors let some categorical language slip through so the time, and I think eductors are doing students a disservice my neglecting it. For instance, I was utterly confused about the "natural" isomorphism between a vector space and its double dual, as well as the "universal property" of tensor products, free modules, etc.. I also saw commutative diagrams in my complex analysis and topology books, but no one bothered to explain where they come from or what the rules are. Much later I finally took the time to study category theory on my own, and I can't help but think the whole experience would have been much more efficient had my professors exposed us to the idea of working with abstract morphisms much earlier.

[thatmathguy]

>Textbook authors let some categorical language slip through so the time, and I think eductors are doing students a disservice my neglecting it.

I agree. When teaching, I try to choose my terms carefully so someone would not go astray from the subject matter. It is remarkable how easy it is to get stuck on a side-quest with looking up abstract terms, while-so missing the main topic. imo the graceful way to handle it is to drop references (and disclaimers in case they are time-consuming) when introducing new things.

Somehow math learning has the property that 'struggle with x, finish x. learn y. if I knew y before x it would've been so much smoother!' sounds correct, but in practice it often does not go so well.