>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.
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.
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.