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Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top (example 1, example 2). In my answers I said that familiarity with categories was necessary to understand. Can answers like that really be helpful to the student? If not then I should stop with it. The student must be central in my thinking when I answer a question. It must not be a way just to expose my knowledge (my flesh is weak).

You know the best what is helpful for your students. So I ask you.

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drhab
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In my opinion the language of category theory is extremely useful. It does take some effort to learn it (considerably more effort than getting the basics of set theory), but the pay-off is considerable. There is however a problem, namely this still seems to be a minority opinion at least as far as textbooks are considered. Having said that, and especially in algebraic topology, I always go with a categorical explanation when (and this is almost always the case) it is an illuminating one.

Remark: This tendency to avoid categorical language in textbooks is well exemplified by Hatcher's book. It's a monumental introduction to algebraic topology with lots of care taken for exposition and explanation yet it avoids category theory. To me, avoiding category theory in algebraic topology is avoiding algebraic topology (or at least its central language).

Ittay Weiss
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    +1 For anyone interested in getting a more modern introduction to algebraic topology, but one which starts from the basics, I strongly recommend the book by Tammo Tom Dieck. – Alex Youcis Sep 26 '13 at 07:18
  • I agree. I don't think there are many areas of math left that do not benefit from category theory at least to a small extend (if nothing else, it will often be a practical language), and a course on algebraic topology will be a good place to introduce some of it to the students. – Tobias Kildetoft Sep 26 '13 at 07:18
  • I think it's even a good viewpoint to have for simpler things like linear algebra sometimes. I never really understood why the fact that $V\cong (V^)^$ for finite-dimensional $V$ was interesting or important when I was an undergraduate (after all, $V\cong V^*$, right?). But a little explanation of how the double dual functor is naturally isomorphic to the identity (ideally without using the words "category", "functor" or "naturally isomorphic") can be helpful - the diagram is clear enough without the general theory. I tried this on some undergrads and it seemed to make things clearer. – mdp Sep 26 '13 at 07:19
  • @AlexYoucis. Thank you for this tip. I found it on internet. Everyone of you: thank you for the enlightening answers. I will go on with it. If they are not (yet) familiar with 'the joy of cats' then let it be an encouragement in that direction. – drhab Sep 26 '13 at 08:04
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To say something contrary, I find that if I write up a proof minimizing or maximizing the amount of categorical language, there's little gained one way or the other. You make the choices you make depending on what you know the audience prefers. But if I start to think one proof is somehow better or more efficient than another, usually that's a reflection of my own indigestion of the material. They're almost always the same proof, just parsed slightly differently, with language hiding quantifiers in various slick ways. In the rare case there's a real or aesthetic difference, maybe you make a choice informed by that.

If you go back and look at some of Whitney's earliest papers in manifold theory, they're stripped of pretty much all formal language. Yet the conceptual content and ramifications were spectacular. You'll always get a broader appeal if you minimize the prerequisites for your writing. If you know you're talking to people who are "converts" to whichever notation you're using, by all means use it. But given a choice I lean more towards minimal prerequisites when I can.

Ryan Budney
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