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I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should every Mathematics student who is not planning to specialize in that area learn ?

I am not sure which area of Mathematics I would like to specialize in, though I guess it would have to be one of Topology or Geometry or Mathematical Physics. Is it a good idea for me to read Categories for the Working Mathematician by Saunders Mac Lane from cover to cover or would so much of category theory be useful only to specialists in Algebra ?

I have some knowledge of basic algebra (Groups, Rings , Fields, Galois Theory, Commutative algebra & basic Homological algebra) from the book by Lang, Manifolds & Differential Geometry (differential forms, de Rham cohomology, connections & curvature on principal bundles) though almost no knowledge of Algebraic topology. Is it a good time to learn more category theory ?

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Well, not every mathematician needs to know the Yoneda lemma, so I'd say the answer is "no more than you already know." On the other hand, the areas you seem to be interested in might be closer to category theory than some other areas are. –  Trevor Wilson Nov 19 '13 at 18:26
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I'm not experienced enough yet do know if I'm "doing it wrong" somehow, but I work with categories of modules all the time, and I don't know Mac Lane cover to cover. I find abstract theorems about categories difficult to understand and remember unless I have an application for them in my work, so I tend to learn what I need when I need it. –  Matt Pressland Nov 19 '13 at 18:28
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Regardless of how much category theory every math student should learn, I'm not sure that MacLane's book is the best possible introduction. –  Najib Idrissi Nov 19 '13 at 18:29
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How much Category theory one must learn to do what? –  Mariano Suárez-Alvarez Nov 19 '13 at 18:42
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Well, study all that and if need arises, study some category theory. –  Mariano Suárez-Alvarez Nov 19 '13 at 18:45
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2 Answers 2

up vote 8 down vote accepted

My personal opinion is that category theory is like set theory; it's a language, everyone should know the basics, and everything in the "basics" is essentially trivial. Here "basics" for set theory means subsets, products, power sets, and identities like $f^{-1}(\bigcap A) = \bigcap f^{-1}(A)$. For category theory, I think "basics" means:

  • categories, functors, natural transformations;
  • duality;
  • basic constructions like product categories, comma categories (at least over- and under-categories), and functor categories;
  • universal properties, representable functors, and the Yoneda lemma/embedding;
  • limits and colimits;
  • adjunctions.

Basically the first 4 chapters of Mac Lane (ignoring the stuff about graphs and foundations). One could probably add "abelian categories" to that list, but I think a homological algebra text is a better place to learn that.

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$f^{-1}(\cap A) = \cap f^{-1}(A)$ follows of course since right adjoint functors preserve limits. :-) –  Martin Brandenburg Nov 20 '13 at 9:14
    
@MartinBrandenburg: Unfortunately, this doesn't explain why it also preserves colimits :-) –  Stefan Hamcke Nov 20 '13 at 14:42
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@StefanH it is left adjoint to $\forall_f(A) = \{y \mid f^{-1}(y) \subset A\}$ –  Yuri Sulyma Nov 20 '13 at 17:30
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@YuriSulyma: Right, I must admit that I had seen this thing before, but I had forgotten... And of course it's a functor since $A⊆A'$ implies $∀_f(A)⊆∀_f(A')$. So THAT is the reason why preimages commute with unions, finally I know it ;-) –  Stefan Hamcke Nov 20 '13 at 20:42
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Let's start to say that I'm one of those people that believe that the more mathematics you know the better it is.

That said based on what it seems that yours interests are I believe that you could be interested in learning quite enough category theory. Category theory was originally developed by Eilenberg and Mac Lane in order to address problem in algebraic topology and homological algebra. These two areas of mathematics continues still now to push forward to develop of category theory and indeed they bring to the study of new structures like model categories for the study of homotopy theory in context different than topological space, allowing also to applies homotopical technique to other field of research.

At the same time both algebraic topology and homological algebra develop using the language of category theory so that it become necessary to well know quite a bit of category theory to work in such fields. Of course if you want to good along in advanced study in geometry is very likely you'll have to look in these fields and so that you'll have to learn some category theory.

Of course there are other part of maths that uses category theory for instance algebraic geometry, in particular scheme theory use massively a lot of categorical tools, and logic too but I don't want to go to deep on this.

Nonetheless I personally give another good reason to study a little of category theory: using category theory teach to think in an arrow theoretic fashion and most importantly using categorical language make clear some intimate connection between different objects in mathematics. Many classical construction which are carried out in some categories very easily generalize to many category (although sometimes is required some additional structure on the category). That means that is possible to generalize lots of result to many context using the language of categories. What's more some times looking definition of object in some context form a categorical point of view can make easier to understand such definition: studying some new subject I've personally experienced a good improvement in the understanding of the new objects studied when I've translate all the stuff in categorical language, because many construction were similar to other construction I've seen in different context. Of course that's just a personal opinion and a personal experience, but I believe that is worth to try learn some category theory.

In the end a suggestion, try to learn as much category theory as you can. If sometimes you get stuck don't bother too much, go along with your study the other stuff will become clearer later when you'll need it or when your mathematics knowledge will grow.

Hope this helps.

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