Category Theory

Is it worth it that I take it as a class? I'm a last year undergraduate.

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morphism
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It depends... What's the course outline? Are you planning on going to grad school? Does it interest you?

mathwonk
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category theory is worth knowing if only to realize it is mostly horse ***. i.e.you cannot laugh at it unless you know something about it.

Hurkyl
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Pah, category theory is the greatest thing since sliced bread. :tongue:

HallsofIvy
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Category theory is so abstract that it applies to every kind of mathematical system. Which means, of course, that it doesn't give any very specific results!

I took it when I was in graduate school. Found it very interesting- and probably more closely related to "foundations of mathematics" than any thing else except, say, a deep course on set and class theory.

Hurkyl
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Well, like set theory and group theory, the basics of category theory are incredibly useful to know. It provides a powerful language for expressing ideas, and systemizes the study of several constructions that are used everywhere in mathematics. For example, the notion of an adjunction was one of the major discoveries of category theory.

And certainly, there are very concrete examples -- for example, the "algebraic" structure of the set of all real matrices of any dimension is that of an (abelian) category.

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mathwonk
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what is an adjunction? adjoint functors?

to me the main point of category theory is that one should always think not merely in terms of spaces but in terms of maps between spaces.

i.e. group homomorphisms aremore important than groups, or at least they are crucial for understanding groups. and simialrly linear transformations are more important than vector spaces.

and of course then functors are more important than categories.

it also helps you realize that an isomorphism is not a bijective morphism, but a morphism with a 2 sided inverse.

another maxim that comes out is that to show two objects are isomorphic, one should look round for a morphism between them.

categorical thinking also helps make it clear why homology groups of homeomorphic spaces are isomorphic. namely homology is a functor, and all functors preserve isomorphisms. the lack of this perspective causes the authors of very old, but sometimes still revered books, like hocking and young, to belabor the proof of such trivialities.
it also provides a framework for discussing singularities of spaces that arise from finite group quotients, like moduli spaces of curves.

namely one should keep the information that singularities of such spaces occur at points parametrizing curves which have automorphisms, and the automorphisms should be part of the structure of the quotient space, niot just the points.

so the moduli space of curves is looked at as a category, where each point is an object equipped with a family of automorphisms.

the only familiarity i have with category theory is from reading the lovely little book by peter freyd, as an impressionable youngster, and hanging around maurice auslander and his descendants.

and of course sliced bread just goes stale faster.

mathwonk
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notice that freyd himself says the main result in the subject at the time was the theorem that all abelian categories can be viewed as subcategories of the category of abelian groups. which pretty much took the fun out of them. i.e. how do you persuade people of the generality and power of axiomatic euclidean geometry when the only euclidean geometry is R^2?

(we'll see if we can wake hurkyl up again.)

Hurkyl
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notice that freyd himself says the main result in the subject at the time was the theorem that all abelian categories can be viewed as subcategories of the category of abelian groups. which pretty much took the fun out of them. i.e. how do you persuade people of the generality and power of axiomatic euclidean geometry when the only euclidean geometry is R^2?

(we'll see if we can wake hurkyl up again.)
Did manifolds become boring once we learned they can all be embedded in R^n? No! In fact the exact opposite is true -- manifolds were originally interesting precisely because they allowed you to get rid of the ambient space!

Oh, are finite dimensional real vector spaces boring simply because they are all isomorphic to R^n?

By casting away all of the irrelevant data, you make things more clear! And there's another benefit: if you look at Abelian categories as subcategories of Ab, that makes it difficult to study functors between them; the 2-category of Abelian subcategories of Ab is a very different 2-category than the 2-category of Abelian categories!

Hurkyl
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what is an adjunction? adjoint functors?
In Cat, yes: adjunctions are pairs of adjoint functors (together with their units).

to me the main point of category theory is that one should always think not merely in terms of spaces but in terms of maps between spaces.
That is one aspect of it, and the one most people seem to focus on. But there is another side to category theory, in that you can define richer structures than you can with set theory.

For example, the "algebra" of real matrices of any dimension is quite awkward to treat set-theoretically, but very natural category theoretically.

And sometimes, one might want to look at the semiring of isomorphism-classes of modules over a ring, so that it fits nicely into a set-theoretic structure. But you retain the entire algebraic structure of R-mod if you instead consider it as a 2-semiring.

In topology, you can consider the fundamental groupoid of a space, where you have points and homotopy classes of paths; this is a little more natural than the fundamental group, because you don't have to do things relative to a basepoint.

Again in topology, you have sites and topoi which can have a much richer structure than you can manage with oridinary topology alone! e.g. you can throw all manifolds together into one big site, and study them all at once, or you have the étale site of algebraic geometry.

Even in elementary geometry: arrows in the plane form a category, and you can naturally do arithmetic with them without having to relocate them to have a common origin. And if you do happen to choose an origin so that you can add points, then you get an algebraic structure called a 2-group.

Chris Hillman
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Ditto Hurkyl, but I think I'll let you guys sort this out.

(Years ago I had a long tutorial on the elements of category theory on-line, but eventually tired of revising it. This featured a long list of examples of adjunctions; this really is one of those useful concepts which every math student should know and use routinely. Basically my tutorial was intended to bring students with a strong math background up to speed without the requirement to take a formal course. The OP is lucky to see one offered but I might agree with mathwonk in suggesting that courses are for the hard stuff; for most mathematicians, category theory is more of an organizing principle used to store and retrieve information more efficiently, but possibly not easily appreciated until one has gotten through at least one year of graduate school.)

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morphism
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category theory is worth knowing if only to realize it is mostly horse ***. i.e.you cannot laugh at it unless you know something about it.
Interesting way to put it! :rofl:

...........

The advice from my advisor was to take the course on category theory to the point where it stopped being interesting and started being tiresome. I'm two lectures in so far, and it's still at the 'very interesting' point (although we haven't really done any more than define a category and a functor and given numerous examples of each). I was interested to learn, for example, that representations are just functors from $$\textbf{Grp}$$ to $$\textbf{Mod}_K$$ where $$K$$ is some field.

Apparently the tiresome part starts somewhere around the Yoneda lemma...

mathwonk
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that's my favorite part! it says any object is determined up to isomorhism by its (functor of) maps into everything. (it was an exercise in my first year alogebra course from mauslander.)