/sci/ - Science & Math

[Anonymous No.6545341]

When is the "right time" to learn category theory?

I see its language thrown all over the place casually, sometimes without any explanation - functors, initial/terminal objects, inverse limits, etc. However, last time I tried learning about it, it was over my head.

I want to study enough category theory to be able to understand the terminology when I come across it, but my end goal is not to study pure category theory. I guess it's sort of like the modernized version of set theory - ubiquitous in the language, but not too popular to study directly.

Does anyone have any good recommendations for sources to learn with my goals in mind? Thanks.

[Anonymous No.6545347]

>>6545341
what are your goals exactly? it's best learnt as you need it.

[Anonymous No.6545360]

>>6545347
I want to study algebraic geometry/algebraic number theory. I know the language of categories gets used all the time in modern geometry.

[Anonymous No.6545370]

>>6545360
where are you at background wise? you might not even have to think seriously about categories for a long time yet.

[Anonymous No.6545424]

You can probably pick up on category theory by taking a course in module theory/homological algebra. Module theory is usually taught in a first year graduate algebra course. For homological algebra try "An Introduction to Homological Algebra" by Joseph Rotman.

[Anonymous No.6545439]

>>6545370
>>6545424
I have some background working with modules, but barely any with homological algebra. In fact, the heaviest category I've ever seen is when I tried reading Dummit & Foote's section on homological algebra. Describing flat modules as somehow being invariant under whatever functor was not helpful at all, since I had no category theory experience.

[Anonymous No.6545449]

>>6545439
functors are about as basic as it gets in categories and shouldn't be a difficult concept at all for someone of your background.
you could probably wrap your head around homological algebra with what you know already and anons recommendation of Rotman is a fine place to start.

[Anonymous No.6545460]

>>6545449
>>6545424
Thanks, I'll try Rotman. It looks like a good place to finally start understanding homology/cohomology. Do you think it will have enough about category theory for my purposes? I see a little at the very beginning with definitions, but don't know how much to expect/how much I should know.

[Anonymous No.6545464]

>>6545460
the material at the start of that book is all you need, probably of more importance is a confidence with modules. once you get through that I'd recommend Weibel's Intro to Homological Algebra which is a bit heavier on the (abelian) category side of things.

[Anonymous No.6546005]

>>6545341
My undergraduate advisor is a categoricalist. Only one I ever met. Here's what he (and everyone else says) - don't learn category theory!

You'll pick up the language very easily from talking to other people. I am told that if you do study category theory there are some non-trivial theorems, however those theorems are never needed for every day mathematics.

So there's your answer. It's a nice language to standardize language but doesn't need to be learned.

[Anonymous No.6546030]

http://archive.foolz.us/sci/thread/6439275/#q6439443

http://math.stackexchange.com/questions/21128/when-to-learn-category-theory

[Anonymous No.6546032]

>>6546030
>d/l links still work.
noice one anon.

[Anonymous No.6546048]

>>6546030
That's a really informative archive post.

[Anonymous No.6546086]

>>6546032
yeah, the thread is just two months old.
The logic oriented thread I refer to at the beginning is

http://archive.foolz.us/sci/thread/6433375/#q6436134

[Anonymous No.6546378]

>>6545341
I agree that you can pick it up as you go along. However, there are lots if materials that make it easy to start early if you want.

Awodey's notes and book are extremely easy, and findable. They focus mainly on groups and posets. The oregon programming school 2012 also has a lecture series by him that goes with the book. It's geared towards programmers, so it's light on the math. This won't give you that much insight into cat theory, but it gives you something to get your hands on.

If you know any type theory or semantics/model theory, some of the underlying themes in Category theory pop to the surface. Borceux also has a very easy to follow writing style. I used JL bell's "local set theories" to get a more rigorous treatment of awodey. The first 50 pages recapitulate all the important stuff: functors, (co)products, exponentials, and more generally (co)limits, and Cartesian closedness.

I also tried to get into cat theory the homological way, which is quite natural, but I found these a bit easier if the other way didn't so it for ye. Also thinking about how it connects to logic makes the whole thing seem a little less mystical.

[Anonymous No.6546395]

>>6546030
>archiving 4chan

[Anonymous No.6546412]

>>6546395
I'm not that guy but there's some useful stuff in the /sci/ archive. Other boards make more use of it than we do though.

[Anonymous No.6546441]

>>6546395
>being this new

[Anonymous No.6546525]

>>6546005
Does your advisor hate his life?

[Anonymous No.6547638]

>>6546378
>Awodey's notes and book are extremely easy, and findable.
yeah his book is good if your math background isn't.

[Anonymous No.6549695]

http://mitpress.mit.edu/books/category-theory-sciences

forthcoming on MIT press

[Anonymous No.6550740]

>>6549695
>for the sciences