I want to do some reading over summer - what are the prerequisites for category theory?

The prerequisites for basic category theory are really just knowledge of basic mathematical notions of functions, composition, etc.
Most categorical notions have motivation from the category of sets, such as product, coproduct, etc.
Better yet you should be familiar with the standard examples of basic categories, Top with continuous maps, groups with homomorphism, commutative rings with identity... so you can see how categorical notions generalize anything at all, namely morphisms and how objects relate to one another via the kinds of morpihsms between them
For this the standard undergrad curriculum will suffice
However you will quickly find what you're reading to be rather useless still if you are not working with subjects where the machinery of categories actually provides clarity.
Algebraic Geometry is a good example. The modern study really begins with the development of schemes, which is motivated by the equivalence between the categories of classical affine varieties and the category of finitely generated k-algebras.

Honestly I don't suggest undertaking category theory unless you are working in disciplines where it is necessary. Its kind of like learning pure topology; its really more of a language than a mathematical study.

Ivery much doubt you have any substantial background in math if you think that category theory is meaningless, useless, easy or whatever you implied in your post.
The most recent (the last 50 years actually) developments in Algebraic Geometry suggest that schemes, the objects that provide unification between geometry, algebra, and arithmetic, are more naturally thought of as functiors rather than sets with structure as most objects in math are described. This is just one example, although I very important one
I suggest you look into Groethendieks 'relative view' for some edifying. Maybe you'll stop posting shit like this on sci

The sure sign that someone isn't cut out for math is when they ask questions about prerequisites.

If you're interested in learning something, pick up a book on it (in this case, Maclane's is still the best), and just start working through it.

If you find some stuff you don't get, you can then refer to other books as you need. If you do this constantly, then you might not be ready for that topic, but you should have a good idea on what you need to know to get ready for it.

Aluffi is a great algebra textbook, but it is nowhere close to being a source for category theory. Its treatment near the end of the most basic of categorical definitions is completely lacking.

Why category theory ? Why not literally anything else ? Do buzzwords and meme-arrows look cool to you ?
Learn more math, then learn category theory. If you have to ask, then you are not going to get anything out of it.

These. Learn about specific categories (typically ones which have something "algebraic" about them), before bothering with category theory, or everything will seem like overkill or fancy terminology for no reason.

Motivating categories for doing category theory:

locales
schemes/rings
R-modules
know an eensy bit about what a sheaf is and what complexes/homology groups are
maybe a bit of algebraic topology

Learn about these things - they are fun and the stuff of "real math" - and learn category theory after/alongside (like with aluffi).

There is no point in learning categorical terminology like "regular epimorphism/monomorphism" if you can't provide its connection to situations where these things have meaning, e.g. congruence relations and substructures in the dual category (e.g. with locales). Why you should care about functors being left or right exact becomes more clear when you know what it has to do with algebraic theories (like from universal algebra) and order theory (locales).

If you are comfortable with things like universal algebra I would maybe dip my toe in with something like learning about triples. One book for this is wells and barr. It's a relatively easy one, so maybe flip through it and if most things look foreign, save category theory for later and do some more algebra first.

>triples
Opinion discarded

>people who care what mathematical objects are called
>mathematicians

pick one

You've never traveled around to different math departments.

Like others said, starting out with category theory without any motivation for it's use in another subject is like learning Italian without the intention of reading Italian books or ever speak to other Italian speaking people.
It will be like graph theory without any ties to combinatorics.

This. Conceptional mathematics is the only book you need to start. It requires no background and builds from first principles.

The only other book id recommend along side conceptional math is Pinters book on abstract algebra

>category theory isn't math

It is but, just like any kind of foundational math, it is boring as opposed to say, topology, algebraic topology, Lie theory, differential geometry, probablity, etc. ie. "actual" math that says something about something.

You don't need any real pre-requisites but some background in math or computer science is recommended. The reason being that said background can give you examples of categories that you can look at properties of. However, a proper category theory course will also give you some more basic toy categories and some insight about how to construct them.

A fun example of a simple closely related to feynman diagrams in physics is called the category of tangles.

That said, being able to intuit and construct categories is far more important than having one good example to always go back to. The latter is like trying to learn set theory by only looking at the set of real numbers. The only people who argue that one should learn category this way are people who don't actually know very much category theory and only really care about some other subject they're applying basic category theory to. No that there's anything wrong with it but it may or may not be what you want to do.

Category Theory is very much like Set Theory in that while it's possible to spend your entire career working inside it, most other mathematicians out there will only ever scratch the surface in order to get some basic grasp of the language and formalism that they can apply to their own fields. Computer science seems to be doing more work in pure category theory these days.

These people are wrong, but they also reflect the opinion of a large number of mathematicians out there who only use a tiny amount of category theory in their work.

That book is shit.

Triple is just an antiquated term for monad. It's still in limited use in some circles.

There's nothing wrong with asking others about their own personal experience with the topic. Stop being pretentious.

>"actual"
I think the word you're looking for is "applied".

it's meta-math

Point-set topology is roughly equivalent to straight up category theory on the dryness scale.

Completely agree in the sense that they're both fairly similar in approach.
I disagree that it's dry. Personally I think both are among the most interesting topics I've studied.

Question, do you actually distinguish between "applied" and "pure" math? If the answer is yes, then you aren't really a mathematician and probably should not talk on matters related to mathematics! What you're looking for is linguistics. They are the ones more interested in words and semantics, and enjoy using flow-charts to explain things. Have fun!

It's not.

It's the difference between talking about doing something and doing the thing. Mathematicians do math, set theorists and category theorists talk about doing math.

Spoken like a true undergrad.

>it is boring as opposed to say, topology, algebraic topology, Lie theory, differential geometry, probablity,
what is pointless topology.

hahahahaha pleb undergrad

this thread pisses me off
all this undergrad shitposting

All math is applied math in that sense. No mathematician ever created a structure unmotivated. All of the most abstract machinery in mathematics are extremely well motivated.

CT is between math and philosophy. Undegrads are not able to grasp this, since all they do is eat lots of classical math without asking questions [and they are too thick to wonder why we do math anyway]

you don't think that concepts like "regular epimorphism" or exactness are more clear if you are familiar with categories where these connect back to traditional mathematical objects?

by no means limit yourself to studying just a few categories if you want to understand categories in their generality, but knowing none seems like a bad idea to me.

OP, ignore the bad advice in this thread.

Start with this text:
amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X

I'd recommend supplementing it with:
amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_1?s=books&ie=UTF8&qid=1466439086&sr=1-1&keywords=pinter algebra

Both are excellent for self-studies as they both start from first principles and build up requiring a minimum mathematical background.

I'd also recommend supplementing the first text with: ncatlab.org/nlab/show/HomePage

by looking up the definitions/theorems as you move along int he first text.

Also, Lawvere is the man:
en.wikipedia.org/wiki/William_Lawvere

One can recommend stuff like this, but nobody in the history ever started with this book (or something similar) and learned with it to the end, or half there, etc.

Don't ignore peoples experiences. As soon as people see no point, they doubt and stop.

>undergrad detected

upd

Good luck communicating to people what you're talking about then.

silly argument. the same word doesn't even mean the same thing in different contexts and is often redefined. neither monad nor triple make it clear if you are even talking about them in cat or another 2-category. some authors use "continuous functor" for "cocontinuous functor", list goes on. It's easy to communicate what you are talking about if you know how the fuck to define it clearly :3

silly argument, silly parent post :3

>input x into f(x)
>output is y
>plug y value into input of g
>get Z
>same as g(f(x))

High level math? This is literally College Algebra.

Not really. To understand a regular epimorphism then you first have to understand epimorphisms/monomorphisms and you have to understand limits/colimits. A regular epimorphism is then just a special application of these concepts.

I think it would be weird to spend time on regular epimorphisms unless you have a specific topic you're interested in.

On the other hand, when talking about families of categories such as exact categories (or similarly such as, braided, symmetric, monoidal, closed, cartesian, abelian, additive, etc.. categories) I think a student should learn about the types of categories they have an interest and background in. I don't think it's useful to tell a student that they should spend a lot of time building up a lot of background just so that they can understand a certain type of category that ultimately won't have anything to do with their interests in category theory. I'm also not convinced that it's more difficult for a student to learn the topic within the context of category at that point (i.e. cross that bridge when they get to it).

tl;dr: When attempting to apply category theory to other topics, it is useful to know those other topics. However, the pre-requisites are flexible and not the same for any student. Also it's probably easier to just learn it when you get there.

...

[spoiler]...he's right you know.
[/spoiler]

A lot of truth here.

>graph theory without any ties to combinatorics.
I'm 100% ok with this.

Me too.

underrated

>and they are too thick to wonder why we do math anyway
We do math to make money, fuck bitches and literally fly to moon you fucking Platonist shitscum.

How would you propose such a study of graphs?

Has category theory replaced knot theory as the meme field of Veeky Forums?

People seem to worship titles of subjects with "theory" attached to it. No one gives a shit about some technical paper in signal processing (say some guy is fucking around with wireless signals), but EVERYONE wants to know about "Information Theory". Just look at class rosters. No one takes technical classes unless there is a lot of hype around it or if the teacher is famous, no one ever gives a shit about whether they're going to learn anything useful in the fucking class.

People have abandoned organized religion and sought refuge in the asylum of mathematics departments.

>function composition
>college algebra
>college

try middle school

>all of those limits completely separate from the definition of a limit.
No.

>People seem to worship titles of subjects with "theory" attached to it.
Perhaps because when things get to the point you can actually call a body of knowledge a "theory" it shows that you're dealing with something relatively well-established, mature, with clear methods and either solidly proven or very probable links to other well-established theories.

>No one gives a shit about some technical paper in signal processing (say some guy is fucking around with wireless signals)
If anyone, then engineers do, and some experimental physicists possibly, if it affects their instrumentation/measurements somehow.

>No one takes technical classes
Not in a fucking mathematics department, of course. If applied and pure maths are under the same dept. then perhaps, but very rarely so in a purely pure department.

Really, I don't see what's the problem here.

except not all categories have extensionality or elements and most categories of interest have a lot more than set theoretic structure even if they do.