Category Theory

Wrong.
Wrong.
Your algebra teacher is only using the language of category theory. You aren't taught any of it. They're doing it to be pretentious.
Too quiet, too long.

Forget everything you know about mathematics.

A Category consists of two kinds of things:
>Objects: denoted by capital letters (eg. [math]A, B, C, \ldots[/math]).
>Arrows: denoted by lowercase letters (eg. [math]a, b, c, \ldots[/math]).

Objects just kind of exist. Each arrow however must start at one object and end at another object (potentially the same one). For instance, given objects, A and B these are some valid arrows:
[math]a: A\to B[/math]
[math]b: B\to B[/math]
[math]c: A\to A[/math]

Now you have some rules.
1) Consecutive arrows may be composed. For instance if you have two arrows:
[math]a:A\to B[/math]
[math]b:B\to C[/math]
then they can be composed to produce a third arrow
[math]ab:A\to C[/math]
or using alternative notation
[math]b\circ a: A\to C[/math]
1.a) Composition is associative. That is to say given
[math]a:A\to B[/math]
[math]b:B\to C[/math]
[math]c:C\to D[/math]
Then
[math](ab)c=a(bc)[/math]

2) Every object, [math]X[/math], has an identity arrow denoted [math]1_X:X\to X[/math].
Here are some example identity arrows:
[math]1_A: A\to A[/math]
[math]2_B: B\to B[/math]
2.a) Warning: Not all arrows from an object to itself are identity arrows.
2.b) Composing (on the left or right) with the identity arrow does nothing. Eg. given
[math]a:A\to B[/math]
[math]1_A: A\to A[/math]
[math]1_B: B\to B[/math]
we have (by composition)
[math]a1_B = a[/math]
and
[math]1_Aa = a[/math]

3) After this one begins to define other types of arrows and begins building up more layers of abstraction by talking about "functors" between categories and "natural transformations" between functors (think of them as higher order arrows). Also special patterns are identified as something called "limits" and "co-limits". This is all still introductory stuff.

Some notes:
There are a few popular notations for composition. The main thing to keep in mind is that in-order composition
>[math]ab[/math] is read as "[math]a[/math] first then [math]b[/math]"
is more popular in computer science while application-order
>[math]b \circ a[/math] is read as "[math]b[/math] of [math]a[/math]"
is more popular in mathematics.

There are many synonyms for the terminology of category theory. In particular arrows may sometimes be called morphisms, maps, functions (don't do this).

Example categories
>Set where objects are sets and arrows are functions between sets.
>Group where objects are groups and arrows are group homomorphisms.
>Top where objects are topological spaces and arrows are continuous functions.
>Tangles which is easier to explain through drawings quora.com/What-is-the-importance-of-knot-theory-to-category-theory
>programming languages sometimes have categories too where the objects are types and the arrows are functions (eg. [math]Integer \, f(Integer \, B)[/math])

That's it. A basic introduction to the concepts of category theory. While we've only scratched the surface, I hope you can see that it gives you another way to approach many areas of mathematics and computer science.

bampu because some retard bumped a bunch of meme threads but forgot this one.

What does this offer that logic doesn't already?
Implication is equivalent to subsets

cheers mate

f^2 + g^2 = (fog)^2

I like this

u are indian :(

That user is saying something that has nothing to do with category theory.

You're welcome.

Up to this point it was taught in our discrete maths lecture (CS), or am I talking about something else? But instead of talking about objects we talked about "mappings", I don't know the English word for Abbildungen.

I meant an object was just a set and these "arrows" were just mappings.

I am
Did not read your post.. ok, I really don't know what's so nice about it, since these are absolute basics which were taught in every math class of us, How did you define functions? We defined every function as mapping with start and destination set

You were taught about functions between sets. This is commonly taught in introductory mathematics. However, it just so happens that sets and functions between them form a category. That is to say they are an example of a category. That said, this is typically not elaborated on very much and it would be disingenuous to say you were taught the basics of category theory. On that note, perhaps a better name for category theory would be "function theory" (though that may also be misleading since often the arrows in category theory do not resemble functions at all).

When one is studying category theory itself one usually is interested in what sorts of property a category has (this is where terminology like cartesian closed category or braided monoidal category come from, to name a few). Doing this allows one to make conclusions about the category. For instance, the category of sets and functions is an example of a cartesian closed category which gives a number of nice but fairly technical properties.

In computer science one often studies the relationship between type theory, category theory, and logic (basically Curry-Howard Correspondence on steroids) in order to talk about structure and semantics of programming languages.

In mathematics it is also common for people to simply use category theory as an alternative (as opposed to set theory) approach to some area of mathematics. The benefit here is that category theory by its nature tends to abstract away a lot of technical but 'not very meaningful' clutter from the mathematics.

>Did not read your post.. ok, I really don't know what's so nice about it, since these are absolute basics which were taught in every math class of us, How did you define functions? We defined every function as mapping with start and destination set

If you look carefully you'll notice I didn't explicitly mention elements of objects. In fact it isn't always the case (depending on the category you're talking about) that objects have arrows or that they necessarily resemble anything like sets (similarly arrows don't necessarily resemble functions, again depending on the category you're talking about).

So that said, suppose we're talking about an arbitrary category. What sorts of things can we define? At first it may seem like you can't do very much but you'll find that in fact you can do a whole lot but you just have to do it at a higher level of abstraction than you normally do. For instance, inverse arrows:
Given
[math]f:A\to B[/math]
[math]g:B\to A[/math]
then if [math]fg = 1_A[/math], [math]g[/math] is a right inverse of [math]f[/math].
Similarly if [math]gf = 1_B[/math] then [math]g[/math] is a left inverse of [math]f[/math].
Moreover, if both of the above are true then [math]f[/math] and [math]g[/math] are said to be isomorphic.
Note that these more general definitions for inverse and isomorphism coincide with whatever other definitions for inverse and isomorphism you've encountered within any category.

This is still very much the tip of the tip of the iceberg of category theory.

>are said to be isomorphisms.
Sorry, typo. My bad.

Thanks for the explanation!

Noticed another typo (give me a break it's early af).
>that objects have arrows
meant to say
>that objects have elements