Most of the people that I've spoken to, including set theorists,

>Why would someone with no knowledge of category theory want to learn it?
its applications to:
algebraic geometry
algebraic topology
abstract algebra
representation theory

>>8390594
It's a handy language to formulate and create analogies between a number of things in the fields described here >>8390632
The reason it seems boring is because it does not seem to talk about anything and it can't really be "applied" to anything. It really just is a different language, but sometimes just formulating things the right way can do wonders.

>>8390632
You forgot logic! Easily the most profound application of category theory IMO. Every type of category gives rise to a logic, and logics give rise to categories.

>>8390594
>Category theory.
>Getting hyped over a reformulation of set theory.

>>8390771
You would have to care about logic though.

>>8390794
Logic is the basis of everything anon.

>>8390632
this. unless you're interested in studying at least two of these, category theory will be completely useless.
even then, one could argue that schemes and generally the level of abstraction in algebraic geometry is so high as to make it basically useless

>>8390810
Seriously? Fermat's Last Theorem was proven using high-powered categorical methods.

There are two kinds of subjects: category-theory-based ones, and ones that category theory hasn't been applied to yet. Even combinatorics has found applications of category theory.

>>8390594
Set theorists generally don't know jack about category theory. However, forcing is a technique that is very naturally described in a categorical framework.

>>8390830

dubious claims

>>8390594
It gives you a different perspective on things and at it's core it's actually very simple. It only seems complicated because it is very conductive to building up many levels of abstraction (eg. algebras over a monad as "basic" example),

I like set theory, I like formal logics (classical and non-classical), and I like category theory. That said, I don't have any real interest in algebra or any of its ilk (eg. algebraic geometry). As such my perspective may differ from that of other responding (a large number of algebra-minded people learn some basic handwavy informal version of category theory solely for the purposes of algebra, that's not a bad thing but it isn't really category theory for the sake of category theory that they're talking about).

>>8390810
Nice meme.

>>8390854
https://en.m.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem

>The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

>>8390783
this is one of the greatest common misconceptions. read mclarty's "abuses of history of topos theory" if you would like to see how this is both conceptually and factually wrong.

>>8390594
The new wave of Bourbakians pushed for category theory to be the setting for their mathematics. Maybe it's a good foundation for other mathematics then?

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It's not boring. It can be used to generalize familiar concepts, it can be used as tool to translate problems from one frameworks language to that of another, and it can be used to extract the essential info from a construction already known.

I'll give a few examples. It generalizes sets, many algebraic structures and topological spaces. It allows one to do algebraic topology with homotopy and (co)homology functors. It digs deep into what makes a product of two objects a product, why a finite dimensional vector space is isomorphic to its double dual in a natural way.

The very beginning is boring, but I just love the stuff I've got myself into by proceeding further. Just give it a try!

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Category theory is realizing that the alphabet song and Twinkle Twinkle Little Star are essentially the same!

>>8390594
>Why would someone with no knowledge of category theory want to learn it?
As someone with no knowledge of category theory, I can say that the name is enough to make me want to learn it.

Say whatever you want about pure math, but they know how to name their subjects to make me want to learn them.

You look into a book and you read "algebraic topology"? Damn right you want to read it.

>>8390594
>composition of functions
>literally drew this same thing in my algebra 1 notes

WOW
SUCH ADVANCED
MUCH ABSTRACT

>>8392107

what's your point kid?

>>8390994
who is this semen demon?

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>>8392119
Maki Nishikino

>>8392107
Who said they were functions?

>>8392134
X is the input into some morphism f, which outputs a value Y, which is the input for a morphism g, which outputs Z

This is the same as morphism g composed of morphism f. It works the same way as composed functions.

Maybe I'm a mathematical genius and didn't know, but I looked up a bunch more of these diagrams and I don't see what could possibly be so hard about cat theory. It's just formalizing stuff with flow charts.

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>>8392142
And the thing you missed is the fact that the morphisms need not be functions. The diagrams are flow charts in a way, yes. I actually visualize pullbacks and pushout as Escherisque aqueducts. Nevertheless, the existence of a unique morphism making something commute in universal properties, for example, is quite a lot, to be honest.

>>8390594
It is a meme theory popularized by hipsters. They think it is cool to draw a bunch of stupid arrows and they try to use it everywhere

>>8392189

this, it is a fad perpetuated by autists in love with flow charts, and the ⊗ symbol. I suspect it will have no more deep or lasting implications than quaternions, which were likewise a fad that burnt out. The hipsters moved on to something else with weird names {the first time I heard of a functor I think I audibly let out a chuckle} and dumb symbols.

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>>8392212
I hope this thread is alive when I wake up. You gave me an idea what to draw, but I'm too tired to do it now. Thus, I shall check this thread out and remind myself of the idea. Thanks and good night!

>>8392212
Functor? I 'ardly knew 'er!

category theory is really easier than bourbakian math. plus it is richer. you mix logic+algebra+topology.

>>8392189
Organic chemistry without the hexagons?

There are less than 7 people on this planet who understand category theory. von Neumann, Terry Tao, and Teichmuller are the only ones. Maybe Andrew Wiles and Perelman, but they're only pretty good at it.

>>8392460
Ironically Tao only recently started really getting into category theory. Before that he was more of a concrete analysis kind of guy.

>>8390854
Look up combinatorial species.