/sci/ - Science & Math

[Anonymous No.8899127]

Whats the best book for category theory?

my class is using 'Categories for the Working Mathematician', but I dont like it. I got a couple of other textbooks which are a bit better, but im still struggling and im hoping even more textbooks will make me understand it even better.

Specifically every time I get to natural transformations i dont know what the fuck is going on, I cant even get through the proof of Yoneda lemma.

And holy shit why is the notation so inconsistent? each textbook I see calls everything something completely different.

[Anonymous No.8899187]

Why don't you like it and what are the other books you have?

The standard, 'Categories for the Working Mathematician', can be a tough intro.
There is a book by Adwodey, but it's for CS people, mostly. Then there is one by Simmons, which does drawing diagrams as a tool very right, but lacks examples.

For the most part, knowing examples is the most important aspect, otherwise it's stale. Understanding the sense in which isomorphisms replace equality in the theory (as a theory itself I mean, written down in logic) is also quite relevant.

I don't know why you have a problem with natural transformations. Then /maybe/ look at functors as a homomorphism for functions an [math] \circ [/math]
[math] F(f\circ_C g) = F(f) \circ_D F(g) [/math]
akin to
[math] \exp(x+y) = \exp(x) \cdot \exp(y) [/math]
and a natural transformation as a homotopy of such homomorphism.
Whatever is the range of the one homomorphism (functor), you can direct it through the homotopy (natural transfomration) to get the image of another.

>Yoneda lemma
Then /maybe/ Try understanding the Yoneda embedding first

Depending on how quickly you need to learn it, I'll discuss category theory in a series of youtube videos this year.

[Anonymous No.8899199]

>>8899187
I may add..
a few years ago I tried to flesh out the common Yoneda lemma proof diagram in a way that displays all it's components and helps visualizing stuff with color. However, I post this only reluctantly since it may or may not be confusing to anyone else

[Anonymous No.8899224]

oh and thirdly, not that instead of the axiomatization of a category via
>objects and hom-sets
there is an equivalent axiomatization via
>arrows, source map and target map

That is to say, objects are a spook, they are optional to talking about categories per se

[Anonymous No.8899455]

>>8899199
Looking good, thou I haven't begin category theory myself, looking at this gets me inspired somehow

[Anonymous No.8899469]

>>8899187
>a good post on /sci/
nice

[Anonymous No.8899695]

are there any interesting categories of numbers?

[Anonymous No.8899775]

>>8899127

how's your understanding of modern algebra?

[Anonymous No.8900620]

>>8899187
>>8899199
thanks, im already using Awodey, ill get the other one too. I have another assignment in a month, but my exams are only at the end of the year, so those vids would help.

>>8899199
ill try to understand this later.

>>8899775
I know group/ring/field theory, also a bit about modules that I quickly went over since a lot of the examples talk about them.

[Anonymous No.8900913]

For what do I use category theory?

[Anonymous No.8900937]

>>8900913
Category arose out of Algebraic Topology, but nowadays it has applications even in dynamics, computer science, physics and pretty much every subfield of algebra. Most theorems can be phrased in the language of category theory.

[Anonymous No.8900945]

>>8900937
But can you give an example of pratical aplication?
This may sound like "I think this is useless", but it really is "I just don't understand how to use it for something in practice".

[Anonymous No.8900952]

>>8900945
http://math.ucr.edu/home/baez/irvine/

[Anonymous No.8900997]

>>8900952
I'm the one who posted the Schreiber pic (>>8899187), but linking to some toy construction on Baez blog shouldn't run as a "practical example".

>>8900945
Category theory is mostly a language with a few practical concepts. On it's own (in particular, without topological spaces of some sort), category theory has almost no theorems - that's why you almost can't see theorems of it applied. But the language and way of thinking about things is prevalent.

It also unified lot's of concepts.

For example, the fact that
functions with pairs as arguments
[math] f: X \times Y \to Z [/math]
[math] f(x,y) := x\, \sin(y) [/math]

are in one-one correspondence with functions on one argument which have functions as output
[math] g: X to Z^Y [/math]
[math] g(x) := y \mapsto x\, \sin(y)[/math]

is the same sort of claim as

[math] 5^(3\cdot 7) = (5^7)^3 [/math]

and this is the same sort of claim as

[math] (X\land Y\to Z) \leftrightarrow (X\to (Y\to Z)) [/math]

>if from X and Y being true, Z follows, then from X follows that given Y is true, Z is also true
>and the other way around

The theory will bring all those together as the same sort of an adjoint in a Carterian closed category

[Anonymous No.8901005]

>>8900952
I'm the one who posted the Schreiber pic (>>8899187), but linking to some toy construction on Baez blog shouldn't run as a "practical example".

>>8900945
Category theory is mostly a language with a few practical concepts. On it's own (in particular, without topological spaces of some sort), category theory has almost no theorems - that's why you almost can't see theorems of it applied. But the language and way of thinking about things is prevalent.

It also unified lot's of concepts.

For example, the fact that
functions with pairs as arguments
[math] f: X \times Y \to Z [/math]
[math] f(x,y) := x\, \sin(y) [/math]

are in one-one correspondence with functions on one argument which have functions as output
[math] g: X \to Z^Y [/math]
[math] g(x) := y \mapsto x\, \sin(y)[/math]

is the same sort of claim as

[math] 5^{(3\cdot 7)} = (5^7)^3 [/math]

and this is the same sort of claim as

[math] ((X\land Y) \to Z) \leftrightarrow (X\to (Y\to Z)) [/math]

>if from propositions X and Y being true, Z follows, then from X follows that given Y is true, Z is also true
>and the other way around

The theory will bring all those together as the same sort of an adjoint in a Carterian closed category

As an application, it's worth noting how some programming languages have stolen the abstractions to make code more slim.
E.g. in haskell, that one is omnipresent

https://hackage.haskell.org/package/base-4.9.1.0/docs/Data-Functor.html

[Anonymous No.8901014]

>>8899224
>>there is an equivalent axiomatization via
>>arrows, source map and target map
what is this one ? are you talking about

[Anonymous No.8901021]

>>8899224
>>there is an equivalent axiomatization via
>>arrows, source map and target map
what is this one ?

[Anonymous No.8901029]

>>8901014
I can't parse the sentence with the "?" placed in the middle,
but in any case I'm speaking of the first definition of category theory as written down here in my notes

https://axiomsofchoice.org/category_theory
here's one from e.g. ZFC
https://axiomsofchoice.org/category_._set_theory

I'll work a lot of this into the youtube series after I covered some Idris in the second half of the year, subscribe here
https://www.youtube.com/channel/UCcrSMnEYhIPX_p127jI23qw/videos

On the wiki I've also made some remarks on the drawing posted above in >>8899199
https://axiomsofchoice.org/yoneda_embedding

[Anonymous No.8901030]

>>8901021
I'm speaking of the first definition of category theory as written down here in my notes

https://axiomsofchoice.org/category_theory
here's one from e.g. ZFC
https://axiomsofchoice.org/category_._set_theory

I'll work a lot of this into the youtube series after I covered some Idris in the second half of the year, subscribe here
https://www.youtube.com/channel/UCcrSMnEYhIPX_p127jI23qw/videos

On the wiki I've also made some remarks on the drawing posted above in >>8899199
https://axiomsofchoice.org/yoneda_embedding

[Anonymous No.8901056]

>>8901030
>I'm speaking of the first definition of category theory as written down here in my notes
but you still use the type ''objects of C''

[Anonymous No.8901106]

>>8899127
Op,

The easiest introduction (but rigorous) to Category theory is "Category theory for Scientist" by Spivak.

Follow that up with "Conceptional Mathematics"

This is the most legit answer in the thread

[Anonymous No.8901549]

>>8901056
No I don't, the sources and targets of the arrows equal identity arrows. Then, here, an identity arrows is one which has itself as source and target (i.e. s and t are endomorphisms and idempotent on the identities).

The definition is opaque for most applications, but its possibility shines light some conceptual issues people often have.

E.g. the
>category of topological spaces
is a shit name. You want to investigate homoemorphisms, or homotopies, or whatnot.
A poset viewed as a category is about the relation of object, not so much about the objects.
Similarly, in some sense the category of sets is "really" the category of functions between sets. The sets (the objects) are just often the thing you define elsewhere and they induce you arrows, but objects are somewhat evil. But don't think I'm dogmatic about it.

[Anonymous No.8901573]

>>8901056
No, the sources and targets of the arrows equal identity arrows. Then, here, an identity arrows is one which has itself as source and target (i.e. s and t are endomorphisms and idempotent on the identities).

The definition is opaque for most applications, but its possibility shines light some conceptual issues people often have.

E.g. the
>category of topological spaces
is a shit name. You want to investigate continuous functions, or homotopies, or whatnot. It's about the arrows. A poset viewed as a category is about the relation of object, not so much about the objects. So, also, in some sense the category of sets is "really" the category of functions between sets in your set theory.
The the objects (spaces, orderable things, sets) are just often the thing you define elsewhere first and they induce your arrows. But objects can be somewhat evil.
But please don't think I'm dogmatic about that perspective.

[Analytical Engine No.8901731]

Can someone explain how one could model the Babbage analytical engine on paper?

[Anonymous No.8901923]

For a first reading I can recommend

http://katmat.math.uni-bremen.de/acc/acc.pdf

But Categories for the Working Mathematician is really top notch and often use it to recall some facts.

Also if you're the YouTube type of guy, then check out the catsters:

https://www.youtube.com/user/TheCatsters

I do not like the presentation, but it is still extremely well thought out.

[Anonymous No.8902240]

>>8901923
I find neither the Joy of Cats not the Catsters to be accessible intros.
Maybe the first few dozen pages of Joy of Cats

[Anonymous No.8902606]

>>8902240
Joy of Cats is a useful reference after knowing basic category theory. It's good because you get a bunch of results about concrete categories - bad because you should already have a strong understanding of universal constructions and nats, adjoints, etc. to see what's special about/what the desiderata are for the concrete cases.