Can someone explain my Category questions

I'll preface this with I DO NOT know Category theory. I am just starting to learn it. I don't really have many people to ask, so hope someone can help.

Here is an example I came across that I get the big idea of but still have some gaps.

First of all I understand Categories satisfy:

A collection of objects that obey,

Composition (associativity)
Identity element
"Arrows" or morphisms from one object to another.

Functors are morphisms between categories.

Let FinSet be the category whose objects are finite sets and arrows are functions.

x is in Ob(FinSe) and suppose x={a,b}

T: FinSet -> FinSet is the functor that takes a set to the set of list on it.

x={a,b}
Tx ={(a,b,c),(a,b,b,b,b),...}

Now if you apply T to itself

You get

T(Tx)=T^2={a,b,c; a,a,b,b,b;, ....}

Which is a set of elements that contain a list of list of elements.

From there somehow the example does "Multiplication" to a set that contains just a list of elements {a,b,c, a,a,b,b,b,..}
I have soooo many questions around this. Nothing was properly defined so I have many gaps here.


First question:
1) Are objects categories? Here the object is finite sets and the category also seems to be the exact same collection of finite sets. I thought categories and there objects were distinct? I'm confused by what consist of purely a category and purely objects contained within them.

Second question:
2) What does a "Functor" do in my example? It seems like a function that applies "multiplcation" to itself and since FinSets are a category and since functors are arrows to categories then it satisfies the definition of a functor. But then why even call that a "functor" instead of a function? Seems confusing. This also goes back to my first question, what is the difference between a category and objects inside the category?

PART II next.

Other urls found in this thread:

byorgey.wordpress.com/catsters-guide-2/
golem.ph.utexas.edu/category/2007/09/the_catsters_on_youtube.html
twitter.com/NSFWRedditVideo

Third question:
3) What is going on with T(Tx)?

Seems like Tx applies a function (is this also a functor?) I'm confused here-- that takes a finite alpha of (a,b) and maps it to list in sets {(a,b,a), (a,b,b,b),..} etc. OK cool. But now how does applying T(Tx) make that a list of list? I'm confused here. I thought Tx was a set that already had list?

What is the real difference between {(a,b,c),(a,b,b,b,b),...} and {a,b,c, a,a,b,b,b,..}?
I don't fully understand the difference between ; and , (the commas) or T(Tx)

Finally,

Assuming all of the above... how do you show this actually satisfies the identity and associativity

>Are objects categories?

You can think of a single object as the category 1.

>What does a "Functor" do in my example?

A functor is a morphism of categories; the structure being preserved here is composition.

T in your example is a functor, taking each arrow in FinSet to its component-wise application, where defined. For example, to use your set [math]x=\{a,b\}[/math] and another set, if [math] f: x \rightarrow y[/math] is a function where [math]a \mapsto a'[/math], then [math]Tf: Tx \rightarrow Ty[/math] will take [math](a,b,c) \mapsto (a',b',c)[/math], [math](a,b,b,b,b) \mapsto (a',b',b',b',b')[/math], etc. You can verify that this does indeed preserve arrow composition.

>What is going on with T(Tx)?

I'm not familiar with the semicolon notation you're using, but it's correct that [math]T^2x[/math] is a set of lists of lists.

Thanks for the explanation.

In:

(a,b,b,b,b) \mapsto (a',b',b',b',b')

what does the ' notation represent here?

I'm not sure what the ; notation represents.

Is it saying the function T take the elements from set {a,b} turn it into set containing list {[a], [b], [a,b,b,b],..} etc and then transforming those list to contain list?

I am having trouble picturing this

What does it look like?

{[[a]], [[b]], [[a,b,b,b]],..} then T(Tx) takes this list of list and collapses it down to a set with a list of elements {a, b, a,b,b,b,..} ?

Would that make sense given this context?

Note my notation... [a] denotes a list and [[a]] denotes a list containing [a].

It's not like nobody here knows Haskell, stick to a notation and proofread your question.
If x={a,b}, why is T producing an c?

I don't know haskell and the reason for the inconsistency in notation is that I do not understand what the original notation represents. My guess is what I wrote here:.

That user defines a' to be the image of the function f when applied to a. He could also write f(a) for it.

The difference between a functor and a function is that a functor not only maps objects to other objects, but also functions between objects to other functions between object. And fumctors fulfill and algebraic property w.r.t. function complositon. In some formulations, functors are merely special functions.

You have not discussed this action of T on functors.

The [] notation is what's used in some programming languages for lists.

T takes
x={a,b} to
T(x)={[], [a], [b], [a,b], [b,a], [a,a,a], [a,a,b], [a,b,a], ...}
i.e. all possible lists of element found in x,
and applied a second time, it takes T(x) to
T(T(x)) = {[[]], [[a]], [[b]], [[],[]], [[], [a]], [[], [b]], ...
...and so on: all possible lists containg lists found in T(x).

But again, a functor also acts on arrows (functions in this case) and you've not talked about this yet.

What's your background?

Okay, this clears up what T is doing.

>You have not discussed this action of T on functors.
>But again, a functor also acts on arrows (functions in this case) and you've not talked about this yet.

So the category is a finite set. T is defined to be the arrows between categories (or in this case from the category to itself) and T is also defined to be a functor (which acts as a function that maps finite sets to a a list of sets).

Is that correct? Did what I wrote above address this?

Additionally you mention:
>i.e. all possible lists of element found in x,
In general how do I generate all possible list of elements found in x also what is the cardinality of that set? is it 2^n?

I'm confused how to generate such an example list from {a,b}

are strings like{[a,a,a,a,a,a,a]} allowed? What are the restrictions or rules to generating all possible elements of a finite set (in general)?

>What's your background?

I'm starting to learn higher level math and programming beyond the computational aspects of calculus I-III, Linear Algebra etc. I'm in the 'transitioning phase' between computational mathematics and math with proofs.

Also learning programming like python + java.

>finite sets
this is not precise. Do you mean finite set in classical set theory or kuratowski finite? do you have a decidable equality on your finite set?

Let's go with classical set theory.

You defined your category to that that of finote sets. The number of objects in this category is not finite.

The fuctors used in programing languages are mostly functors from one caregory to itself (because for more restricted codomains you'd need to implement subclasses).
Such functors are called endo-functors btw.

You will probably not need the perspective that functors can be viewed as arrows.
Functors carry over a whole lot of data, and again, hard to implement categories within a language (as opposed to viewing your expressing as thing inside a category).
Come back to categories as objects at a later time

>which acts as a function that maps finite sets to a a list of sets
No, as an endo functor, it maps back to finite sets.
Sets that conain a finite number of (possibly infinite) lists.

>what are the rules...
I'm not going to write down the rules of how those sets can look down more formally than I did. It's just finite sets of lists that contain, as elements, a and be. In a set, there is no order (if you do constructive math or envoke le axiom of choice, you can order it, though) or multiple elements, while a list alwas has an order and repetition is allowed.

>In general how do I generate all possible list of elements found in x
Does it matter?
>what is the cardinality of that set? is it 2^n?
No, if we're talking lists as finite sequences of elements, it's infinite, [math]\aleph_0[/math] to be precise.

>finite sets
this is not precise. Do you mean finite set in classical set theory or kuratowski finite? do you have a decidable equality on your finite set?

learn category theory with eugenia
byorgey.wordpress.com/catsters-guide-2/

I remember them talking about a monad which turns things into list.

...

yeah so watch the course on monads

golem.ph.utexas.edu/category/2007/09/the_catsters_on_youtube.html

not learning cat thorugh these books

Ok you guys lost me. I don't think I understand anything from the example at all.

I'll watch this then

I am ordering all three of these books. Thanks

Which one should I start with?

start by the beginning if you are lost

byorgey.wordpress.com/catsters-guide-2/

what you need is the notion of a category, of a functor, of a natural transformation, then you see the monad.

Oh, I just realize a problem with your initial defintion just now: Indeed, you T, if it maps a set to all lists with those elements, doesn't have FinSet as codomain.
You must fix this codomain or your definition of what T does

>No, if we're talking lists as finite sequences of elements, it's infinite, ℵ0 to be precise.

Why is a list of finite sequences infinite? What is the order you can list them? What are the rules for generating this infinite sequence?

Are the below valid iterations from the sequence {a,b}

{null, a, aa, aaa, aaaa, aaaaa, aaaaa,...,aaaaaaaaaaaaaaaaaa, ...}

{null, b, bb, bbb, bbbb, bbbbb, ...., bbbbbbbbbbbbbbbb, ...}

{null, a, ab, abab, ababab, ....., abababababababababababa, .... }

or no?

Ok, Thanks I'll start from the beginning.

This doesn't make sense to me. What are you saying in dumb terms?

T: Fine Set -> *, where * is what exactly?

I am totally lost. I don't even understand my own example anymore. You guys confused me way more.

Can someone take my example and define/example this in the dumbest way possible?

I thought I had a grasp on it but I guess not.

I'm ordering all three books here and reading through them the entire summer.

Watching the videos. What does ∃! mean?

There Exist... ! means what there?

unique

Don't order all three of them. Start with the leftmost and Awodey's Category Theory (not pictured). If you read along as you watch the Catsters lectures, you should be an alright category theory enthusiast by the end of summer.

>Why is a list of finite sequences infinite?
Because whenever you have a sequence of a's and b's, you can always get a new one by appending yet another a or b.

Thanks.
Ok cool. So start with:
Conceptional Mathematics & Awodey's Category Theory & watch the Caster's videos?

Ok this is what I will do this summer.

I'm saying the two definitions, of where T maps to and what T is supposed to do, are not consistent.

PS don't watch the catsters videos. They are too hard.
PPS books are online, in one way or another

He's mapping sets to *sets* of lists. The codomain is totally right.

I'm saying the two definitions, of where T maps to and what T is supposed to do, are not consistent.

I can't tell you what * is, just like I can't tall you what the codomain of n mapsto n^7 is. The range of the functor is a category of sets, however.

PS don't watch the catsters videos. They are too hard.
PPS books are online, in one way or another

In the OP, the OP declared
>T: FinSet -> FinSet is the functor that takes a set to the set of list on it.
Emphasis on ->FinSet

>PPS books are online
What is PPS?

Can you take what I did and re-define my example? I thought I understood it but now confused.

I need to see what the category, fucntor, function, and a worked out example is....

People pointed out so many things I no longer follow. I need a holistic example and definition of my example..

post post scriptum

Your example is ill-defined. The problem is
>T: FinSet -> FinSet is the functor that takes a set to the set of lists on it.
The set of lists over a nonempty set of elements is not finite.

So what is the mapping?

Taking a finite set and generating a finite sequence of list? This is infinite?

Then how do you fix it?

The easiest fix will be to consider T a fuztor from Set to Set, dropping any 'finite' requirement.

Just google list- functor mate.
You seem to really want to know this, so stop getting fed.
"T maps sets x to sets T(x) that contain lists, where the elements are elemets of x"
is a clear enough specification of how T acts on objects.
How it acts on functions was also explained to you in that 3rd post.

I haven't really thought about this, but I guess you can define T : FinSet → FinSet as taking
- a set A to the set of sequences of length 5 (or any other fixed number) over A, and
- an arrow f : A → B to something like "replace x's with f(x)'s in these sequences".
You might want to check the laws hold, draw a picture or whatever.

OP here,

Instead of trying to learn Cat. theory from random examples I'll learn it from the ground up.

I'm getting the recommend books, will watch the recommended videos and continue onwards.

Thanks for the help.

>actually buying a book on category theory
>not just picking it up through Algebra, Algebraic Topology, Algebraic Geometry,etc.

>I'm starting to learn higher level math and programming beyond the computational aspects of calculus I-III, Linear Algebra etc. I'm in the 'transitioning phase' between computational mathematics and math with proofs.
There's really no point in learning category this soon then.

Why spemd money when you'll drop it 3 weeks in?
You can download the book by Adwodey or how the fat man is called and learn from this.

I won't drop it.

Already teaching myself Abstract Algebra in conjuction to it.


Based on this discussion it seems like:

The category is the object of a finite set {a,b}
T is the function that maps finite sets to a set of list.

The arrows are T, which is the functor & function that maps from the category to itself.

T(Tx) is collapsing the set of list of list into a set of list.

Is this correct?

Why do you want to learn category theory? Category theory for the sake of category theory is dumb and it's hard to understand unless you have examples (e.g. those from algebraic topology) that you can study the concepts with.

Can you address

No, because I don't want to take the effort to understand the non-standard language that you are using.

Just know that there is no need for you to study category theory until you understand why structure preserving maps (at the very least continuous maps in topology, homomorphisms in algebra, linear transformations in linear algebra/functional analysis) are important to begin with. In order to understand generalizations / abstractions you have to have enough specific examples to work with so that you get an intuitive sense of what the abstraction is actually about.

Yeah you're unhelpful which is consistent with Veeky Forums.

Not that guy but I can't make heads or tails of your example, so I'm going to provide a different but slightly related one.

Any category with one object is equivalent to a MONOID -- which is typically depicted as a set with a binary operator on it satisfying associativity and identity axioms, but in a category we have a collection of arrows Mor(C) with a binary composition operator instead. Since the morphisms always have the same start and end point (the object itself), composition becomes a well-defined function Mor(C)*Mor(C) -> Mor(C). (I'm assuming you know what a monoid is -- if not, look it up before reading further.)
The simplest example of a monoid is the category of natural numbers.
It contains a single object, which we denote "N"
And infinitely many morphisms "N" -> "N", one for each natural number 0,1,2,3,...
Composition takes any two arrows m,n and returns the arrow m+n (their sum, e.g. 5 is the composition of 2 and 3).
This is associative since (m+n)+p = m+(n+p).
The identity morphism is 0, since m+0 = m = 0+m.

Another example of a monoid, closer to yours, is the set of all words formed from the characters {a,b}.
As a category, it has a single object which we denote {a,b}*
And infinitely many morphisms {a,b}* -> {a,b}*, one for each word a,b,aa,ab,ba,aaa,...
Composition takes any two words W,X and returns their concatenation WX (e.g. "ababbbb" is the composition of "abab" and "bbb")
This is associative, and as identity morphism we take the empty string.

(cont 1/2)

(2/2)

A functor is a map from categories to categories. A simple example of a functor is "len" which maps from the category of words on {a,b} to the category of natural numbers.
To define a functor we must specify its value on all objects and morphisms.
For monoids it is trivial to specify its value on objects, since there's only one: len({a,b}*) = len(N).
For the value of len on morphisms, simply map each word to the number of characters in it, i.e. len(b) = 1, len(ab) = 2 etc.
And of course, len(empty string) = 0.
This is one of the conditions a functor must satisfy -- it must respect the identity law. The other condition is that it must respect associativity: len(WX) = len(W)+len(X) for any two words.
Indeed this condition holds since we can count the number of characters in WX by counting the characters in W and X separately, and then summing. So len is a valid example of a functor.

Thanks for these examples and explanations! :D
This is what I was looking for throughout the entire thread.

No problem.
However, note that this is hardly the full story, since my examples are not really categories but monoids. (Assuming objects are equal is an evil thing to do in category theory, though I can't really explain why in a 2000-character post.)

As the other posts in this thread have been repeatedly pointing out, you don't really learn category unless you have some concrete examples of 'things' that you want to convert into categories and analyze in that way.
(Well, you can, but you'd be fighting a seriously uphill battle, since category theory has been considered too abstract even for pure mathematicians to wrap their heads around, hence its moniker 'general abstract nonsense'...)

Great response. I'm learning going through an Abstract Algebra text this summer as well to motivate the examples.

the fuck??
Veeky Forums has put a lot of effort in the answers in this thread

but this () makes it apparent OP has some serious problems understanding concepts, even if explained to him back and forth discussions

He's right you know. Abstraction for abstraction's sake isn't a great thing.

No, it highlights the people here haven't explained what OP had questions on. This is a pretty shit place to ask for help.

>>He's right you know. Abstraction for abstraction's sake isn't a great thing.
until it makes things far simpler and you understand that people who loath abstractions are just resentful because they cannot follow what goes on.