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/sci/ - Science & Math

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Category theory Anonymous Sat Nov 2 06:45:29 2019 No.11111875 [Reply] [Original]

Hello I'm trying to learn category theory and I have a simple question.
If in a category there exist two morphisms:
a : A -> B that is surjective and
c : C -> B
Is it true that there is then a morphism
σ : C -> A
so that a after σ = c?

>>	Anonymous Sat Nov 2 06:55:58 2019 No.11111893 >>11111875 It's trival.

>>	Anonymous Sat Nov 2 06:56:58 2019 No.11111895 >>11111875 Yes in SET, no in general. In the category of topological spaces with continuous maps, let A be the real line, let B and C be the circle (R/Z), let a be projection and c be identity.

>>	Anonymous Sat Nov 2 06:58:57 2019 No.11111899 >>11111895 Ah thanks a lot

>>	Anonymous Sat Nov 2 07:02:03 2019 No.11111909 >>11111875 Just want to emphasize that an object is not a container of element. I say this because I'm not sure you're sure how you define "surjective".

>>	Anonymous Sat Nov 2 07:11:09 2019 No.11111924 >>11111909 Oh yeah you're right. I always think of surjective in set theoretic terms but I should be thinking of epimorphisms.

>>	Anonymous Sat Nov 2 07:15:16 2019 No.11111930 File: 211 KB, 976x906, undergradcattheorist.png [View same] [iqdb] [saucenao] [google]

>>	Anonymous Sat Nov 2 07:16:19 2019 No.11111931 File: 103 KB, 938x584, undergradcattheorists.png [View same] [iqdb] [saucenao] [google]

>>	Anonymous Sat Nov 2 07:19:51 2019 No.11111938 >>11111930 Are you saying I should first be focusing on other branches of math and then learn category theory to connect them instead of learning category theory in the beginning and damaging my views of math permanently?

>>	Anonymous Sat Nov 2 07:22:40 2019 No.11111941 >>11111938 Exactly.

Anonymous Sat Nov 2 07:26:59 2019 No.11111944

>>11111875
this is true in Set, but it's a very special property of this category. when you put B = C and c = id, your question is whether every epimorphism a: A -> B has a section s: B -> A. this is of course obviously true for sets, you just pick an arbitrary preimage in A for every element in B (and this is possible precisely when the map is surjective). but it stops being true once you require this choice to be continuous or smooth, preserve some algebraic structure etc.

>>	Anonymous Sat Nov 2 07:29:03 2019 No.11111946 >>11111941 I was thinking of learning category theory early to speed up learning about other branches. Hmh never heard that "the undergrad category theorist" is a more common phenomenon. Thanks for informing me anon

>>	Anonymous Sat Nov 2 07:36:55 2019 No.11111956 >>11111944 Thank you that definitely clarified it for me

Anonymous Sat Nov 2 07:39:52 2019 No.11111962

>>11111946
>I was thinking of learning category theory early to speed up learning about other branches.
not him, but don't do this. you will become the person who raises his hand after each definition or theorem to ask:
>can we express this somehow in categorical terms ?
and I tell you right now that the answer will be always the same:
>we probably can, but the effort to do so completely misses the point of the theory we're building here

>>	Anonymous Sat Nov 2 12:58:13 2019 No.11112435 File: 136 KB, 907x1360, maclane.jpg [View same] [iqdb] [saucenao] [google] >>11111875 All the answers about mathematics can be found in pic related.

>>	Anonymous Sat Nov 2 13:08:08 2019 No.11112446 >>11112435 can you post first few pages ?

>>	Anonymous Sat Nov 2 13:10:01 2019 No.11112451 >>11112446 no

Anonymous Sat Nov 2 13:12:57 2019 No.11112457

Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G.[3][note 3] In other words, for every x in X, there is exactly one element y such that the ordered pair (x, y) belongs to the set of pairs defining the function f. The set G is called the graph of the function. Formally speaking, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph. Functions are also called maps or mappings, though some authors make some distinction between "maps" and "functions" (see section #Map).

In the definition of function, X and Y are respectively called the domain and the codomain of the function f. If (x, y) belongs to the set defining f, then y is the image of x under f, or the value of f applied to the argument x. Especially in the context of numbers, one says also that y is the value of f for the value x of its variable, or, still shorter, y is the value of f of x, denoted as y = f(x).

Two functions f and g are equal if their domain and codomain sets are the same and their output values agree on the whole domain. Formally, f = g if f(x) = g(x) for all x ∈ X, where f:X Y and g:X Y.[4][5][note 4]

>>	Anonymous Sat Nov 2 13:15:31 2019 No.11112461 >>11111946 >>I was thinking of learning category theory early to speed up learning about other branches. yes this is good, also proofs in cat th are easier than bourbaki maths.

>>	Anonymous Sat Nov 2 13:17:13 2019 No.11112466 >>11112461 >hurrrgghhhhdurghryggggghhhhlearncategorytheorybeforeotherfieldsofmathhugggghhhhhhbrrrrggghhhg

>>	LMC !!6bFXHDKL67m Sat Nov 2 13:20:34 2019 No.11112472 File: 186 KB, 580x852, mac lane categories 44.png [View same] [iqdb] [saucenao] [google] >>11112446 >can you post first few pages ? (1/4)

>>	Anonymous Sat Nov 2 13:21:44 2019 No.11112474 File: 191 KB, 1761x690, 1544946412413.jpg [View same] [iqdb] [saucenao] [google] >>11111875 all the books to learn about category theory

>>	LMC !!6bFXHDKL67m Sat Nov 2 13:22:05 2019 No.11112476 File: 50 KB, 605x911, mac lane categories 45.png [View same] [iqdb] [saucenao] [google] >>11112446 >>11112472 (2/4)

>>	Anonymous Sat Nov 2 13:22:50 2019 No.11112478 File: 209 KB, 1000x1200, 1547924727245.png [View same] [iqdb] [saucenao] [google]

>>	LMC !!6bFXHDKL67m Sat Nov 2 13:23:06 2019 No.11112479 File: 101 KB, 599x870, mac lane categories 46.png [View same] [iqdb] [saucenao] [google] >>11112446 >>11112472 >>11112476 (3/4)

>>	Anonymous Sat Nov 2 13:23:08 2019 No.11112481 >>11112476 stop, this is fucking pathetic.

>>	LMC !!6bFXHDKL67m Sat Nov 2 13:25:06 2019 No.11112483 File: 189 KB, 591x858, mac lane categories 47.png [View same] [iqdb] [saucenao] [google] >>11112446 >>11112472 >>11112476 >>11112478 (4/4) >>11112481 lol viva la matematica

>>	Anonymous Sat Nov 2 14:44:35 2019 No.11112627 >>11111875 FUCK OFF TRANNY. this is the epitome of the undergrad cat theorist post. I can gaurantee with 100% certainty that you are at least one of the following. 1. Attention seeking pseudo intellectual tranny 2. Undergrad 3. CS major.

>>	Anonymous Sat Nov 2 16:16:11 2019 No.11112813 File: 20 KB, 400x304, 4195627.jpg [View same] [iqdb] [saucenao] [google] >>11112627 call me whatever you want, it won't cure your intense and overbearing autism

>>	Anonymous Sat Nov 2 16:58:06 2019 No.11112874 >>11112813 well learning category theory wont make you seem smart to anybody except for the brainlet friends you hang out with. birds of a feather flock together as they always say. people like you are a perfect example of the degradation of fan base

>>	Anonymous Sat Nov 2 17:15:47 2019 No.11112904 File: 95 KB, 486x330, 1572457702239.jpg [View same] [iqdb] [saucenao] [google] >>11112874 >I'm angry because I've lost control of my life I'll just chalk it up to bad schooling. /sci/ has truly flayed your mind.

>>	Anonymous Sat Nov 2 17:32:26 2019 No.11112939 >>11111875 No, just consider the category where the only morphisms are a and c.

>>	Anonymous Sat Nov 2 17:59:51 2019 No.11112983 >>11112939 lmao

Anonymous Sat Nov 2 18:00:25 2019 No.11112985
File: 90 KB, 866x1300, 22812010-young-businessman-in-dark-grey-coat-and-steel-blue-shirt-standing-isolated-on-white-background.jpg [View same] [iqdb] [saucenao] [google]

>>11112481
>stop, this is fucking pathetic.
Lol I fucking read all four of those pages but I still can't figure out what triggered this nigga

>>	Anonymous Sat Nov 2 18:09:19 2019 No.11112997 >>11112472 >>11112476 >>11112479 >>11112483 >>11112985 can I get a proper mathfag to explain what's going on in these pages? I tried really hard.

Anonymous Sat Nov 2 18:35:58 2019 No.11113036

>>11112997
Are you really that fucking stupid that you can't understand something as simple as few arrows. It's literally showing you what it is. Someone created a math field so brainlets can have drawn images to capture it better. Everything is a category and can be represented that way.

>>	Anonymous Sat Nov 2 18:36:13 2019 No.11113038 File: 65 KB, 463x380, 1552648922450.jpg [View same] [iqdb] [saucenao] [google] >>11112874 >people like you are a perfect example of the degradation of fan base

>>	Anonymous Sat Nov 2 19:03:51 2019 No.11113096 >>11113036 >I'm not sure what it is about, but it's supposedly trivial. Please be patient with me. I can't figure out how to "read" the arrows like you do anon. Could you please try to point out the exact part that upset you?

>>	Anonymous Sat Nov 2 19:12:44 2019 No.11113112 >>11112472 >>11112476 >>11112479 >>11112483 Isn't this from the book Mathematics Made Difficult? I could swear I saw the exact same diagrams in there, but it was supposed to be a joke.

>>	Anonymous Sat Nov 2 20:33:15 2019 No.11113289 File: 97 KB, 1300x866, 1572392800109.jpg [View same] [iqdb] [saucenao] [google] >>11113112 What is the joke? Why is >>11112481 so deeply offended by it? I've never heard of "category theory" before this thread.

>>	Anonymous Sat Nov 2 21:41:27 2019 No.11113396 File: 831 KB, 2404x1260, swfu1m3tcvy01.jpg [View same] [iqdb] [saucenao] [google] >>11113038

>>	Anonymous Sat Nov 2 22:20:18 2019 No.11113458 Analysts are seething in this thread because their field of study has been trivialized by the algebraists

Anonymous Sat Nov 2 22:28:30 2019 No.11113470

>that one ineffectual anon who reposts the same memes in every thread because the chads in his state school program goof off with category theory and get more pussy than he does
>>11111930
>>11111931
I will grant you that the Haskell/CS babies are annoying tho
>>11111946
>Hmh never heard that "the undergrad category theorist" is a more common phenomenon.
It's not-- that's just a /sci/ regular that is particularly butt blasted about it for some reason, he posts in every thread it comes up in

>>	Anonymous Sat Nov 2 22:55:48 2019 No.11113519 >>11111930 >burned out mathematical beauty receptors that unironically worries me

>>	Anonymous Sun Nov 3 03:45:13 2019 No.11113933 >>11113519 anon if you have any papers on your bucket list then I really think you'd better start writing them now

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