Mathematics - 2015-05-20 (page 1 of 5)

Ted Shifrin

00:00

I would start with a disclaimer, then ...

My students trip over this a lot.

Stan Shunpike

So suppose $X$ is a metric space with metric $d. Let $\phi : X \mapsto X$ and let $c<1$ st $d(\phi(x),\phi(y)) \leq cd(x,y)$ for all $x,y \in X$.

So $\phi$ is a contraction because....it keeps reducing the distance between $x$ and $y$ by that amount?

Hmm...Rudin then has a theorem this implies it must contract to a point.

That's cool!

@TedShifrin is this relevant for the implicit function theorem or inverse function theorem?

I'm trying to figure out why you and Rudin both talk about it. And now that I sorta get the gist, I think I can now ask that question

Ah....*things start to click*

@TedShifrin what do you think of Serge Lang's books?

copper.hat

00:18

a complete metric space, i presume...

you can prove the inverse/implicit function theorems using a contraction.

Stan Shunpike

@copper.hat yes complete. Really? Hmm. I think this is what Rudin does for the inverse function theorem. If i recall correctly, you told me this the other day.

copper.hat

if i recall correctly, the map $x \mapsto x-Df(x)^{-1} f(x)$ is a local contraction.

sorry, the $x$ in the derivative should be fixed at $x_0$.

sorry gotta go in a hurry.

Karim Mansour

back

back @TedShifrin

which rudin is that

baby rudin or grad rudin ?

Mike Miller

elements of real analysis

Karim Mansour

that is a must read

I glanced over it last semester

very nice

Stan Shunpike

00:27

@KarimMansour i am starting to get more out of it as i learn more.

Karim Mansour

yeah

Stan Shunpike

I am reading Principles of Mathematical Analysis. I dunno if that's the one you meant

Mike Miller

oh, yeah, that

Ted Shifrin

The most natural way of proving the inverse function theorem is to get the inverse as the solution of a fixed point problem, @Stan. There's also a proof of Contraction Mapping in the first section of what I sent you.

Mike Miller

i got the structure right

Karim Mansour

00:28

yeah

Mike Miller

___ of ____ analysis

Karim Mansour

I saw the proof before that I liked

1 moment

for inverse function theorem

Stan Shunpike

@TedShifrin i probably wouldn't have read about it except it was in what you sent :P so then i inquired a bit. I think I am starting to get the idea.

@MikeMiller lol

Karim Mansour

@TedShifrin do you like analysis?

Stan Shunpike

@TedShifrin I guess I don't really understand what the inverse function theorem is saying....or rather why it's important. When i learned about vector spaces, we assumed the existence of inverses. This comes up in abstract algebra too. So why do functions have this theorem? Can't functions be vectors? What does the inverse function theorem give me that i couldn't get just by working with a function space?

Ted Shifrin

00:33

You're mixing apples and oranges, @Stan. Go back to single-variable calculus and ask when a (differentiable) function has a (local) inverse. Try to generalize.

copper.hat

@StanShunpike: i would echo what @TedShifrin wrote above. you can also find a 'parameterised' verssion of the contraction mapping theorem that is useful in many contexts, such as proving continuity of solutions of an ode with respect to some parameter.

Ted Shifrin

Yes, @Karim.

copper.hat

@StanShunpike: think of linear problems. if $A$ is invertible, then $Ax =b$ has a local inverse (hence global, of course, since linear).

same for the implicit function theorem. If Ax+By=0$, and $A$ is invertible, then you can write $x = - A^{-1} B y$. see the resemblance to the ift formula for the derivative.

evinda

Hello @copper.hat
Are you familiar with algorithms?

Stan Shunpike

@TedShifrin @copper.hat okay, if the idea is to make claims about the relationship between the derivatives of two inverse functions, then i get it.

copper.hat

00:39

@evinda: that's a bit broad, and i am without pencil & paper :-), what sort of algorithms?

Ted Shifrin

BTW, @Stan, there are plenty of non-invertible linear maps, so I don't know what your linear algebra statement meant. Indeed, the point of the inverse and implicit function theorems is that for $C^1$ functions, what is true for the derivative at a point is locally true for the non-linear function locally.

No, @Stan, the point is to claim existence locally of an inverse function.

copper.hat

@StanShunpike: it is an extremely useful result.

Ted Shifrin

Note the example at the beginning of section 2 of what I sent you. For a differentiable but non-$C^1$ function, even $f'(0)>0$ does not imply a local inverse function for maps $\Bbb R\to\Bbb R$.

evinda

@copper.hat Undirected graph G=(V,E), weights of edges and a subset T of the vertices.

Output: The minimum spanning tree at which the vertices of the set T are leaves( The tree can also have leaves from the set V-S)

I want to write an algorithm that solves the above problem and runs in time O( E logV)
Could you give me a hint what I could do?

Stan Shunpike

@TedShifrin yes i remember that. I will reread it again now that I understand contraction mapping better and see what new things I learn this time around

copper.hat

00:41

sorry @evinda, without my cormen rivest & stein nearby this is not my natural domain :-(.

evinda

@copper.hat Ok.. no problem... :) Which is your domain?

copper.hat

somewhere in the control/optimization (as in continuous) domain. or analog circuit design...

there is a connection :-)

Ted Shifrin

It helps to do some rigorous multivariable analysis before jumping into this stuff, @Stan :) BTW, did you settle your econ problem yet?

Stan Shunpike

No, I just finished a set due today. Hiatus. I plan to do it tonight :D hence i am asking questions

to try to understand B better

@TedShifrin What do you mean by rigorous?

I mean, I can do surface integrate, line integrals, etc

And total derivatives

Ted Shifrin

I mean the rigorous definition of derivative in the multivariable setting, plus some understanding of the chain rule, norms, etc.

Stan Shunpike

00:47

And that sort of thing

Ted Shifrin

Integration is irrelephant here.

copper.hat

@TedShifrin: i like that misssspeiling

Ted Shifrin

I did it on porpoise.

copper.hat

:-)

Stan Shunpike

Yes, having read Rudin, I feel I have some things to learn to up my calculus to this level. But that's why I'm wrestling with this stuff. Learning :D

Ted Shifrin

00:48

Some sense of (aqueous) humor helps.

Stan Shunpike

Lolol

copper.hat

@StanShunpike: have you dealt with lagrange multipliers?

Ted Shifrin

That's what landed us in this mess, @copper.hat :)

Stan Shunpike

Yep, tho @TedShifrin may disagree

Ted Shifrin

I gave him some interesting exercises on that.

copper.hat

00:49

ahh, i see. i was going to suggest it as a good example of the use of the ift.

Ted Shifrin

more really the implicit function thm, unless I'm missing something you're thinking of.

copper.hat

thats what i meant, sorry

i think of them as being much the same

Ted Shifrin

ok, @Stan, I'll leave you in pieces for now.

copper.hat

a commodian?

Semiclassical

02:24

the contrast between the thrust of this answer, and the name of its author, gives me a bit of a chuckle:

1

A: Can I combine the wave and heat equations?

No, one cannot obtain a solution of this PDE by adding a solution of heat equation to a solution of the wave equation. (With linear PDE, we can combine solutions of the same equation to make new ones; but your situation is different). Your PDE is known as damped wave equation and is solved here....

octatoan

02:51

@KarimMansour no, I will

octatoan

03:03

Extremely pressing question: how does one say $\sup$? "soup" like Spivak or 'sup?

user147690

Well Supremum(atleast in Australia) is said starting with 'Sup', so I would say it is 'sup', like from 'supper'.

user147690

'sup' 'prem' 'em'

Ted Shifrin

No, soup, not sup.

Like superior :)

user147690

@TedShifrin What? You say superior with 'soup' at the beginning, in America?

Ted Shifrin

Youp.

user147690

03:16

Not sure if joking, can't tell on internet, but the last thing makes me almost certain you are

pjs36

Come on, Alex, you're making it really hard for me to not make any comments about taking pronunciation advice from an Australian xD

user147690

:'(

Ted Shifrin

Nope, I never joke. Well, not about this.

user147690

I still can't tell lol

user147690

dictionary.reference.com/browse/supremum

user147690

03:17

@ted Do you say it like that?

pjs36

Aww, too far? It was all in good fun. But in the States, even my funky Russian professor said "soup" for sup(remum)

user147690

@pjs36 No not too far haha, the face wasn't serious. Weird, we say it exactly as the audio clip in that link says it

user147690

Also what would you call a list like:

Commutative rings $\supset$ integral domains $\supset$ Integrally closed domains $\supset$ unique factorization domains $\supset$ principle ideal domains $\supset$ Euclidean domains $\supset$ fields $\supset$ finite fields

Mike Miller

i wouldn't

user147690

A chain of properties or something?

Ted Shifrin

03:20

I've never heard superior pronounced suh-perior.

user147690

That's how we say it here in straya

Ted Shifrin

@AlexC: I'd be sure to spell principal correctly.

user147690

@TedShifrin oxfordlearnersdictionaries.com/definition/english/superior_1 third pronunciation

user147690

@TedShifrin Sure

user147690

Noone knows what to call such a list?

user147690

03:23

List of mathematical objects? Feels very strange, but wiki says: "Principal ideal domains are thus mathematical objects"

octatoan

I remember someone begging for something like that for topological spaces or something on /r/math.

user147690

Chain of class inclusions it says in one of the subpages. Is that good?

Mike Miller

why do you care

i mean, i guess you could call it that, but I don't really see why you would ever name it

user147690

Why not? If I talk to people about it, how would I refer to it? How do I write my notes up about it?

octatoan

"All these kids going on about inconsequential things early in the morning . . ."

Mike Miller

03:27

about what? the study of chains of class inclusions?

user147690

@MikeMiller No about the properties of each of the 'classes(?)' and how all of the latter properties are properties of the former (in the relevant chain of class inclusions)

Mike Miller

"Every Euclidean domain is a PID"

user147690

Indeed

octatoan

Quick question: can I use $(ab)^n = a^n b^n$ in the exercises of the first chapter of PMA?

user147690

Yes

octatoan

03:33

And $(a^b)^c = a^{bc}$?

user147690

If you look back, you will see what he has shown, I can't remember for sure what he has given you, but I imagine all of the normal properties you would work with are given

octatoan

I don't think this is proven back there. Regarding exponents, he only shows $[\exists!y] \;y^n=x$

and $(ab)^n=a^nb^n$

for integer $n$

user147690

Well $a^b=a\times a\times \cdots\times a$ so $(a\times a\times \cdots\times a)^n=a^n\times a^n\times \cdots\times a^n=a^{(n+n+\cdots+n)}=a^{bn}$

user147690

So it comes straight from that property you were given

octatoan

^ Yeah, I did it.

user147690

03:41

Cool

octatoan

I meant $(a^b)^c = a^{bc}$ for real $b,c$ at first though, but what you wrote was how I started the solution.

I think I'll skip the $\mathbb{C}$ and $\mathbb{R}^n$ problems. We did some in school.

Although the ones near the end are hard.

Mike Miller

similar argument for rational exponents; for real exponents you need to take limits, but it should be fine

octatoan

Yeah.

Kaj Hansen

03:57

@MikeMiller, I have a question

Mike Miller

hey

Kaj Hansen

This question here: http://math.stackexchange.com/questions/1290573/how-to-prove-a-linear-transformation-is-onto-if-it-is-one-to-one

Would one way of approaching this be to recognize that a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a special type of group homomorphism preserving the additivity of $\mathbb{R}^n$? Hence, we could just apply the isomorphism theorems?

Mike Miller

There are lots of injective, non-surjective group homomorphisms.

Samuel Yusim

why not just use the isomorphism theorems for vector spaces?

Kaj Hansen

Oh yes of course. I'm just so used to thinking about homomorphisms surjectively by throwing away everything but the image.

My linear algebra is really rusty too. I desperately need to review things.

Samuel Yusim

04:03

I actually learned the first isomorphism theorem as a statement about vector spaces before I learned it for groups

Kaj Hansen

@SamuelYusim, could you be more specific? I see generally where you're coming from, but haven't thought about the statement for vector spaces, at least I don't think I have.

The Rank-Nullity theorem, though, sort of resembles it.

Samuel Yusim

well, you can define a quotient of a vector space by a subspace in a really intuitive way if you're good at group theory

and yeah, you can prove it from rank nullity

Kaj Hansen

To be honest, I'm not sure I've ever "grokked" quotients in terms of group theory. The coset definition doesn't seem to be related to the quotients from, say, topology or ring theory where you're essentially "gluing" points together.

Samuel Yusim

so basically let $V$ be a vector space. then if $T: V \to T(V)$ is a linear map then $V/ker(T) \cong T(V)$ exactly as you'd expect

I don't know topology so all my intuition goes through cosets

but really what's going on is you're setting a bunch of stuff to be "equal" and dealing with the consequences just as you would with rings

Kaj Hansen

Ok very nice, I like that statement. How should I be thinking about $V/\ker(T)$?

Samuel Yusim

04:10

if $V$ is an $F$-vector space, then just think of it as $F^{\dim V - \dim \ker(T)}$

pjs36

But Kaj, with quotients of a group, aren't you just "gluing" everything in a coset together, so you forget, say, $gN$ exists other than as a solid blob in $G/N$?

Kaj Hansen

That's a fair point @pjs36. Oh this is definitely making more sense!

That makes sense as a way of thinking about it since the cosets partition the group.

pjs36

Wohoo! I'm finding things that I was only computationally proficient with are making more sense now, intuitively.

Unfortunately, I'm not gaining any new ground, but I'm appreciating the ground I've been on much more :)

Kaj Hansen

Are we talking about the analysis realm @pjs36 ?

pjs36

To a small extent, yes. But more so some basic algebra. But I should emphasize some.

Karim Mansour

04:16

hi guys

Kaj Hansen

Hey

pjs36

I was pleasantly surprised I actually answered a very basic analysis question the other day; it was really to convince myself that I could.

Karim Mansour

lol

Kaj Hansen

I dislike analysis :P

Samuel Yusim

that's the only reason I've ever done analysis

Kaj Hansen

04:18

Well, I find it rather dry I guess I should say

pjs36

Yeah, I'm terrible at it. I've been trying to convince Ted to sell me his multivariable math book for cheap on the downlow, but to no avail.

Karim Mansour

we should go to ted and tell him we must have it for free

pjs36

I was always able to get by, but it was always on a really basic symbol-pushing level; I just had an idea about how to get all the symbols to line up, not so much why they were doing that.

Samuel Yusim

basically my problem is that it's far too technical for what it is and that all the proofs use a different magic trick

Karim Mansour

intuition to doesn't come easily in analysis

Samuel Yusim

04:20

someone once said that if you eat corn by going along the cob horizontally then you're an analyst and if you eat corn by rotating the cob vertically then you're an algebraist

Karim Mansour

lol

pjs36

At my school analysts ate in spirals, algebraists in horizontal rows... it must all be a scam then. Or location dependent :)

But that really was a conversation though, so that's pretty funny it was elsewhere

Samuel Yusim

I guess it's just a weird part of the folklore

pjs36

Now that makes more sense

Kaj Hansen

I have Ted's multivariable book :)

octatoan

04:25

And logicians eat corn in loose spirals. Link

pjs36

Cover and all, @Kaj?

Kaj Hansen

LOL, yes @pjs36

It's his algebra text that is falling apart on me.

pjs36

haha, that's funny yet terrible. I imagine the construction quality was brought up in class several times, or just here on MSE chat?

Kaj Hansen

Oh it was indeed

I've used the algebra book very, very heavily. So maybe that makes sense.

Karim Mansour

I should also read ted algebra book 2

Kaj Hansen

04:31

@Karim, I wouldn't recommend it if you've already taken algebra. It's truly for a first go around.

Karim Mansour

oh

Kaj Hansen

But if you haven't taken algebra yet, it's really not bad at all.

Karim Mansour

yeah I took 2 algebra classes did pretty good but want to perfect my algebra skills this summer by reading DF

@KajHansen

Kaj Hansen

I found DF to be reasonably approachable after 2 semesters of algebra.

Karim Mansour

yeah I read it like a comic book now

Kaj Hansen

04:33

I haven't read even 5% of it yet, but every now and then I reference it when I need a little more depth on something, and it's always there for me.

Karim Mansour

I want to cover until algebraic geometry this summer @KajHansen

pjs36

I've been attempting Aluffi, but Isaacs has some extremely terse algebra books, if you're into that thing - group theory, algebra, and character theory, at the least

Karim Mansour

so far I finished the first 2 chapters

Kaj Hansen

I really, really like Artin's 3rd edition. Especially the Galois theory chapter was very lucid.

Karim Mansour

oh I see I heard alot of people talk about artin I should look at it too

ted told me it takes a geometrical approach

Kaj Hansen

04:35

haha, Ted and his geometry

Karim Mansour

but @KajHansen I would def recommend DF its very very nice also its problems are very good since he makes you do some problems that come later in text.

Kaj Hansen

mhmm, I have a copy of my own

I'm taking algebra for a 3rd time this fall. I'm very excited; it's been my favorite subfield of math thus far.

pjs36

Is it a "topics" kind of class, or what's the deal, @Kaj?

Samuel Yusim

I don't get to take any algebra this fall

Kaj Hansen

@pjs36, it's a graduate course. I'm excited, but also fairly anxious since I've only ever taken undergrad courses until now.

Samuel Yusim

04:38

what's it on?

Kaj Hansen

@SamuelYusim, the course description is:
"A course in groups, fields and rings, designed to prepare the student for the algebra prelims. Some topics covered include the Sylow theorems, solvable and simple groups, Galois theory, finite fields, Noetherian rings and modules."

octatoan

@KajHansen 3rd edition?

Samuel Yusim

interesting

pjs36

Interesting indeed. We had Alg I and II at my school, then it was all "topics" class after that

Samuel Yusim

that'd be a really useful course

pjs36

04:40

But I'm sure you'll do great, @Kaj, I wouldn't be too anxious

Samuel Yusim

yeah, those things are all reasonable if you make sure not to lose track of what's going on

Kaj Hansen

Well I'll be @octatoan. I could've *sworn* the 3rd was the latest. I was referring to this: http://www.amazon.com/Algebra-Edition-Featured-Titles-Abstract/dp/0132413779

Which I guess is in its second.

octatoan

So there's no 3rd edition, right?

Samuel Yusim

also I don't think I've looked at a comprehensive exam in algebra that didn't have a sylow theorem question

Kaj Hansen

I guess not @octatoan

On the bright side I recognize all those words and can say something about them. It'll just be in a lot more depth than what I already know right now.

Samuel Yusim

04:43

yeah, that's why I think it'd be a great course

octatoan

@KajHansen That's a really great feeling.

Karim Mansour

yeah I am taking that course too in the fall @KajHansen

Kaj Hansen

Very exciting @KarimMansour. Hopefully we'll see each other in here from time to time and compare experiences

octatoan

"I resisted including group representations, Chapter 10, for a number of years, on the
grounds that it is too hard. But students often requested it, and I kept asking myself: If the
chemists can teach it, why can't we?"

Gigili

hey

Karim Mansour

04:48

yeah that would be gr8

@KajHansen

Gigili

where is @DavidWheeler

Kaj Hansen

lol, what's that quote from @octatoan ?

octatoan

Artin 2e, preface

Gigili

@robjohn how can I increase the amount of bounty I put on my question?

Karim Mansour

actually group presentation is presented in section 1.2 of DF

hahaha

section 1.2

xD

Gigili

04:49

@anon could you help?

Kaj Hansen

I love the quotes in Artin

pjs36

Karim: Artin was talking about representations: Presumably these are presentations, but the second time around (I kid)

Karim Mansour

haha

isn't presentation same as representations @pjs36 ?

pjs36

Presentations are the $\langle x, y : x^2 = y^n = 1\rangle$ ways of defining a group, representations are homomorphisms into groups of matrices

Karim Mansour

I see

pjs36

04:56

Like writing elements of cyclic groups $C_n$ as $\begin{bmatrix}\cos 2\pi/n& \sin 2\pi/n \\ -\sin 2\pi/n & \cos 2\pi/n\end{bmatrix}$, that's a representation of $C_n$

Karim Mansour

yh

pjs36

Man, I shouldn't attempt to tex at night, I'm just not up to it...

Karim Mansour

its alright @pjs36 it happens xD

anon

05:11

@Gigili what's up?

presentations are representations in the colloquial, nonmathematical sense of the term "representation," but not in the technical, mathematical sense of it

Gigili

@anon I need your help

anon

okay

Gigili

Asked a question, awarded a +150 bounty... still nothing

LeGrandDODOM

@Gigili wtf ? math.stackexchange.com/users/181854/gigili

Gigili

Haha!

No idea really

Karim Mansour

05:23

why stalking people @LeGrandDODOM xD

Gigili

Doesn't matter right now

The only thing that matters is my thesis defense :|

@anon Have a link?

anon

yeah I'm looking at it

LeGrandDODOM

but this works math.stackexchange.com/users/181853/gigili which is really weird

anon

@Gigili I am not familiar with things like dictionary learning (or much CS at all really). My advice would be to add in background on what the concepts in the paper actually are (like what dictionaries and dictionary learning are, etc.) one stage at at time for the next few days (i.e. in multiple edits spread out over time), even if only to get more exposure. also reach out to anybody who can help you IRL.

Sawarnik

@LeGrandDODOM :O !

LeGrandDODOM

05:29

@Sawarnik :O

Gigili

@anon Sounds like a good idea.

Sawarnik

Can anyone think of a simple solution to this? math.stackexchange.com/questions/1281770/…

Gigili

OK thanks

anon

sure, sorry I can't help more

Gigili

@LeGrandDODOM Also, "wtf" is a bit rude.

Sawarnik

05:30

@Gigili Are you doing your Phd?

Gigili

No, master's thesis

anon

@Gigili not really, it just expresses surprise, and it's not even directed at you or anything you've done but the situation with your accounts

Sawarnik

@Gigili :D .. which univ?

Idris

@paramanand r you here?

Soham Chowdhury

^ Are you by any chance the Idris of the language Idris?

This?

Vrouvrou

05:42

@robjohn are you there ?

@robjohn in the sequence $u_n(x) =|x|^{\alpha}$ when $|x|\geq \frac1n$ and $n^{-\alpha},$ for $|x|<\frac1n$ so when $n$ tends to $\infty$ we have $|x|^{\alpha}$ for $|x|\geq 0$ and $\infty, $for $|x|<0$ we know that |x| is positive so the secand case is refused and then the limite of $ u_n$ is $|x|^{\alpha}$

i don't understand how we use the monotonicity convergence in order to find the limit of u_n ? @robjohn

Paramanand Singh

hello Idris

Idris

hello Paramanand

I will try to copy then past my text for you

Paramanand Singh

Sorry man for thinking you as a student. You should give some more info on your profile

Idris

dont worry!

here is the text Dear Paramanand, I\ wish to show how one can compute the limit B of problem
1 only by using basic limits! Assume that the basic limits (which can be
computed by LHR) are known:%
\begin{equation*}
\lim_{x\rightarrow 0}\frac{\sin x-x+\frac{x^{3}}{6}}{x^{5}}=\frac{1}{120},%
\text{\ \ \ \ \ \ \ \ \ and\ \ \ \ \ \ \ \ \ \ }\lim_{x\rightarrow 0}\frac{%
\sin ^{-1}x-x-\frac{x^{3}}{6}}{x^{5}}=\frac{3}{40}.
\end{equation*}%
Note that one can write%
\begin{eqnarray*}
B &=&\lim_{x\rightarrow 0}\frac{2x-\sin x-\sin ^{-1}x}{x^{5}} \\

Paramanand Singh

looks very good

I think you should pose this as an answer to math.stackexchange.com/questions/437926/…

Kaj Hansen

06:02

Apparently "soberification" is a mathematical term. Hilarious. math.stackexchange.com/questions/1290662/…

Idris

Oh thank you, this post was unknown to me

i updated my profile

i think i have done the answer you were looking for when you did post your question!

Paramanand Singh

yeah

Soham Chowdhury

@KajHansen The more you know . . .

Idris

well i will format the answer to be posted in a moment

Paramanand Singh

ok i will upvote when it comes up

Soham Chowdhury

06:25

Is it correct to say "$f'$ is a section of $\pi_A$"? (Terminologically, I mean.)

Balarka Sen

@KajHansen Indeed.

Armstrong has some nice quotes too.

The coolest one I recall was Poincare's, which is in French, but the gist of it is "Geometry is making good logic out of bad figures", iirc.

Soham Chowdhury

@BalarkaSen can you answer my previous question?

Balarka Sen

Which question?

Soham Chowdhury

Is it correct to say "$f′$ is a section of $\pi_A$"? (Terminologically, I mean.)

This.

Balarka Sen

Depends on what $f'$ and $\pi_A$ are.

Soham Chowdhury

06:40

Random function and the natural projection respectively.

Balarka Sen

I dunno what you mean by "terminologically"

Tobias Kildetoft

@SohamChowdhury Also depends on what you want it to mean

Soham Chowdhury

As in, is that sentence correct?

@TobiasKildetoft Right-inverse:

Balarka Sen

@SohamChowdhury A section is usually supposed to be of a short exact sequence

Or fiber bundles.

Tobias Kildetoft

@SohamChowdhury then yes

Soham Chowdhury

06:41

@BalarkaSen I suppose it'll take me some time to understand what you mean.

:)

Tobias Kildetoft

(noting that it should also have whatever extra structure the others functions have)

Soham Chowdhury

@TobiasKildetoft $f' : A \rightarrow A \times B$ and $\pi_A: A\times B \rightarrow A$

Tobias Kildetoft

@SohamChowdhury that is not what I mean

Balarka Sen

Although section is essentially a right-inverse, we assign the name to the things I mentioned above. The origin of the name comes from seeing what it does to vector bundles, really.

Soham Chowdhury

Can you explain, @TobiasKildetoft?

Tobias Kildetoft

06:43

I mean that for example a section of a continuous function should itself be continuous

Soham Chowdhury

There is no other "structure" AFAIK.

Oh, that.

No, no other such properties.

@BalarkaSen Cuts them up?

Balarka Sen

Yes, kind of.

Soham Chowdhury

I know nothing about bundles yet, though.
Anyway, is there any, um, structure that fails to be a category because $\text{Hom}(A,B)$ and $\text{Hom}(C,D)$ aren't disjoint (for unequal $A,C$ etc. obviously)?

Tobias Kildetoft

@SohamChowdhury If you have such a thing, you can always turn it into a category anyway

by just attaching the domains and codomains to all morphisms

Soham Chowdhury

@TobiasKildetoft Can you explain?

Can you add data to a morphism like that?

Tobias Kildetoft

06:49

@SohamChowdhury Just replace the elements of $\operatorname{Hom}(A,B)$ by triples $(A,f,B)$

Idris

it is posted with some extra explanation of how the computations are computed and do not fall from the sky!

Soham Chowdhury

@TobiasKildetoft Oh, I understand.

Thanks.

Tobias Kildetoft

@SohamChowdhury in practice, this is never an issue

Soham Chowdhury

This works because the set (or class?) of morphisms can consist of anything, right?

Not just ordered pairs.

Idris

@paramanand it is posted with some extra explanation of how the computations are computed and do not fall from the sky!

Tobias Kildetoft

06:51

Right, it could be anything at all

Soham Chowdhury

I see.

Balarka Sen

Homs are just sets

They needn't have any extra structure.

Tobias Kildetoft

@BalarkaSen they need not even be sets

though I am not sure I ever saw anyone consider categories where the Hom's did not form proper classes

Balarka Sen

ugh

Soham Chowdhury

So a hom can consist of dissimilar things?

Tobias Kildetoft

06:52

and of course, we almost always restrict to locally small categories

Soham Chowdhury

Locally meaning?

Balarka Sen

what's your definition of dissimilarity?

Tobias Kildetoft

@SohamChowdhury Yeah, they can really consist of anything. The important thing is that we have rules for composing them

that for any objects $A$ and $B$, $\operatorname{Hom}(A,B)$ is a set

Soham Chowdhury

@BalarkaSen Like not all n-tuples, or not all diagrams, or whatever.

Tobias Kildetoft

(a small category is one where the objects also form a set)

Soham Chowdhury

06:53

@TobiasKildetoft Yes, I know that.

What's "locally"?

Balarka Sen

I don't get it. I don't know how you can define a good notion of similarities unless your Homs are categories too

Tobias Kildetoft

@SohamChowdhury the thing I said after "that for" (I forgot to add what that was in response to)

Balarka Sen

(In which case, you get something close to a 2-category)

Soham Chowdhury

@BalarkaSen Basically, what I meant is this: can the same hom contain a 2-tuple and a 5-tuple?

Balarka Sen

so you're specifying that your morphisms are tuples?

Soham Chowdhury

06:55

@TobiasKildetoft Oh.

Tobias Kildetoft

@SohamChowdhury Sure, there are no rules other than that we must be able to compose them in some way we specify and that composition must follow some rules

what the homs "actually" are is not important at all

Soham Chowdhury

@BalarkaSen Not necessarily, just that they are all instances of the same thing (like 2-tuples for example, or diagrams, or whatever.)

Balarka Sen

oh, sure.

Soham Chowdhury

@TobiasKildetoft Yes, that answers my question.

So it's not necessary that the morphisms are all similar, right?

Balarka Sen

but it still won't make sense to say "same" thing unless you give your Homs extra structure.

Tobias Kildetoft

06:58

@SohamChowdhury The reason they often tend to be similar objects is that then it is easier to give a uniform description of the compositions, rather than having to describe it for each pair

Soham Chowdhury

Okay, I see.

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$Mathematics$ Mathematics