Mathematics - 2017-12-01 (page 1 of 6)

If all theorems can be expressed in such clean fashion as categories, there will be no more headache on what it means

MrAP

oh thanks

Our natural language is sometimes just too wordy to read

@MatheinBoulomenos are coequalizers and quotients the same?

@LeakyNun I don't see how the last row is any particular isomorphism theorem. But the fact that $PGL(V)/PSL(V) \cong F^*/(F^*)^n$ follows from the diagramm (this is an instance of the so-called "nine lemma")

12:22 AM

8 mins ago, by Leaky Nun

the last row is the second isomorphism theorem / nine lemma

:)

ah yeah

@LeakyNun I don't know a definition of "quotient" in a general category, so I'd say no

What does the isomorphism theorems look like in commutative diagrams?

@MatheinBoulomenos in groups?

@Secret first iso:

why do we need the two nodes of zeros at the top of the quotient group?

@LeakyNun the coequalizer of $f,g: A \to B$ is the quotient of $B$ by the normal subgroup generated by all elements of the form $f(a)g^{-1}(a)$, yes

12:27 AM

@Secret if $0 \to A \xrightarrow{f} B$ is exact then $f : A \hookrightarrow B$ is injective

if $A \xrightarrow{f} B \to 0$ is exact then $f : A \twoheadrightarrow B$ is surjective

third isom ^ (@MatheinBoulomenos amirite)

@MatheinBoulomenos but can I have $A/\ker(f)$?

What do you mean?

you can also make a diagram such that 3. iso follows from the nine-lemma

oh really

this is beautiful lol

@MatheinBoulomenos right, I think I see it, just need to put some $\operatorname{id}$ somewhere

yup

that's the thing

Hmm... I am missing something here...

@Secret what is it?

12:35 AM

f does not look really injective to me

the first colum is $1 \to N \to N \to 1 \to 1$

↪ is a cover, right?

I am not familiar with the use of $\twoheadrightarrow$

↪ is for monomorphisms and ↠ for epimorphisms

@LeakyNun yup, that's it

12:38 AM

@Secret no, the opposite (embedding)

@Secret exact means image = kernel, right

so the intersection in the middle has to be the same

i.e. the thing is continuous

(the thing = line above, cone below)

@MatheinBoulomenos but the previous one looks much more beautiful

6 mins ago, by Leaky Nun

@MatheinBoulomenos I prefer $0$ because abelian categories

O I see, it's the image of the maps equal to the kernel of the next map, not every domain maps to the kernel of the next domain

right

@MatheinBoulomenos how would I do second iso though

no idea

Sometimes I do wonder whether we can visualise those maps directly without drawing the domains, but guess not since each map is uniquely specified with its domain in addition to its image. which mean for any diagram that is used to visually illustrate a given map, it always consists of 3 glyphs, corresponding to the domain, the map and the image

but no diagram compares to the butterfly lemma

$B(A \cap C)/B(A \cap D) \cong (A \cap C)D/(B \cap C)D$

12:51 AM

Actually... I might be able to do it this way...

so the maps to and from the trivial objects constraint the other maps

@LeakyNun that diagram I posted is actually super useful

@MatheinBoulomenos what is that theorem?

@Secret zero objects

it's not a theorem

what is the use?

@TedShifrin hi

@Secret I don't understand the diagram

12:55 AM

hi Leaky. If you're trying to understand $d$, $d^2=0$, and vector calculus, you should watch one (or more) of my relevant lectures. I did that all very carefully.

it allows you to reduce statements about exact sequences of the form $A \to B \to C$ to statements about exact sequences of the form $0 \to A \to B \to C \to 0$

rerehi @Ted

I've lost count, Mathei. I'm only here briefly before I retire to the kitchen.

Ye

Ugh cheating

Oh, good grief, Balarka is unsleeping badly.

I am going to fix my sleep schedule today

Going to keep awake all day

12:56 AM

Yeah, sure.

@MatheinBoulomenos how?

As the saying goes, promises, promises ...

I sense disbelief in your voice

In my voice and in my fingers.

Thy shall not be disappointed

12:57 AM

@LeakyNun the exactness of the middle sequence is equivalent to the exactness of the three diagonal sequences

@BalarkaSen you have sleep schedule? I don't believe you

Not that Mathei is doing much better.

Kevin Driscoll

@TedShifrin What do you usually cover in difftop after Mayer-Vietoris?

why coker(f) = im(g)? and did you spell coker as koker?

That was near the end of my G&P course, Kevin. Remember I wasn't teaching a grad course. ... I was in the hospital with anemia last time and someone covered for me and didn't even do what I wanted.

I'm German

12:59 AM

@MatheinBoulomenos but did you spell koker as coker?

It's not a category theory diagram, but a visual illustration. The idea is that cones represents the kernel of the map (since everything on the LHS will map to the zero object), parallel lines represents bijections, frustums represents a neither injective nor surjective map, and arrows represents mapping from zero object to zero object.

So if we are given the maps in blue, and the requirement that the maps form an exact sequence, then the black glyphs have no choice but to be constrained by the blue maps, which is why when you compose the two maps in the top row, you get two arrows thus gu…

German doesn't use c like that.

Ich habe cein Idee. Um, no.

lol

Kokaine