>tfw family will never know what cohomology is... >they'll

>>7802572

Algebraic Topology is overrated

>>7802579
>he things functors and cohomology belong to the field of algebraic topology

>Most people think mathematicians deal with numbers, and think mathematicians are like advanced accountants

>>7802572
>Most people think mathematicians deal with numbers, and think mathematicians are like advanced accountants

This actually triggers me.

>You study math?
>Oh, you must be good in numbers

I fucking hate. Last time this happened, right after that I was asked an arithmethic question. Saying he was old and then was wondering how many years had passed since someday. He asked me to do the math for him.

I GOT FUCKING TRIGGERED. I HAD TO DO IT RIGHT. If I didn't then he would think I was a fucking moron. I did the math in like half a second, fearing that if I took more time, he would think I had a two digit IQ.

After saying it, I did the math like 10 times, confirming my answer was correct. As the 'confirming' cycles went by, my state went from really fucking triggered, to slightly uncomfortable, to okay, to 'fuck yeah'.

So much pressure man. The high expectations everyone has of me.

>>7802754
I can't remember the last time I actually dealt with real numbers.

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>>7802780
Good. Real numbers are bullshit.

>>7802780
>>7802787
>his research doesn't involve number fields
Fuck any and all transcendental numbers, though.

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>>7802572
>tfw I make nearly double straight out of college what my dad does now

I really don't know how to feel about this. Like, what has he even been doing all this time?
>inb4 "raising you, faggot"
Not really

>>7802801
The best reals are those whose existence is unprovable, such as [math]0^\#[/math]. I bet you love those.

>>7802801
Why even bother studying things as boring as little numbers when you can study pure structure?

>>7802831
what do u do anon?

>>7802845
Aerospace engineer, $80,000

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>>7802579
>He's never studied étale cohomology, sheaf cohomology, de rham cohomology

>>7802909
>étale cohomology
>algebraic topology
Nigger what the fuck are you doing.

>>7802572
> cohomology

Freshman level shit bro.

>>7802572
Cohomology is seriously overrated it's a fuckload of abstract nonsense with almost no meaningful application

Functors are easy as shit. You know what a linear map is, right? A linear map is something that maps some object (vector space) to some other object of the same kind (vector space), which might look different than the first object. Now a functor is the next step in abstraction. A functor is a map that maps similiar things (e.g. vector spaces) to other things that are possibly of a different kind(e.g. topological spaces). However the cool thing about functors is that it also maps maps between the first kind of things (i.e. linear maps) to the kind of maps between the second kind of things (contious maps).

>>7802572
Cohomologies are a meme. Just admit you only learn them to be pretentious about it on the internet.

>>7802970
So, in short: A functor is just a map that maps maps.

With some properties such that the whole thing makes sense. In fact if you would define a functor as just that "a map that maps maps in a way that makes sense" you could probably derive the formal definition on your own.

>>7802970
>>7802988
That's pretty cool. Is there a book out there for us plebs who aren't pure mathematicians to learn more about this (and it's applications)

>people ask me to calculate big numbers
>no one asks me about algebraic varieties

>>7802932
That was my point retard. He thought all cohomology was a subset of algebraic topology, making his limited knowledge obvious.

>>7802932
Is that the only one you noticed? None of them are algebraic topology things.

>>7802831
He got married and had kids. It's just a losing proposition.

>>7802761
I had a coworker ask me about the stock market because he though all math majors did was talk about finance and interest. Yeah. :)

>parents have PhDs in biology fields
>all family friends are colleagues at their institution
>I'm a 1st year PhD grad student
>despite being in research/academia, I have almost nothing relatable with them (mostly occurs during family events)
>doing theory as well so my studies and research interests really fall on deaf ears if I get even slightly technical (i.e. mathematical)

>>7802761
autism speaks

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>>7802988
>A functor is just a map that maps maps. With some properties such that the whole thing makes sense. In fact if you would define a functor as just that "a map that maps maps in a way that makes sense" you could probably derive the formal definition on your own.

We can make it precise quickly. If you have a set [math]S[/math] and some maps
id : S -> S
f : S -> S
g : S -> S
h : S -> S
on it, i.e.
[math] id \in S^S [/math]
[math] f \in S^S [/math]
[math] g \in S^S [/math]
[math] h \in S^S [/math]
then they form a monoid where the multiplication is function concatenation [math] \circ [/math] :

[math] g \circ f \in S^S [/math]
[math] h \circ g \in S^S [/math]
[math] f \circ id = f [/math]
etc.

A set valued functor [math] F [/math] acting on those maps is a homomorphism in that the concatenation [math] \circ' [/math] of the new maps
[math] F(id), F(f), F(g), F(h), ... [/math]
on the set (which may call [math] FS [/math]) are given by the old concatenation

F(f) : FS -> FS

and

[math] F(f) \circ' F(g) := F(f \circ g) [/math]

A general functor of sets is now exactly this, except we don't necessarily require that we only deal with a single set S but instead the domain and codomain of the maps may be different.

If
f : X -> Y
g : Y -> Z
then
[math] g \circ f : X \to Z [/math]
and a functor is still able to map them

[math] Ff : FX \to FY [/math]
[math] Fg : FY \to FZ [/math]
[math] F(g \circ f) : FX \to FZ [/math]
where
[math] F(g \circ f) = F(g) \circ' F(f) [/math]

Form a topos perspective, the most important functor may be the hom functor for a set T which takes domains/codomains S to the set of function [math] S^T [/math] and which maps
maps
[math] f : X \to Y [/math]
to maps
[math] F(f) : FX \to FY [/math]
i.e.
[math] F(f) : X^T \to Y^T [/math]
which work as follows:

If you have two function
[math] x \in T\to X [/math] and [math] y \in T \to Y [/math], then
[math] F(f)(x) := f \circ x [/math]

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>>7804017
The point is that you make functions f between sets into function F(f) between function spaces.
Those are morally better objects because properties like „homomorphism“ and „continuous“ (which algebra and topology is really about) is not a property of the elements of sets, but of functions between set.

Algebraic topology and many cohomology theories are about using such a homomorphism F (of a non-total monoid, i.e. of a cateogry) to pass from a world of topological spaces to a world of algebraic objects.

Topos theory is about realizing this insight and defining topology (and even stuff like differential geometry) in an algebraic way in the first place.

>>7803002
I feel there is no simple and comprehensive into, if you're a mathematican maybe it comes naturally after ring theory (not my thing)

here are some directions

Categories_for_the_Working_Mathematician, Mac Lane
(abstract algebra)

Simmons - Introduction to Category Theory
(algebra, strange but with pictures)

Awodey - Category_theory-Oxford_University_Press,_USA
(Comp-Sci, more modern)

Robert_Goldblatt - Topoi_The_Categorial_Analysis
(logic, the simplest intro do categories, but it introduces functor very late and beyond page 150 it's higher order logic)

Sheaves_in_geometry_and_logic - Saunders_MacLane, Ieke_Moerdijk
(topos theory, not a good starting point)

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>>7804017
Should have ended with

If you have two functions
[math] x : T \to X [/math],
[math] y : T \to Y [/math],
i.e.
[math] x \in X^T [/math],
[math] y \in Y^T [/math]
and if
[math] f : X \to Y [/math]
note
[math] f \circ x : T \to Y [/math]
and then
[math] F(f) : X^T \to Y^T [/math]
given by
[math] F(f)(x) := f \circ x [/math]

>>7804028
Thanks. Top tier post. I love you Anon.

I'm gonna start with the one that has pictures. Looks like it's freely available in nice LaTeX as well.

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How pseudo-intellectual are the posters ITT that they are honestly bragging about knowing what a functor or what a cohomology is? While you're at it, would you like to brag about your knowledge of calculus and quadratic equations as well, jerking off each other over several posts explaining in retard terms what a derivative is or how to complete the square?

Undergrads pls go. This is a serious academic discussion board.

>>7804053
Pretty sure no one is bragging, they're sad/frustrated that they can't talk their friends and family about their field.

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>>7804045
Simmons book with the pictured is nice but you'll need to draw your motivations from somewhere else. I think you'll need to have a gial to grt into it in the first place. The theory is formal and abstract and you should always try all definitions on some categories of your actual interest.
What the book does a very good job at is explqining universal arrows, limits and colimits, which are superimportant, I think, to understand all the connections nw between different subjecs and different definitions in mathematics. It might be most interesting if you have a want to understand formal logic - at least for me it is. It seems like almost all important concepts are universal morphisms in some way.
Strangely, the age old book "categories by the working mathematican" still works as into. You'll have to read several books and as I said tbink of your personal applications and needs in mind.

>>7804053
That's interesting, I sometimes feel I'm among the very few who try to bring some content to the threads (maybe except for the explicit homework threads)

>>7804056
>>7804088
Read the thread. Do we seriously need pseudo-intellectual wank posts like >>7802970 >>7804017 when the concept of a functor can be read within 1 minute on wikipedia? You're not even making it simpler to understand. You're just masturbating publically.

>>7804088
Thanks for the tips.

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>>7804095
I (guy with the street pic) wrote the second post you linked to, actually. Yes, it's super basic algebra definitions, expressed ad hoc and sure it has been done better before.
People are also more involved if they talk to another person instead of just sitting around and reading. They need to do that eventually, of course, but that's why I also make a reading list part of the explanation.
I'm really just biting my time on a snowy Saturday in a cafe and motivate some people who have interest in a subject I find interesting. If you're gonna say "you can look it up", then you only leave meme posts to /sci/. The merit of 4chan is that it's funny, but if the only thing coming from this particular board were black science man images then I'd just stick to /tv/ or some other board where there is also funny stuff and pretty girls as well as "information" about upcoming entertainment.
Nobody here has the dedication for working for more difficult stuff, and there are too few people for actual higher level discussion (I've tried making threads again and again), so writing down motivational intros at least has the merrit that I'll copypaste it to my notebook (or save some other anons recommendations, pic related on categorical logic) and will in 10 years still be able to guide people to it, if they feel they need some perspective that comes from a human being who wrote it (not from MacLanes book).

>>7804053
>consciousness
>transhumanism
>time travel
>doctors are low IQ
>reminder that Einstein is a "kike fraud"

high level academic topics alright

>>7804095
It feels like you're saying we should only discuss bleeding edge topics.

I think discussing calculus is fine for example.

An undergrad discussing chain rule differentiation with another helps both with the learning process. Even more advanced students explaining it to them helps them crystalize it better.

If they were bragging/circlejerking sure shoo shoo to reddit, but I really didn't get that tone ITT, maybe I'm reading differently just because I'm in a different mood.

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Doing physics.

First I could talk about some cool stuff but the longer I studied the more abstract it got and now I can't even start talking about anything without going deep into the mathematical formalism and most of my friends are literally retarded ("I study economics because of >muh money ")

It's frustrating.

>B - B- Bu - But anon how about the physicists at your uni?
They are autistic sperglords. Even the mathematicians are less socially retarded at our uni

>mfw

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>>7804146
>An undergrad discussing chain rule differentiation with another helps both with the learning process. Even more advanced students explaining it to them helps them crystalize it better.

Yeah and we can even use that topic

[math] f(g(x))' := f'(g(x)) \cdot g'(x) [/math]

i.e.

[math] (f \circ g)' := (f' \circ g) \cdot g' [/math]

as a model for functors and cohomology.

From the Wikipedia article on then chain rule

>In abstract algebra, the derivative is interpreted as a morphism of modules of Kähler differentials. A ring homomorphism of commutative rings f : R → S determines a morphism of Kähler differentials Df : ΩR → ΩS which sends an element dr to d(f(r)), the exterior differential of f(r). The formula D(f ∘ g) = Df ∘ Dg holds in this context as well.

>The common feature of these examples is that they are expressions of the idea that the derivative is part of a functor. A functor is an operation on spaces and functions between them. It associates to each space a new space and to each function between two spaces a new function between the corresponding new spaces. In each of the above cases, the functor sends each space to its tangent bundle and it sends each function to its derivative. For example, in the manifold case, the derivative sends a Cr-manifold to a Cr−1-manifold (its tangent bundle) and a Cr-function to its total derivative. There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. This is exactly the formula D(f ∘ g) = Df ∘ Dg.

and then pic related

>>7804158
In the first year my electrical engineering friend learned about Hilbert spaces and enjoyed it a great deal - they went into Fourier transforms and he soon knew stuff beyond me. Then their math lectures just stops and I remember talking with him in the third semester, writing Lagrange equations on a baloon at a party and he didn't knew anymore what I was talking about.

>3rd year comp-sci
>mom still doesn't know what comp sci is

On top of that, my school likes to emphasise theoretical comp sci. It's hard to explain it to someone who thinks their browser is "Google Google" when she actually uses IE.

>their browser is "Google Google"
?

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>>7804262
>mom calls me up asking how to attach a signature or something to an email
>I asked what browser she was using so I could give her exact instructions
>"Google"
>like Google chrome?
>"no Google Google"
>what's the icon you click on to get for the internet?
>"the one with the blue e"
>mfw

>>7802973
I learn knot theory to do that

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>>7804017
>>7804028

No.1 post right there, buddy. Thank you for taking over for me after I went to bed.

>>7804095
Fuck off, nigger. Dude wanted to know what a functor is and the poster explained that very well without first going the long way over categories and kept everything minimal such that it stays understandable for people with little mathematical background.

This thread is miles better than the other popsci and shitposting threads on /sci/.

>Family will forever think my Comp Sci degree is a degree in computer repair and internet reconnections.

>>7804292
Why would you need to know what browser she uses to attach a signature or something to an email? If she said "google" and the subject was emails, wouldn't you think gmail and just need instructions how to do it with that?

>>7804361
It was mostly to rule out whether she was using a dedicated client like outlook, or a web client, not to mention it was office365.

And I can relate to those feels.
>be in uni first year living on res
>house full of guys in comp sci
>get knocks on the door from other students asking if we could set up their printer or fix the wifi
>roommate would never say no, so he got cucked the whole year

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>>7804335
>without first going the long way over categories
in fact at one point I found a first-order logic definition of the theory in the dungeons of the nLab, pic related

I believe it stems from Lawveres et al.'s autistic ECTS projects etc..
In particular I find it admirable that (I think I read it in MacLane) he attempted and succeeded in defining adjoint functors, previously only taken to be the bijection of hom-sets (emphasise on sets), without reference to sets. I mean he had to do it to set up non-setty foundations. Thus we now have the purely algebraic unit/co-unit definition (which is my favorite pov, actually)

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>>7802973
I am really tired of this algebraic geometry meme on /sci/ these days.
I'm sure there are some competent people actually working (ie. doing their PhD) on these things here but for every one of these we get a thousand undergrads who just learned what a *insert algebraic/categorical/topological term here* is and cannot wait to tell the world about it.
Curiously, you never hear people bragging about knowing PDE, complex dynamics, functional analysis, stochastic analysis or whatever other topic.
For some reason, every loudmouth math major here seems to want to do algebraic geometry.
I *really* wonder why that is..

>>7804415
>tfw I study stochastic analysis
>tfw nobody on /sci/ ever talks about it

>>7804415
>Curiously, you never hear people bragging about knowing....functional analysis

>Banach space thread at least once a day

>>7803350
Sheaf and de Rham are absolutely and definitely alg top things, nyugguh.

>>7804454
>de Rham are absolutely and definitely alg top things

Learned de Rham cohomology in my vector analysis class. It obviously has significant connections to alg. top., but is not a purely alg. top. subject.

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>>7804426
Then why not start a thread?

I'd have questions on concise definition regarding the partitions necessary for defining the Itō integral.
Also I have worked out what I think good motivations for physicists and engineers to make it more accessible.

>>7804415
>>7804432
I'd argue current PDE, harmonic analysis, functional analysis problems (unless it's about numerics) are more of academic interest than all the type theory and cats stuff. Why would someone honestly ask a question about PDE's unless he or she had to solve some for a course?

>>7804470
Ultimately it's semantics, but I consider differential topology a subset of topology, and it is the business of algebraic topology to assign algebraic invariants of topological objects (a topological space admits at most one structure of a smooth manifold, right?).

>Work in set theory
>Have absolutely no idea what 95% of mathematicians are talking about and consequently feel dumb

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>>7804488
https://en.wikipedia.org/wiki/Exotic_R4

>>7804045
I remember I wrote another basic motivation for the functor I used in that example (except here it's functions into some space "· -> Y", not out of some space "X -> ·", where they are called set-valued sheaves. fun fact: those objects all exist as primitive notions in Haskell)

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Grad student in nuclear physics. My family is completely ignorant when it comes to science.

At thanksgiving, I mentioned how I got to handle a piece of radioactive americium for an experiment. One of them was worried I should see a doctor for exposure. Another was surprised they let a clumsy person like me hold the sample, since if I dropped it, it might explode the whole town.

How am I supposed to have a meaningful conversation about my research on quark gluon plasma when I'm dealing with that level of retardation?

>>7804558
Ah fuck you're absolutely right, I'm such an idiot (this is even more pathetic since I took a minicourse on Seiberg-Witten theory a couple months back). De Rham still only depends on the underlying topological space, and it's better to view it as a nice way to compute singular cohomology.

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>>7804558
ah another one, here that functor in the context of harmonic analysis

I'd argue that a lot of category theory is the formalization of notions that previously have been done in the meta theory. If you look at the historical conception of the subject, this seems to be the case too

>>7802572
I've explained (co)homology to my artist/writer wife.
I'd not expect her to remember, but she understands what a topological space is (not the exact definition, but say a simplicial complex).
I then explained homotopy as paths mod wiggles, and (co)homology as the relationship between subspaces and their boundaries.

Having your family to understand the details would obviously be nice, but unrealistic, but you'd be surprised how much somebody with no maths knowledge can understand intuitively.
Not many people will appreciate "connected components of the oo-groupoid of (homotopy classes) of morphisms" definition, of course.

>>7804612
Algebraic topology nice like that because it is extremely spatial, so one can communicate the basics to non-math people in a way that they can intuitively understand.

The same certainly doesn't hold of many other fields of math.

>>7802970
>Cohomology is seriously overrated it's a fuckload of abstract nonsense with almost no meaningful application
What about all algebraic topology and all (modern) algebraic geometry?
We also have Galois cohomology in number theory, which was used in a fundamental way (along with techniques like Étale cohomology from algebraic geometry) in the proof of Fermat's last theorem.
There's also group cohomology, that is used in the classification of finite groups, graph cohomology, Lie cohomology...
There are loads of applications in modern physics, and even applications in robotics in things like positioning of arms.
Here's a list of other applications - I won't clog up the thread correcting what I hope is just a shitty attempt at trolling - hopefully someone will have found this interesting.
http://mathoverflow.net/questions/60108/occurrences-of-cohomology-in-other-disciplines-and-or-nature

>>7804630
I agree.
The book "A Topological Picturebook" is fantastic for that.

As >>7804502, this thread is the kind of thing I'm talking about.

I have almost no idea what cohomology even is, even though it seems common knowledge among so many other mathematicians.

>>7804644
One application I know of fairly well is in CFT where you differentiate between physical and nonphysical states by whether or not that are BRST cohomology classes. Where BRST cohomology is like de rham cohomology but with a Noether symmetry operator, the BRST charge, instead of the exterior derivative.

>>7804654
You can do BRST cohomology in a lot of situations, but it does seem to crop up more if CFT/ST.
>>7804650
It might be easier to avoid the topological side of it and just look at the homology of chain complexes.
In this situation, the difference between homology and cohomology is immaterial (just depends on the direction of the arrows, but a simple relabelling will fix that)You have a series of abelian groups, [math] ... C_3 \rightarrow C_2 \rightarrow C_1 \rightarrow ...[/math]. You form from this a series of groups, [math]H_i[/math] by taking quotients [math]ker d_{i} / im d_{i+1}[/math], where [math]d_i[/math] is the map from [math]C_i to C_{i-1}[/math].

In topology, you assign a chain complex (the above chain of groups) to a space, and though this assignment isn't topologically invariant, the homology groups are.
Cohomology is very similar, but instead you have groups [math]C^i[/math], and maps [math]d^i[/math] that increase degree, instead of decrease it, for example by replacing the groups [math]C_i[/math] by [math] Hom(C_i, G)[/math] for a fixed abelian group G. This is a trivial change on the level of homology groups (they determine each other), but it turns out if you take cohomology with coefficients in a ring instead, you can define a product on the direct sum of the cohomology groups, which carries extra info about the space.

Anyway, its the abstract notion of extracting data from a chain complex that is useful in applying the theory to other areas of maths, by looking at functors from some category you care about to the category of chain complexes, you have cohomology of lots of things. You can also replace the abelian groups by objects in an arbitrary abelian category, which is essentially an abstract thing with just enough structure to make the above work (you can define kernels and images, and hom sets have an abelian group structure).

>>7804745
Wow, that was actually extremely illuminating. Thanks!

>>7804650
It was so fruitful in topology that people in most areas of math have their own version. It's just that chains of maps between objects are helpful in studying the objects themselves.

What really is the differen e between homology and cohomology? It seems like the co-theory is just better..

>>7805644
The arrows go in opposite directions.

>>7805644
cohomology is the dual of homology

>>7805661
>>7805665
okay, but homology seems to be weaker or less often applicable somehow

why?

just "dual" can't be it. Or even if, what does change so much by flipping the arrows?

>>7802572
> be me, love algebraic stuff, studying cohomology
> decide not to fell for the meme and do a probability PhD
> get paid 100k for a research job at a private company, love what I do
> browse /sci/
> can no longer understand the cohomology meme
it hurts sometimes. I wish I was still poor and stupid enough to have fun with the cohomology meme...

>>7804426
ma nigga, I have the same feel
currently studying Skorokhod convergence/invariance principles and stochastic calculus with jumps. Pretty technical huhu

Trying to discuss the Fourier series with people who can't understand basic algebraic manipulation.

>>7806608
>love what I do
Do you really though? Or is that just a way to deal with your regret of not being paid to study cohomoly in academia?

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>>7802836
[math]0^{#}[/math]
0 is covariant? 0 is in the natural coframe? 0 is a p-form?


I'm very tired.