The h Bar

user434058

04:57

-3

A: Why equations of motion doesn't work here?

You cannot use results that exclude drag to describe the motion of a projectile with drag. You need to go back to Newton's second law. You have two forces, weight = mg (downward), and drag = c*|v|^3 * v/|v|. I am assuming that upward is +. c is a constant of proportionality to make the units ...

user434058

What should I do, take back my downvote or wait till the OP inproves the post by adding MathJax and then take back my downvote? Also the comments are more suited as a discussion on a Meta rather than a comment thread on an answer. However I don't want them deleted so I am not flagging them, but to be fair, they are off topic comments.

John Rennie

The problem with a downvote is that it feels like a slap in the face. ggcg has put effort into their answer, and while that effort didn't extend to using MathJax it's still a genuine effort to help the OP.

There are differing views on the use of downvotes for answers. My own view is that I'm reluctant to downvote anyone who appears to making a genuine effort to help.

Yuvraj Singh...

05:12

@ACuriousMind @AaronStevens yes time refer to the time when the matter changes it 's state and properties.

I think that it was logical to relate the nature of particle using time and energy as @JohnRenniementioned about energy of a photon.

@JohnRennie what you are thinking about the statement?

14 hours ago, by Yuvraj Singh...

hi everyone today I was watching a film where one of character as a physicist says 'energy and time is the best two things to define a state of matter or a particle .was he right ?if we know about energy and time of a particle can we define it's state or would we need some other factor too

John Rennie

@YuvrajSingh... I would have to see the program as I'm not sure what it means to talk about the time.

Yuvraj Singh...

i have mentioned !

John Rennie

Which film was it?

Yuvraj Singh...

movie was a south indain movie

@JohnRennie forget about movie sir, what I am asking is is it possible to define nature using these two quantities?

John Rennie

No

Yuvraj Singh...

05:19

why not ?

John Rennie

In classical mechanics the fundamental property is the Lagrangian or Hamiltonian and they are formulated using position and momentum.

In quantum mechanics the fundamental property is the wavefunction.

Yuvraj Singh...

05:34

ok @JohnRennie

skullpatrol

06:37

user434058

07:09

@JohnRennie I definitely value ggcg's efforts and that's why my downvote is temporary. I expected that a few downvotes on one of many ggcg's badly formatted (math POV) answers, might make them consider learning MathJax. ggcg is consistent with the poor math formatting and his answers (which include math) have either been edited to add MathJax or are in a ugly state (math POV).

user434058

ggcg has also admitted that they should learn MathJax and use it, in comments in an older post, however they have never done so. It is the consistency of not using MathJax which made me cast a temporary downvote.

user434058

Thanks for the edit. I need to learn mathjax — ggcg May 16 '19 at 12:55

user434058

And now I honestly can't take back the downvote and neither am I really keen to take it back. So I'll wait, if the answer gets edited, I will surely reverse the downvote.

user434058

@skullpatrol Is this paper some sort of a hoax? I don't believe that a few people could alreadyhave predicted this that early...

John Rennie

@FakeMod it's just that downvotes are rarely a good way to motivate people. On the whole they demotivate people. In a comment ggcg says he can't be bothered to edit the post since it has been downvoted.

2

@FakeMod HIV probably originally came from people killing and eating monkeys. It has been well known for decades that eating wild animals can cause infection from xenoviruses.

skullpatrol

07:21

@FakeMod they have been ready for this for 13 years, have a look at thier numbers:

user434058

Hmmm...

skullpatrol

now that is effective flattening of the curve

you gotta think ahead of the curve to flatten it

not run out and buy toilet paper :-)

user434058

AFAIK, due to the strict nature of Chinese government and probably some data hiding,they were very effective in "fighting" against the coronavirus.

John Rennie

The best example is South Korea. They have also managed to contain the virus very effectively.

They did it by a massive investment in tracing contacts and testing.

The UK on the other hand did very little testing, and that meant when someone got infected they couldn't find and test all the people the ill person had been in contact with.

The result is 12,000 deaths and counting.

ACuriousMind

@FakeMod Why would it be a hoax? We had the first SARS, also zoonotic from bats, in 2002 - it doesn't take a genius to say "this could happen again, just worse".

skullpatrol

07:36

@JohnRennie that, sir, almost sounds like a violation of our freedom of rights to not reveal who we have been in contact with (not that I agree with with-holding such information during a pandemic)

John Rennie

@skullpatrol there is inevitably a conflict with what we regard as our rights in these situations. For example I could regard it as my right to leave the house whenever I want so I shouldn't be restricted by the lockdown.

Whether that would make me a libertarian or a dick depends on your view.

skullpatrol

Exactly.

Or the right to bear-arms against mask wearing Asians?

John Rennie

Or burn down 5G radio masts because you think they cause coronavirus.

One wonders sometimes why evolution put all that effort into evolving human brains.

4

skullpatrol

When we end-up giving in to our primeval instincts.

Meanwhile, trump wants to lift the lockdown in two weeks?

/rant

Slereah

08:44

Predicting that pandemics can happen isn't an extremely bold act of fortune-telling since pandemics have been around for thousands of years

By now we know what to look out for

If there's a lot of infected people/witches casting the evil eye, this is a bad sign

ACuriousMind

Maybe we shouldn't have gotten rid of all the witches, they'd know some spell against this

Slereah

only evil ones, I'm afraid

ACuriousMind

How do you know? Are you a witch, hmmm?

Slereah

Nah, too heavy

ACuriousMind

09:02

@ZeroTheHero Not too bad, but it's not going to become a staple for me

Secret

09:34

I am a witch, at least in my delusional conspiracy theory of trying to unmake the world just to get back at a person who upsets me 10 years ago

The fact is, I don't think our universe is supernatural enough to support any statistically significant magical influences

so I am not a witch

Or more accurately:

$\lvert \psi_{\text{me}}\rangle = \frac{1}{{}^{6}\sqrt{\frac{1}{0}}} \lvert \text{witch}\rangle \pm \lvert \neg\text{witch}\rangle$

Anyway, Australia seemed to be flattening the curve somewhat now

Slereah

10:38

Science in the mail!

Boy oh boy

it's Gourghoulon's numeric GR book!

And Lichnerowicz GR book!

Lichnerowicz is another one of those 1930's cheap books that aren't properly cut

So pages are still stuck together

I gotta cut 'em meself

Who even sold this?

Did someone keep that book since 1939 and never read it?

ACuriousMind

is it really an original printing from '39?

Slereah

10:59

Let me check

yep

Printed 19-8-1939

ACuriousMind

neat

Yuvraj Singh...

12:02

@ACuriousMind string theory is a theoretical concept right, which has not been proved unlike the GR right?

Then why some physicists says it is well accepted theory in community!

ACuriousMind

@YuvrajSingh... "Theoretical" does not mean "without evidence". Both GR and ST (and QFT, etc.) are theories, or rather theoretical frameworks in which one can design specific models/theories (e.g. specific universes, quantum electrodynamics, the Standard Model, etc.).

What's true is that there is no experimentally tested prediction of any string theory model (that is not also trivially a prediction of QFT/GR). We don't even have a stringy model that faithfully reproduces the Standard Model without extra stuff (this is why e.g. detecting supersymmetry at CERN would've been a good hint that the QFT we see is the reduction of a string theory)

Yuvraj Singh...

12:32

Ok

Slereah

@ACuriousMind I don't know if I would say "trivially"

Nothing trivial about it :p

ACuriousMind

@Slereah I mean "trivially" because it would have been obtained by first going to the low-energy effective QFT and computing something there

The process of computation is of course not trivial, but that the result is both a prediction of plain QFT as well as of string theory is

tpg2114

Sigh... all of my bosses (like, all 6 of them) are starting to be efficient with teleworking tools and since they are only managers, they have no actual work to do except call meetings and micromanage. Productivity has plummeted...

ACuriousMind

That sucks.

tpg2114

Yeah. I've got at least 3 hours a day of meetings now, plus all the time that goes into prepping for those

Yuvraj Singh...

12:43

@ACuriousMind is you city is is under lock down? How are you spending your days?

ACuriousMind

Depends on what you mean by "lock down", we're still allowed to leave the house, but not to congregate in groups.

As for how I spend my time:

Apr 12 at 12:21, by ACuriousMind

@dumbasarock Well, since I can work from home the routine hasn't changed much. Most of my in-person RPG groups have shifted to happening online, and I finally can stay home and play video games the rest of the time while claiming to benefit society, so...it's not so bad ;)

Yuvraj Singh...

Ah! Here we are not allowed to go outside our house!

Only when we have something very very very urgent work which includes medical and some groceries and for eggs, bread and milk.

@ACuriousMind

I am mentally disturb because I couldn't not concentrate on my studies because of crowd around me (I mean my family members) every day my frustration increase by a level.

ACuriousMind

@tpg2114 The advantage of online meetings is that no one can check that you're really doing something else during them ;)

I've had some very productive meetings, just what I produced wasn't related to the meeting at all...

tpg2114

@ACuriousMind Indeed -- meetings have turned into "mute everything and play FTL" or, sometimes, mute everything and keep working

Yuvraj Singh...

Any help?

Please!

ACuriousMind

12:51

@tpg2114 My work laptop doesn't have games on it because I still want to get some work done :P

Yuvraj Singh...

Because guys like you have experience to stay at home and utilize best of your time to work!

ACuriousMind

@YuvrajSingh... I've always had my own room where I could lock the door if I wanted, so I'm afraid I don't have much to offer there

tpg2114

@ACuriousMind I discovered the in-home streaming on Steam... huge mistake. On the plus side, the video and sound still play on the monitor of my personal desktop, so it's great for scaring my wife

ACuriousMind

But I'm also an only child so there weren't many people to disturb me to begin with when I lived with my parents :P

@tpg2114 Don't tempt me!

bolbteppa

13:15

Is the Lichnerowicz thing his thesis?

André Lichnerowicz

André Lichnerowicz (January 21, 1915 – December 11, 1998) was a noted French differential geometer and mathematical physicist of Polish descent. == Biography == His grandfather fought in the Polish resistance against the Prussians. Forced to flee Poland in 1860, he finally settled in France, where he married a woman from Auvergne. Lichnerowicz's father held agrégation in classics, while his mother, a descendant of paper makers, was one of the first women to earn the agrégation in mathematics. André attended the École Normale Supérieure in Paris, gaining agrégation in 1936. After two years...

'His Ph.D. students included Thierry Aubin, Albert Crumeyrolle, Edmond Bonan, Marcel Berger, Yvonne Choquet-Bruhat, Yvette Kosmann-Schwarzbach, and Thibault Damour'

'In 1967 the French government created the Lichnerowicz Commission made up of 18 teachers of mathematics. The commission recommended a curriculum based on set theory and logic with an early introduction to mathematical structures... These reforms have been called a new math and have been repeated internationally' so he's responsible

Slereah

13:30

@bolbteppa Maybe?

Slereah

14:06

How nlab explains the Ward identity

I should learn to read nlab someday

ncatlab.org/nlab/show/geometry+of+physics

This is the intro, apparently

bolbteppa

You can't really understand a concept until it's been strained through the BV differential into a BRST concept

Slereah

Ah, I think BV is the bivariational complex

Slereah

14:25

Ah, there's the Secret of Physics :

"The answer is known, alternatively, as (higher) functorial geometry (Grothendieck) or synthetic differential geometry in gros toposes (Lawvere), or variants thereof."

aaaaaaaaaaaaaah

ACuriousMind

@Slereah It's Batalin-Vilkowisky, but yes, complexes are involved :P

Slereah

@ACuriousMind Dang it

Abhas Kumar Sinha

writings.stephenwolfram.com/2020/04/…

Big Claims require bigger proof!

But, worth reading...

Slereah

arxiv.org/pdf/1601.05956.pdf

ACuriousMind

@Slereah Physicists might know the BV-formalism as "field-antifield formalism", at least that's the name it appears under in Hennaux/Teitelboim

Slereah

14:33

A paper written in the nlab style

Mine eyes

Abhas Kumar Sinha

Good one by Stephen Wolfram

Slereah

@ACuriousMind Is it the spooky ghost field?

ACuriousMind

No, the "antifields" are not exactly ghosts

Abhas Kumar Sinha

Spooky graph transformations

Quantising space like dis is the most beautiful thing, I've ever seen!

But, raises question on existence of dimensions (I dun like that part tho)

Ama go now, bye.

ACuriousMind

It's an alternative formulation of the BRST procedure and isn't really restricted to gauge theories only, but I have to confess I remember too little of it to say more without rereading these chapters

Slereah

14:37

"diffiety geometry"

What's a diffiety

"A well-kept secret of the traditional formulation of variational calculus on jet bundles is that it does notin fact allow to properly formulate global aspects of local gauge theory. Namely the only way to make thefields of gauge theory be sections of a traditional field bundle is to fixthe instanton number (Chern class) ofthe gauge field configuration."

Bundle conspiracy theories

B. Brekke

In a simple electron hopping on a lattice model, e.g. Hubbard model, in the interaction picture

The creation and annihilation operator must always have the same imaginary time for a finite temperature case, right?

ACuriousMind

@Slereah It's true though. Varying the instanton number isn't really well-defined even in the conventional "rigorous" setting because you're changing the topology of the principal bundle itself :P

Slereah

@ACuriousMind Well you could argue that those are different theories altogether!

ACuriousMind

@Slereah But people regularly do stuff like "sum over instantons"

Slereah

maybe

never really looked much into instantons

ACuriousMind

14:44

But in principle each of the $A$ or $F$ appearing in the summand live on completely different bundles and are, effectively, incommensurable

This doesn't mean what people do is "wrong" by the standards of typical physics rigor, but the nPOV has a nPoint here

Slereah

Aaaah

This paper has the jet bundle of the jet bundle

Have mercy

ACuriousMind

Ah, the JJ bundle

Slereah

ACuriousMind

I think I read this once, too, that looks familiar

Slereah

arxiv.org/pdf/1601.05956.pdf

that's the one

bolbteppa

14:49

My vague understanding is that Lagrangian's are immediately relativistic, Hamiltonian's don't exist for such Lagrangian's so Dirac analyzed the Hessian condition leading to constrained Hamiltonian systems, then FV took that Dirac constraint stuff and set up the S matrix for Dirac's constraint theory, and BF then found the Lagrangian analogue of this stuff, imagine being burdened by topos theory trying to discover this

Slereah

@bolbteppa How can you do anything if you don't immediatly consider the higher category version of it

bolbteppa

haha

There's no 'immediate', if you didn't start from it it wasn't an honest understanding

Slereah

It's getting a bit too categorical for my blood

also french gerbe means vomit

ACuriousMind

@Slereah It means 'sheaf' and gerbes are a generalization of sheaves.

Slereah

I'm getting anxiety just looking at it

bolbteppa

14:56

'"Gerbe" is a French (and archaic English) word that literally means wheat sheaf.'

Slereah

Yes I am aware

But in more modern times it mostly means vomit

and how serendipitous

bolbteppa

"Bundle gerbes have been used in gauge theory and also string theory. Current work by others is developing a theory of non-abelian bundle gerbes"

I guess calling this stuff disgusting would make some sense then :p

ACuriousMind

Don't be so dismissive. Gerbes are just a formalization of what is e.g. in the context of gauge theory we were discussing the "bundle of possible principal bundles".

If you accept that bundles are good, and that we need to be able to talk about switching between different ones, then this is really not so outlandish a construction

Slereah

I know

Just a bit hard to grasp :p

hard to find a decent introduction

bolbteppa

I would prefer to do this BF/BV/FV stuff without any bundles :p

Slereah

15:06

Sorry, mandatory bundles

ACuriousMind

@Slereah Yes, this is the main reason I haven't done more with this stuff. There seems to be almost no text with a setting between "extremely elementary" and "everything is an $\infty$-category".

Slereah

@ACuriousMind pretty much

Also they throw around a lot of terms without defining them

ACuriousMind

They usually expect you to be fluent in higher category theory and its applications to geometry, yes

bolbteppa

"In this generality, the gauge principle of physics is the mathematical principle of homotopy theory: in general it is meaningless to say that some objects form a set whose elements are either equal or not, instead one has to consider the groupoid which they form, whose morphisms are the equivalences between these objects."

Slereah

Do you think that the nlab people would get angry if I showed them the set theory axiomatization of category theory

ACuriousMind

15:12

@Slereah They would just counter with the category theory axiomatization of set theory :P

Slereah

Although to be fair I don't think you can axiomatize higher category theory in ZFC

ACuriousMind

Category theory in general is not really in standard set theory because the morphisms get "too big"

Slereah

you'd probably need Von Neumann–Bernays–Gödel set theory or something

bolbteppa

One positive of this discussion is I found a Dirac paper on constraints which probably forms the basis of his little book on this stuff

ACuriousMind

If you want to pretend everything is a set the most popular trick is to have a sequence of nested Grothendieck universes

bolbteppa

15:14

Inverse limits, there's a set thing I still have no idea about

ACuriousMind

But in reality, the most popular trick is to not care at all :P

Slereah

The most popular trick of set theory for the average person is to pretend sets are bags

Put the banana in the bag

your set has now a banana

ACuriousMind

what if my set eats the banana

Slereah

That would be the empty function

people would probably deal with ZFC more easily if the average shopping bag could contain $\aleph_0$ items

The axiom of choice is then easy to illustrate

Is it a crime to answer to a PSE question by using the NPOV

"Why does a ball fall?"

"Let's consider the higher topos of quantum field theory describing a ball"

"A general context for spaces is a big (∞,1)-topos H\mathbf{H}."

It big

bolbteppa

15:30

I would love to see some basic projectile/inclined plane mechanics problems solved with this stuff tbh

Slereah

Famously Arnold's book uses fiber bundle for the pendulum

But that would be interesting

Take the toposiest prequantum field theory functor, and turn it into an actual solution of the pendulum

ACuriousMind

I think it's a bit like coordinate-free geometry - in the end, when you solve a concrete problem, you still have to pick coordinates and compute stuff

Slereah

Sure but you can do it easily

Just pick your basis of the tangent bundle and a coordinate chart

and bam

even better you can do one according to a specific observer!

I haven't seen anyone do a top-down derivation of anything using category theory like that

I should probably just look at like

A big list of simple topoi

bolbteppa

it would be a great way to make some of this stuff understandable tbh

Slereah

To get a good feel of the topos

yeah

ACuriousMind

15:38

This sounds perfect for another project I'll start and never finish :P

Slereah

@ACuriousMind Please use M-theory with the NPOV to describe a free fall

bolbteppa

'To solve this problem we need to consider the prequantum field theory of bundles over $\mathbb{R}$ with fibers ...????'

JingleBells

Does anyone know what the mechanism or device is called when for example in airplanes, there's a remote controller of some TV but it's not with Bluetooth but with a rollable cable that it clicks when you get it out fast... I can't explain it exactly

John Rennie

This is on the HNQ? Really?

JingleBells

then you have to move it fast to re-click it again so it starts rolling it up

ACuriousMind

15:41

@JohnRennie "Isn't physics wrong?!" is an all-time favourite.

Slereah

@ACuriousMind back in the days they just burned scientists as witches

much simpler

"A category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise."

nlab uses the word "big" also

JingleBells

Btw I'm ordering a 3D printer tomorrow. Big investment for me but I'll be doing it right instead of selling crappy recycled molded plastic

Slereah

are "big" and "large" two different notions

ACuriousMind

@Slereah usually yes - categories are large and small, while sites and toposes are big and small.

@JingleBells I don't think there's a proper name for these things, it's just a (automatic) rewind mechanism

PM 2Ring

15:57

@JingleBells The term for the mechanism that does that with car seat belts is called "inertia reel". That terminology is also used for safety lifelines.

@JohnRennie I already downvoted for lack of prior research &/or being written in a confusing way.

Slereah

I don't know why I still try to learn category theory to read nlab, outside of the belief that if I do, I will become a wizard

the manifold is a $(\infty, 1)$-topos, apparently

I guess I'll have to read about $(1,1)$-topoi, $(2,1)$-topoi, etc

it may take a while

ACuriousMind

16:37

argh

Slereah

there's that joke!

I guess a groupoid is a group where we're not imposing that the group operation is closed?

ACuriousMind

not really

at least I don't see how that works

I think of them more like state machines: At every object, a groupoid consists of a group (the automorphisms of that object) and some invertible ways to "transition" into a different group (the morphisms to other objects)

So it's really a collection of groups with invertible transitions between them

This fits nicely with the canonical example of the fundamental groupoid: It consists of the fundamental groups at every point, together with morphisms between them given by the homotopy classes of paths between the points

Slereah

Ah yes

I can see that

I should try to make an article mb

Category theory physics but not terrible

might be challenging considering that the starting point is the topos, and that's pretty far down the line

But I guess I can cut away all the stuff not useful for physics

John Rennie

16:59

From this week's New Scientist magazine:

Slereah

@ACuriousMind I am reading that the elements of a set are usually done as arrows on objects, but I have also seen it done as every element being an object

Is there like

an equivalence between the two?

some manner of unspooling

I guess maybe it can be done via the category of morphisms of a category

bolbteppa

He referenced the Yoneda Lemma in setting up the Euler-Lagrange equations in that pdf...

> In his Algebraic Geometry class a few years back, Ravi Vakil explained Yoneda's lemma like this: You work at a particle accelerator. You want to understand some particle. All you can do are throw other particles at it and see what happens. If you understand how your mystery particle responds to all possible test particles at all possible test energies, then you know everything there is to know about your mystery particle.

10

Q: Cayley's Theorem and the Yoneda Lemma

Hi, from wiki, I know that yoneda lemma is the generalization of Cayley's theorem. But I am not quite understand the intuition behind that. Anyone can help me with that? Cheers!

ct.category-theory

ACuriousMind

@Slereah I don't really understand how the latter is supposed to work - what are the morphisms between the objects?

Slereah

@ACuriousMind Depends on the category

bolbteppa

So from the fact a group is isomorphic to a group of permutations, if you know all permutations, you know your mystery particle...

Slereah

17:11

I recall seeing $\mathbb{N}$ as $0 \to 1 \to 2 \to \ldots$

Where the arrows are the successor

ACuriousMind

I'm not really sure what you're talking about. "The natural numbers" are usually an object, not a sequence

Slereah

I suppose, yes

Hm

How would you prove $1 + 1 = 2$ in category theory

I guess we have $1 : 0 \to N$ and $2 : 0 \to N$

And $s : N \to N$

And by definition $2 = s \circ 1$

Addition is a functor I guess?

ACuriousMind

You have to be a bit more precise what category you're working in here

Slereah

The category of natural number objects, I suppose?

ACuriousMind

That's not a thing :P

Slereah

17:19

How do you define $\mathbb{N}$ then?

ACuriousMind

"natural number object" is a special object in a category, not something a category can consist of

Slereah

what's the smallest category where you can do Peano's axioms then :p

ACuriousMind

Depends on the flavour you want :P You could for instance define the natural numbers as the free monoid on the singleton set

(this is a category-theoretic definition if you construe "free" as the adjoint functor to the forgetful functor from monoids to sets)

Slereah

@ACuriousMind I assume they are all equivalent

ACuriousMind

Or you could say it's the initial object in the category of rigs

Slereah

17:25

Rigs?

ACuriousMind

rings, but the additive part is only a monoid and not a group

Slereah

ACuriousMind

These definitions are obviously not "equivalent" because they talk about two different objects in different categories

Slereah

But aren't they isomorphic in the category of categories???

ACuriousMind

But they are consistent in that the forgetful functor from rigs to monoids that forgets about the multiplicative structure maps the "natural numbers" in the rigs to the natural numbers in the monoids

@Slereah Certainly not, rigs have more structure than monoids and hence will heuristically have "more" objects and "fewer" morphisms

Slereah

17:29

alright

I wish there was the category equivalent of metamath

I'm trying to find a proof that 1 + 1 = 2

Is it so much to ask

ACuriousMind

Yes, you first have to define what you mean by that :P

Slereah

link.springer.com/article/10.1007/BF00370829

could be this

ah yes, he defines sequences of objects there

this seems to be addition

Now how to actually perform an addition...

I can kinda see how it may work?

Not sure

I'm not even sure, actually

I'm guessing it's gonna be something of the form $(1,1) \circ + = 2$ via some diagram chasing, but those don't have the correct types

Slereah

18:22

@ACuriousMind I need your help

Take this gun and shoot me if I ever try to learn category theory again

ACuriousMind

lol

Slereah

Reading category theory makes me realize how much we take set theory for granted even though we rarely use it explicitely

They should make a Simple Nlab like how wikipedia has a simple wikipedia for dumdums

Qmechanic

18:40

Abstract nonsense

In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, “abstract nonsense” may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself. == Background == Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content. As a result, mathematical proofs that rely on category-theoretic...

Slereah

It's not really specific to category theory

You get the same sort of feeling if you do say, lambda calculus instead of set theory

"An $(\infty,2)$-category is the special case of $(\infty,n)$-category for $n=2$."

Can I punch nlab

bolbteppa

You're not allowed to do Lagrangian mechanics until you've framed the Euler-Lagrange equations in terms of Left Kan extensions and applied the Yoneda lemma P7

Slereah

@bolbteppa That's the easy part, they didn't even include constraints!

Hm

Maybe something I should do to help is like

Write down the entire diagram of a category

ie write down all the relevant objects and morphisms for say, $\mathrm{Smooth}$

the category of smooth jazz

bolbteppa

"In particular, the physics literature is secretly well familiar with smooth∞-groupoids in their infinitesimal incarnation as $\infty$-algebroids: these are equivalently what in physics are called BRST complexes. What are called ghosts in the BRST complex are the cotangents to the space of equivalences between objects, and what are called higher order ghosts-of-ghosts are cotangents to spaces of higher order equivalences-of-equivalence"

Obviously the fermionic path integral representation of a determinant involves the cotangents to the space of equivalences between objects

Slereah

18:55

double ghost

very spooky

"Smooth manifolds are equivalently the 0-localic CartSp-generalized schemes of locally finite presentation."

Category theory is basically like deep sea marine life

It evolved away from normal math for too long and now it's barely recognizable

ACuriousMind

I think the problem is that you also not familiar with its "precursors" like algebraic geometry. For instance, "schemes of locally finite presentation" is not a notion unique to abstract category theory, it also occurs in algebraic geometry

Slereah

Could be, yes

Not really a point in its favor, though!

bolbteppa

Homological algebra really seems to be the thing that started category theory and algebraic geometry sent it into marine world

Slereah

It's independent from algebraic geometry so that it shouldn't be necessary to know it

ACuriousMind

And a scheme is locally a spectrum of a ring, just like a manifold is locally a Cartesian space. If you were an algebraic geometer that had never learned anything about differential geometry, you would probably find the "smooth manifold" part of this sentence the inscrutable one! :P

bolbteppa

19:05

Algebraic geometry is also impossible

22

Q: "Pick up a homological algebra book and prove all of the theorems yourself" (exercise from Lang's Algebra)

There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in the third revised edition, and I'm wondering if it's still in the third revised edition, if it's o...

homological-algebra soft-question books

ACuriousMind

(and you would find this actually a nice analogy to get started thinking about manifolds :P)

Slereah

@ACuriousMind it like tiny $\mathbb{R}^n$

it would help if I could find something that was non-abstract

like studying the same category all the way through

bolbteppa

19:19

7 page summary by Dirac of his constraint theory!

Also skimming this Dirac paper on applying it to GR, looks very simple compared to what you'd think

> The exact Hamiltonian for the theory of gravitation, given by equations (28), (40), (41), turns out to be rather simpler than one might have expected. One starts with ten degrees of freedom for each point in space, corresponding to the ten $g_{\mu \nu}$, but one finds with the method here followed that some drop out, leaving only six, corresponding to the six $g_{rs}$.

> This is a substantial simplification, but it can be obtained only at the expense of giving up four-dimensional symmetry. I am inclined to believe from this that four-dimensional symmetry is not a fundamental property of the physical world

That really is a big leap

> The present paper shows that Hamiltonian methods, if expressed in their simplest form, force one to abandon the four-dimensional symmetry. From the mathematical point of view the loss of four-dimensional symmetry is to be regretted merely because it means a loss of transforming possibilities in the equations. It is amply compensated for by the increase in transforming possibilities arising from one’s being able to make contact transformations in the Hamiltonian equations

Apparently there are issues with what he did but not sure, simply amazing how he's able to get results that could, you know, question Einstein's basic assumptions

ACuriousMind

Dirac is just making the well-known argument that "gauge symmetry is not a real symmetry", just for GR instead of a generic gauge theory and a bit melodramatically :P.

Which is funny because we often use "you can keep manifest Lorentz invariance" as an argument for gauge theory in different contexts, not against it.

bolbteppa

19:42

You mean being able to do coordinate transformations is not a 'real' symmetry?

> One gets a simpler scheme by choosing the system of co-ordinates $x^{\mu}$ such that the surfaces $x^0$ = constant are all space-like, and then considering only states on these surfaces and using the co-ordinates $x^1, x^2, x^3$ as parameters to label the points on these surfaces. The symmetry between the four $x$’s is completely destroyed, but the resulting simplification makes this sacrifice well worth while.

ACuriousMind

I think Dirac is just getting over-excited about what is essentially the ADM formalism

Slereah

The ADM formalism is very exciting

ACuriousMind

@bolbteppa Also, note that coordinate transformations are not the same as the gauge symmetry in GR, no matter how often you will read this confusing claim. Cf. physics.stackexchange.com/q/346793/50583 and its linked posts

bolbteppa

Well if so, that paper was from 1958 and ADM "was first published in 1959" :p

I don't really really understand why coordinate transformations are so different to a gauge transformation of a field, I get why people confuse this stuff

Slereah

Depends how you define "gauge transformation"

They are certainly transformations that leave the action invariant

ACuriousMind

19:56

@Slereah Yes, but every proper physical theory has an action that is invariant under coordinate transformations.

Slereah

@ACuriousMind What about Aristotle's physics

@ACuriousMind Also the fact that it is so doesn't mean it's not important!

ACuriousMind

GR is special because it is invariant under the gauge transformations that act like a coordinate transformation on all dynamical fields, but don't actually transform the coordinates. You don't see this in the notation because we often suppress the dependence on the point

Slereah

The tetrad rotation you mean?

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