Mathematics

What is a family? I read on wikipedia that it's a collection of subsets of a set, so is it correct to say that a family is a set of subsets of a set? Perhaps I make things too complicated...is something more behind families?

Semiclassical

wee mathematica

Zee

4:26 PM

@Felix.Cdeoends on context but usually it means a collection

s.harp

A universal $G$ bundle is a $G$ bundle so that any other $G$ bundle is isomorphic to a pullback bundle over this bundle. Can this be expressed in a way that makes this seem like a universal property?

Mike Miller

it's a choice of representation of the functor that sends a homotopy type to its set of principal G-bundles

hTop -> Set

Typhon

@Fargle because that requires the assertion that the world is real. Nobody can claim that statement with 100% certainty. You say real world must be what is observed. I claim the lesser requirement of studying what is around them or the world they are a part of. After all, if there were a fictitious world, why wouldn't the residents of that world be justified in studying science?

s.harp

usually ive seen universal properties to be expressed via a diagram, for example the stone cech compactification is universal in that any map from a space into a compacta factorises uniquely via the stone cech compactification

isnt this usually the meaning of universal?

(also cofunctor)

Fargle

@Typhon Let me clarify, then: by "what we observe" I mean what comes to us through the senses, and therefore through experiment, and by "the real world" I mean that which can be said to exist independent of human observation.

Typhon

4:32 PM

@Fargle ah ok fair enough

Fargle

Science makes the tacit assumption that these two realms line up with one another.

Typhon

i was thinking that like if it was a computer real world would only refer to the outside world even if there is a world inside a computer

Fargle

Not necessarily.

s.harp

so like there is some diagram which only this guy can fufill

Fargle

Also, to clarify my own position, I think this is a perfectly fine assumption to make, but it is still answering a question of philosophy.

Mike Miller

4:33 PM

no

Typhon

@Fargle i never said it wasn't. i just said claiming its "realness" can be speculative. To some, what is imaginary is real. It is all relativity.

Mike Miller

i don't really see the point in trying to make this fit with some language you know when you know precisely what the space does

classifies G-bundles, dawg

s.harp

:D

Mike Miller

i'm really hungry

s.harp

its just my association with the word universal is so entrenched with diagrams

Typhon

4:36 PM

@MikeMiller then eat food :D

@s.harp universal means applicable everywhere

Mike Miller

i think that's bad

s.harp

@Typhon yes but ive rarely seen something called a universal construction that did not have some kind of a categorical definition via uniqueness of some such diagram

Typhon

@MikeMiller then go hungry

@s.harp im just referring to the word universal...

Mike Miller

but in any case form the category of spaces with free proper G-action. the projection to the quotient shows that this is the same thing as a principal bundle

maps are G-equicariant maps

in the language of G-bundles, the maps are maps f: X -> Y with isomorphism f*P' = P

then EG is the terminal object in the homotopy category of this

s.harp

siiiiick

Mike Miller

4:39 PM

"Every G-bundle has aj wqiicariant map to EG, unique up to homotopy"

But that's the same as saying that the pullback of the universal bundle is isomoephic tonyour bundle

and the uniqueness statement says that there's only one map to BG doing that, up to homotopy

Balarka Sen

What are we talking about?

Typhon

@BalarkaSen geometry IIIIN SPACE!

Balarka Sen

It's a beautiful result that [X, BG] = Vect_G(X). Please don't use the Vulgarization Functor on this

s.harp

whats the vulgarisation functor?

Balarka Sen

One which takes beautiful geometric theorems and turns it into obscene algebraic symbols

Functorially

s.harp

4:47 PM

so something like riemann roch gets turned into $\frac{\tilde\Xi}{\bar \Xi}$?

Typhon

sure

s.harp

its better if you write it on paper

Typhon

@BalarkaSen english please

:p

Balarka Sen

@Typhon Nyet.

s.harp

niemals!

Typhon

4:48 PM

nada

Fargle

@s.harp Would have made a great rap duo--they got bars

Typhon

-1

Q: Is there a multivariate integer function f(x,y) that returns the number of factors of y in x with a closed form?

I'm looking for a simple function in terms of the elementary and exponential functions that when passed an integer x and integer y, it returns the number of times that y can be divided from x such that the result is an integer. I believe that it is something along the lines of $\lfloor\log_y(x)\r...

help?

:p

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$Mathematics$ Mathematics