@ACuriousMind You can see GR by looking at the sky.

@0celo7 I don't care in the least who the average person has heard of. I don't take my cues from them.

@0celo7 oO Where?

@ACuriousMind Precession of Mercury?

@0celo7 But...it's difficult to even explain to people wtf that has to do with space or spacetime being curved!

@dmckee That's a non-answer.

@ChrisWhite What's your impression on the Hilbert thing?

@ChrisWhite You sure it says astronomer on your job description? ;)

@0celo7 Look at it this way: I know that the best painter of the impressionist era was Gustave Caillebotte, without a doubt and bar none.

And yes, I've been to Paris and seen the Degas and the Monets and the van Goghs. They are breathtakingly brilliant and still second best.

So why would I can about Joe Q. Random's knowledge of mathematicians?

@dmckee All of my wat :P

They're much less... "dreamy"

different strokes, I guess

(see what I did there?)

@ChrisWhite There are a couple at the art Institute in Chicago, too. So you don't have to get all the way to Europe.

But you should go all the way to Europe :D

@Danu Heck, yeah.

But do not go to Amsterdam's Van Gogh museum---it's really not that great.

@ChrisWhite and since you are in europe, there are many better painters to see ;-)

I really like some of the expressionist paintings that we have here in Munich: With the Blaue Reiter movement that was based here there are some really really nice things to see: Lots of stuff by Marc & Kandinsky, for instance.

@yuggib Fan brushes at dawn!

@dmckee what does that mean?

@dmckee What??

What is the dual of a Hilbert

@ChrisWhite topological or algebraic?

Is there a difference?

@0celo7 the same Hilbert

^that

(topological dual)

Seriously, what's the difference

Anyway, I have to go administer a test and then talk about the kinetic theory of gasses.

you once explained the dual of a top vec space

it was the exact same thing as an alg vec space

or something

what is an alg vec space

@dmckee At 6PM?

@ChrisWhite :-D if you are willing to replace dual with adjoint, then functor is an option as well

@0celo7 It's not hard. The dual is the set of morphisms from the space into the base field. The algebraic case has as morphisms linear homomorphisms, the topological case has linear continuous homomorphisms.

whenever you say "it's not hard"

I prepare for the worst

@ACuriousMind Is there a category of abelian groups?

@0celo7 probably there are more than just one

it depends on the arrows you take

@0celo7 Yes

@yuggib what

@yuggib It's pretty unique, "the" category of abelian groups is the one with group homomorphisms as morphisms

@0celo7 a category has objects and arrows (morphisms)

@ACuriousMind it has not to be unique...but I agree that this is the most used one

@yuggib I know

I don't do PhD level category theory

but it seems not useful to introduce a category without specifying the arrows...you could as well call it a set/class

@yuggib Not sure which other one would see use.

@yuggib yes

Lee defines $\mathsf{Ab}$ as abelian groups and homomorphisms

ah, so a functor is a map between categories

and a natural transformation is a map between functors

so why are the morphisms of the abelian groups homomorphisms

@yuggib Or "a 2-morphism in the 2-category of categories" ;)

@0celo7 it is a choice compatible with the definition of category?

@ACuriousMind exactly ;-)

but I have to say that these things seem less worth to me than large cardinals and set theory :-P

@0celo7 Well...the morphisms in a category should be thought of as maps that "preserve the structure" you are interested in. The group homomorphisms are precisely those maps that preserve the group structure, so they are the natural choice for the morphisms of the category

Ok, what exactly does it mean that a homomorphism preserves the group structure?

I see that thrown around but I've never really understood it.

@0celo7 It's kind of a tautology ;)

@ACuriousMind ?

A map "preserves the group structure" if it does what a homomorphism does - commute with the group operation: It doesn't matter whether I first multiply two group elements and then apply the map or if I first apply the map and then multiply the two images

(If you consider a group to be itself a category with one element, a group homomorphism between two groups is precisely a functor between their two associated categories, but I think one only finds that appealing if one has already drunk the kool-aid of category theory ;) )

I was just looking for the shortest way to define a vector space :o

...wat

something like $(V,+)\in\mathsf{Ab}$

If the word "category" appears anywhere near that, you're looking at the wrong place :D

until now I never appreciated the difference between "category" and "set"

@0celo7 a vector space has two operations

@yuggib yes, scalar multiplication

and the field has two operations

so it cannot be just an abelian group

I know

Obviously, a vector space is an object in the category of vector spaces :P

when did I say that>

@ACuriousMind ...you're insane

@ACuriousMind of course

what is the category of vector spaces

The category of vector spaces is really boring, though. It's not ugly enough to be interesting

vector spaces + vector space isomorphisms?

@0celo7 Exactly.

@ACuriousMind that's like first semester linear algebra boring

even my shitty class did that

Wait, no, not isomorphisms

All linear maps (or vector space homomorphisms) are allowed morphisms.

The isomorphisms are just the, well, isomorphisms :D

vector spaces are boring if they do not have infinite dimensions

@yuggib The infinte-dimensional ones only become "interesting" with a topology, I think.

it depends

you can define a unique (up to *-isomorphisms) Weyl algebra starting from (real) vector spaces with a symplectic form

@ACuriousMind what's a vector space homomorphism

but no topology

@0celo7 A linear map.

@ACuriousMind no shit

@yuggib Okay, but you need some added structure.

don't know why I asked that question

oh, why are those the morphisms

^ that's a case of fancy terminology to say a trivial thing

@ACuriousMind a symplectic form is a "trivial" structure :P

@0celo7 Again, because you define them to be - and they are the natural choice because they "preserve the vector space structure".

@yuggib Tell that to the Hamiltonian mechanists/symplectic geometers ;)

> "preserve the vector space structure"

What does that mean

the linear map of a vector space is a vector space?

@0celo7 Uh, no, it's the same idea as in the group case - it doesn't matter whether you first apply a linear map to two vectors and then add the images, or whether you add two vectors and then apply the map (and the same for scalar multiplication)

@ACuriousMind :-P

That this "doesn't matter" is precisely the definition of a linear map.

@ACuriousMind I saw a stack of these journals on one of my prof's desks

@ACuriousMind ...ok

next one

$\mathsf{Top}$ are the topological spaces and the continuous maps

why

Well...maybe you begin to sense a pattern here ;)

What will my first response here be?

IN WHICH CATEGORY

Uh, no

What have I responded to the question why the morphisms in $\mathsf{Ab}$ and $\mathsf{Vect}$ are what they are?

something something preserves the topological stucture

Exactly!

but I don't know what that means

@0celo7 definition of continuous map?

(between topological spaces)

open sets --> open sets

the preimage of open sets ....

@yuggib is open

so?

what was that face for, Bajoran?

what characterizes the topology?

@0celo7 open sets ---> open sets is wrong, if taken in the naive sense.

A continuous map is not necessarily open, and an open map not necessarily continuous.

@ACuriousMind I've been meaning to ask that...why?

@yuggib the open sets

@0celo7 Because the image of open sets isn't always open under a continuous map?

so a continuous map preserves the topological structure, for open sets have open sets as preimages

Just take your favourite bijective continuous map that isn't a homeomorphism to see that.

@ACuriousMind really

@ACuriousMind don't have one :(