/sci/ - Science & Math

I remember when you posted this as a challenge to /sci/, implying that those who couldn't solve it were lacking intellectually.
Your deception has afforded you no help from me.

>>4859620
If it would be a quiz, I'd answer a sufficient condition is for B to be the trivial group, harhar.

No idea for a real answer though, OP.
What are your ideas so far?

>>4859559
It's hard to even tell what you're looking for. What about the specific case where A and B have two connected components each, and the connected components of A and B are groups? Can you give an example of the type of conditions you're looking for for that case? If not, why do you expect there to be any useful criteria for the general case?

I'd try answering the question in the case of topological groups first and then only moving on to the case of topological groupoids.

>>4859620
I posted it before (no one answered) - I don't remember implying that though; if anyone's lacking here it's me.
>>4859632
Posted it on stack exchange and they recommend <span class="math">B[/spoiler] being skeletal and connected - I don't see how this is sufficient though...

http://math.stackexchange.com/questions/167264/groupoids-with-all-components-isomorphic

>>4859658
>What about the specific case where A and B have two connected components each, and the connected components of A and B are groups?
What prevents <span class="math">F[/spoiler] mapping one component of <span class="math">A[/spoiler] to one component of <span class="math">B[/spoiler] and the other component of <span class="math">A[/spoiler] to the other component of <span class="math">B[/spoiler]?

>It's hard to even tell what you're looking for.
>Can you give an example of the type of conditions you're looking for for that case?
Firstly, I want the condition to be as weak as possible - keeping <span class="math">A[/spoiler] and <span class="math">B[/spoiler] as general as possible - it's really <span class="math">F[/spoiler] I wanted to restrict. Stuff I would be looking for is; <span class="math">F[/spoiler] defined as a universal functor, full, faithful etc - that kind of shit.
>If not, why do you expect there to be any useful criteria for the general case?
I'm not sure there is any useful criteria at all.

Thanks for the help.

>>4859706
If Qiaochu Yuan doesn't have an answer for this kind of question, I'd not get my hopes up on /sci/.

>>4859831
Fuck.

Looking at groupoids as oriented graphs I'm sure one could come up with something. Groupoid morphisms look particularly nice when proceeding visually and your condition on F (images of components being equal) too. Looking at a few morphisms this way might yield a nice way of restricting F.

>>4859831
seriously does that guy ever stop internet-math-ing?

>>4859962
I'm well aware of the visualisation - it hasn't really helped me.
>>4859948
Him and Zhen Li answer nearly every question I post on stack exchange - they are fucking machines.

>>4859948
I'm aware of the visualisation - it hasn't really helped me.
>>4859962
Him and Zhen Li answer nearly every question I post on stack exchange - they are fucking machines.

Very interesting question, OP.

I'm clueless on it, but how did you even think of this?

So I don't think there will be an answer OP.
Maybe there is no good answer without any information on A and B.

Nevertheless, regarding your pic, I really like this:

http://www.youtube.com/watch?v=kya8TP9sTVA

also, check out ma pic, hehe

>>4859559
I have no idea what a groupoid is, but I like graph stuff.

So just for information to udnerstand the question:
A connected component is an equivalence set of vertices where there is a path from each one to any other, right?

What is meant with a component? Any possible subset of vertices?

>>4860037
Part of paper I am working on for summer project. Trying to make explicit the information captured by a transform. Long story - very boring.

>>4860071
>Maybe there is no good answer without any information on A and B.
Let me get back to you on that one - I'll see if I can put more in.

It's always amazed me that Stone looks so photogenic at all times.

>>4860096
>What is meant with a component?
Component is just short hand for connected component - in category theory anyway.

>>4860096
>Any possible subset of vertices?
One that is connected in the sense that everone goes back and forth form all the vertices.
The problem with thinking in graphs only is that there is a structure in the category which isn't visible if you just draw a picture. Namely which arrow u followed by which arrow v is which other arrow w. There are rules like <span class="math">u=v\circ u[/spoiler] for all the arrows u,v,w,x,y,z,... .
Also, and that's the main question here, a functor doesn't destroy the structure when he goes from one category to the other. Therefore, the question is difficult to answer without knowing A and B. There is probably no good general anwser (except for the one I mentioned, where there is only one dot with one arrow pointing back to its starting position.)

>>4860101
except for the ones in which she looks like a frog.
Nevertheless, I never expected her to be so young and good looking when I learned about Stones theorem on one-parameter groups.

>>4860096
again. As my knowledge of category theory is virtually nonexistant, I'll just intuitively guess in what ways it behaves like graph theory.

So let's just take a simple graph where a points to b but b does not point to a. They would be seperate connected components. Would they be isomorphic?

If not, then every component of A is completely seperated and you would get F(A) is seperated and that no two components of B could be isomorphic, else you could just map the components of A to different components of B.

Whelp, maybe I'm just totaly wrong. Should start to delve a bit deeper into algebra.

van kampen's theorem

>>4860133
>except for the ones in which she looks like a frog
Dammit I would have preferred to stay ignorant.

>>4860164
By components are isomorphic. I mean that if we consider the components as categories there's an invertible functor between them.

Bump.
>>4860172
No. Guess again.

>>4860334
Yes.

http://ncatlab.org/nlab/show/homotopy+hypothesis#for_groupoids_25

>>4860365
Still no.

Last bump

Forgive me, but what exactly do you mean by (F-)images?

Are you talking about homotopy or essential images? I'm assuming the former, right?

>>4860660
Essential image.