/sci/ - Science & Math

Well, you can just embed your sheet R^2 in your large space R^4, and however the line R^1 was lying on the sheet, after this the line will lie in R^4 somehow. Without having said much about what it means to "change an embedding", let me capture that idea by

forall k, n, m
h(R^k, R^n) -> h(R^k, R^m) <=> h(R^n, R^m)

>>7197070
Depending on what level you already are, why don't we try to clarify questions here, anyway?

I say this with caution, because I don't know if it helps or unhelps, but for intuition there is this informal introduction
http://ncatlab.org/nlab/show/Leinster2010

and a perspective which I really like, but which seems not completely mainstream and very geometric is
http://ncatlab.org/nlab/show/space+and+quantity

I've put together some notes on the latter, which I feel motivate the space argument very well - in particular, if you understand the contravariant hom functors action on arrows well, it should make the core message of the Yoneda lemma for categories of sheaves clear.

>>7199019
I'm not from Germany.
Thanks.
And what's that supposed to mean? I know what I want to learn in the future and I know I want to escape the grand application & publishing pressure hell avaiting everyone in academia. Just because I don't know at what firm I'll earn my one-two bucks in 6 months doesn't mean I'm unfocused.
(Btw. "categorical quantum mechanics" doesn't seem right, I'm somewhat interested in differential geometry in topoi. In any case, while cats are cute, I'm more interested in (>>7199136) kind of stuff.)

>>7199022
I wonder why algebraic geometry is so overrepresneted on other parts of the internet, though. MathOverflow in particular.
I think the interest in topoi proper, in your sense, is to the largest part motivated by the application of sematics for type theories now. So it's obscure from a math career perspective, ironically.