/sci/ - Science & Math

>>9398876
Thanks. I could try to make myself comfortable with manifolds the next spring. You mentioned geometry. Which kind?

HoTT :^)

Maybe categories, but only if you already know stuff. What you seem to be looking for, atleast to me it seems so, is something analogous to the idea that certain statements are independent of context: for any two contexts in which you may say something, this something is true in either iff it is true in both of them. This kind of an idea is there, an underlying idea in CT. The category you work in is your context, but the universal language is that of CT.

It is also easy to study it by yourself, but without a proper background in different fields of math, it is quite dull and lacks depth. Luckily, you can easily have this fixed by studying some topology and algebra, both easily studied by oneself. Maybe a bit of logic, too.

A quick remark, though. CT doesn't solve all the problems by itself, nor does it give you any immediate results. It is not suitable for all situations (my guess is that analysts dislike it because their stuff can break down in composition), and it should be used as a tool instead of considering it to be the endgame of math.

Have fun!

>>8755642
This, then, induces cochain complexes

[math]0 \to C^0(X; G) \to C^1(X; G) \to \cdots[/math]

and

[math]0 \to C^0_k(X; G) \to C^1_c(X; G) \to \cdots[/math].

Defining things the usual way, one gets [math]H^k(X; G)=H^k(C^*)[/math] and [math]H^k_c(X; G)=H^k(C^*_c)[/math].

That this works for the case with compact supports follows from the fact that [math]\text{supp}(\delta^k\varphi)\subset \text{supp}(\varphi)[/math].

I think this is cool, and more intuitive than the construction of for example singular cohomology.

>>8474435
Thanks!

>>8448000
Draw pictures. It makes several claims pretty intuitive, for example how the Möbius strip can not be embedded into R^2. An example fit for this thread now that it has been mentioned several times.

Easily. No analysis is required for point set topology. Just know a bit of set theory.

It is not poor nutrition that causes it. I rarely eat anything, but I can say for sure the increase is (400-250)/250*100%=60%.