/sci/ - Science & Math

I just read the useful answer at http://math.stackexchange.com/questions/458975/help-understanding-cw-complex-construction, and it sounds like I was just misinterpreting the attaching maps as being indexed by (n-1)-cells. Is this correct? Are they being indexed by an arbitrary index? Hatching just leaves it "implicit," so I'm not sure.

Go ask stackexchange. /sci/ is just a bunch of undergrads masturbating to IQ threads.

>>7406934
I think I got rubber-ducked just by asking the question here. I have it now. I stopped when I got to characteristic maps before, because I didn't really see how he built up to it, but the characteristic maps made me see how the definition he gave in my screen shot their ends up defining the space we are CW-complex'ing by "realizing" every attaching map as a gluing of some disk.

Also, don't forget that /sci/ has some smart people. I'm going to start a /research/ thread, because people are always eager to share what they are passionate about! Have some Grothendieck and enjoy your day, anon.

>>7406912
The attaching maps are indexed by the n-cells. Think of it as \del D^n \to X^{n+1} rather than S^{n-1}. That's where you got confused.

We haven't had an AT thread in a while and this is as good a place as any considering OP's question has been answered.

So what is everybody else working on?

Personally, I'm working my way through Ravenal's orange book right now.

>>7407160
I am the OP; I have a good command of undergraduate topology, and as I have said, am now starting Hatcher. Where do you recommend I move from here if I am interested particularly in relationships between homotopy theory and motivic cohomology? (This is specific because I am having more troubles going into higher category theory without the topological motivations to accompany it.)

>>7407197
Higher category theory is one of those things that's not particularly interesting in and of itself. You shouldn't necessarily try to learn it for its own sake; learn it if it comes up.

Once you finish hatcher you should become familiar with k-theory, spectral sequences, localization, characteristic classes, and bundles in general. To that end Milnor and Stasheff is suprisingly readable as is Atiyah's book on k-theory.

After you've covered all the classical material you'll probably be reading paper's on homotopy theory if that's what you're interested in.

>>7407270
Cool, thank you for the suggestions! I took a screenshot to refer back to. What sorts of stuff are you studying in algebraic topology?

>>7406941
You should start that thread. I'm interested to hear what you're working on.

>>7407197
>I am having more troubles going into higher category theory without the topological motivations to accompany it.
There's this misconception that category theory only has topological motivations. This is false.

>>7407270
>Higher category theory is one of those things that's not particularly interesting in and of itself.
This is an opinion.

>>7407322
I was planning to do so when I get home; keep an eye out in a few hours.

>>7407329
I understand that there other motivations because I got into category theory from a formal logic perspective to begin with, but this doesn't change the fact that many good researchers approach higher cats from a topological perspective primarily. At least, so it seems.

>>7407329
It certainly is. Personally I find it to lack the spark of fun that much of other math has.
Also the divergence in definitions once you get past 2-categories is kind of annoying (from an aesthethic viewpoint).

Overall I find that whenever I need higher category theory, the result I need is self-evident in my particular case. Essentially it provides useful words, but their aren't many theorems that I can call upon to do some heavy lifting in a proof.

Of course it's still a branch it math, and some people may enjoy it, but I find that it contributes little in the way of results and more in the way of "viewpoint" to areas outside of itself. Thus, if your goal is to study something outside higher category theory, then it isn't particularly useful to know more than definitions and the general thread of results unless you need something in particular.

>>7407384
Are you trying to sound like you're saying something profound? There are many fields that contribute little in the way of results to other fields at the highest level, and only have interplay at lower levels. The fact that you'd think to say that just for category theory only shows how influential it is.

>>7407420
Profound? quite the opposite I am giving my own silly reasons for my somewhat irrational preferences. At the end of the day math is pragmatic, you use what works because it works.

>>7407420
I like the view that category theory is to mathematics as mathematics is to physics: it gives a strong framework in which one can study their field, and ties everything together in a very nontrivial way. I feel that category theory is incredibly interesting to study outright, because it lets us classify huge overarching patterns in mathematics itself as objects of study (universal constructions). I will not be surprised if mathematics looks very categorical fifty years from now, because this is the direction mathematics has been evolving in since Cantor and Hilbert.

>>7407469
>I like the view that category theory is to mathematics as mathematics is to physics: it gives a strong framework in which one can study their field, and ties everything together in a very nontrivial way.
That's fine, but notice that at the same time category is also a field of study on its own, just as people study pure mathematics with no (immediate) application to physics.

>>7407473
Of course; I never implied otherwise (au contraire, I am one who studies category theory in its own right). I am giving the reason that I study it this way. It feels very profound and fundamental to study category theory.