/sci/ - Science & Math

>>8755694
That's algebraic topology. You take a topological space, you try to do something using the tools of topology, realize it's too hard, and you have two choices: give up and get depressed, or try solving your problem using a bit of algebraic structures.

Assuming you chose not to give give up, you would then have a nice arsenal of different methods at your disposal. Among these you have homology and cohomology theories that share the core idea but differ otherwise. In a homology theory, you would construct an algebraic structure (groups, modules, vector spaces) for every non-negative integer n such that your construction somehow resembles the notion of dimension, or n-dimensional holes. This allows you to sort of decompose the space into a chain of algebraic objects. You will also have a fixed map from the object of dimension n to the object of dimension n-1 such that the composite of any two subsequent maps takes everyting to 0. This sounds complicated but makes proving stuff a lot easier.

Cohomology is the same in the sense that you, again, construct an algebraic structure for every non-negative n, but this time the associated maps go from n to n+1, like with Alexander-Spanier. This is more complicated than homology, but also yields more powerful results.

I hope this clarifies things a bit. It's hard (for me) to explain this properly when you don't know topology or algebra.

>>8755705
That you don't want to discuss it doesn't mean no one would.

>>8402369
It has firepower. It has a lot of firepower. Yet, it is kinda like the abstraction of [math]Hom( \mathbb{Z} , G) \cong G[/math] for abelian groups. [math](^{*} \smile ^{*})[/math]

>>8284825
>hey, let's fuck up the ecosystem of some island no one gives a shit about and see what happens
Sounds fun af desu