Sorry I haven't been able to get into Brown yet. I'm rather ill.
Hopefully soon I'll catch up with y'all.
@Narcissus Cellular is a tool to compute singular homology rather than an entirely new homology theory
But it's construction is rather curious. It's a "first order" approximation to a certain spectral sequence. I can try to elaborate if you're interested
@BalarkaSen Are you saying that there is some construction of the 'tool' cellular cohomlogy by means of the hypercohomology of some (single complex from a) double complex or something?
So say $X$ is a CW-complex. That means $X$ is a space obtained inductively as follows: you start with a bunch of points, which is the 0-skeleton, denoted as $X^0$. Then glue to it intervals, or 1-disks by maps $D^1_\alpha \to X^0$, to form the 1-skeleton $X^1$. Then glue to it 2-disks by maps $D^2_\beta \to X^1$, and so on.
This gives a filtration $X^0 \subseteq X^1 \subseteq X^2 \subseteq \cdots \subseteq X^n \subseteq X^{n+1} \subseteq \cdots$ of $X$.
So, what you ask is the following. Can you somehow compute the homology of $X$ just out of this filtration/CW-structure? In singular homology, you look at the chain complex $C_\bullet(X)$ whose chains groups are free abelian groups generated by maps from $\bullet$-simplices to $X$.
The drawback is that the chain groups are massive; it's impractical to compute anything from the singular chain complex.
There is a more concrete version of this in simplicial homology, however. If $X$ is a simplicial complex, you look at the chain complex $\Delta_\bullet(X)$ whose chain groups are free abelian groups generated by the literal $\bullet$-simplices of $X$.
You can easily compute the homology group of eg the sphere using this simplicial chain complex; give it a small simplicial structure (of a tetrahedron, say), and just write down the chain complex and the boundary maps
The idea of cellular homology is to do something similar. Instead of using the big ass singular chain complex, use a chain complex whose chain groups are free abelian groups on the $\bullet$-cells of $X$
Now, you can do this by hand, but the big theorem would then be to prove that this "cellular homology" is isomorphic to the singular homology groups $H_\bullet(X)$
Let's try an example. Consider the torus, with CW structure given by attaching two 1-disks to a point, and then attaching a 2-disk to the wedge of two circles obtained from that by sending the boundary of the disk to aba^-1b^-1
Now, I want to build a chain complex out of these. What are the boundary maps $d_n : H_n(X^n, X^{n-1}) \to H_{n-1}(X^{n-1}, X^{n-2})$, is the next question, of course
It's actually pretty simple. There's the snake map $\partial : H_n(X^n, X^{n-1}) \to H_{n-1}(X^{n-1})$ coming from the long exact sequence of $(X^n, X^{n-1})$. There's also the map $j_* : H_{n-1}(X^{n-1}) \to H_{n-1}(X^{n-1}, X^{n-2})$ coming from the long exact sequence of $(X^{n-1}, X^{n-2})$. Define $d_n : = j_* \circ \partial$
At least, that's what it algebraically looks like
You can check that $d^2 = 0$ holds no biggie. It's a commutative diagram argument y'all algebraists love and adore
But the bottom line is, you have this chain complex $$\cdots \to H_{n+1}(X^{n+1}, X^n) \to H_n(X^n, X^{n-1}) \to H_{n-1}(X^{n-1}, X^{n-2}) \to \cdots$$
You take homology of this chain complex to get the cellular homology $H^{CW}_\bullet(X)$ of $X$
What always fascinated me about this is that it's taking "homology of homology" in some sense. Indeed, what you can do is take "homology of homology of homologies ..." to get pages of successive homology groups arranged in a chain complex obtained from computing homology of a chain complex of homology groups in the previous page
And that giant book of informations is known as the spectral sequence of a filtration
Cellular homology is the first page of that spectral sequence
@Narcissusjewel Better to say, the "third page". First page is just full of LES's of the singular homologies, second page is homology of that, so full of LES's of cellular homologies, and so on
But yeah, all the pages give isomorphic homology theories
It's just that the second page, $E^2$, is extremely helpful in computation
I proceeded a little while blackboxing the Serre spectral sequences (which somehow gives a way to compute homology of a space $E$ when it's sandwiched between a fibration $F \to E \to B$, given homology of $F$ and $B$)
But I don't understand the essence of the theory. It feels more magical
