$\newcommand{\Fib}{\mathrm{Fiber}}\newcommand{\map}{\mathrm{Map}}\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\deloop}{\mathbf{B}}$

The main trouble is that ordinary homology has a certain bluntness to it. Of course this is partly due to the relative efficiency of Smith’s Algorithm: if it’s easy to collect a system of generators for the kernel and the image of a differential, then it’s easy to reduce the (Abelian) group presentation to a sum of prime-torsion cycles; in full generality, homotopy is not computable. There is a rich hierarchy between the easy and undecidable, whose details remain for the most part mysterious.

A fibration (and every map factors as $ f \circ c_0 $ for a fibration $f$ and cofibration-equivalence $c_0$) has locally-equivalent fibers. Or, better,
$$ f : A \to B \vdash A \simeq \sum_{b : B} \Fib( f \vert b) $$ where $\Fib$ is the homotopy-universal choice of a space for every point of $B$ generated by the tautological preimages under $f$. In coq style, we might say

```
Inductive Fiber { A B : Space } (f : A -> B): B -> Space :=
tauto (a : A) : Fiber f (f a).
```

In any case, however, it has been found useful to study relations among the homologies of $A,B$ and the fibers of whatever map $f : A \to B$.
Now, the ordinary homology of a space $X$ can, by (If I Recall Correctly) Dold+Thom, be *defined* as the homotopy groups of the abelian topological group generated by $X$: that is, as the colimit of (ordinary) quotients $X^n / \mathfrak{S}_n$ having group operation defined by the natural maps $ X^n\times X^m \to X^{n+m}$. Since I’m not sure how to make the ordinary quotient a sensible thing in homotopical language, and I don’t even want to *think* about smash products of (connective) spectra, I will prefer to work here with *cohomology*; but for a homological hint at the general idea, note that a finite sum of (signed) points of $\sum_{b:B} \mathrm{Fiber}(f|b)$ “is” a sum finite sums of (signed) points of $\mathrm{Fiber}(f|b) $ over some finite set of points $b : B$.

To discuss the cohomological side of things, it helps to study the delooping functors just a little. Particularly, for whatever pointed space $X$, the $n$-fold loop-space is $n$-times deloopable, and furthermore $\deloop \Omega X$ is the component of the basepoint of $X$:
\[ \deloop \Omega X\to X \] Similarly, the second delooping of the second loopspsace of $X$ maps to the basepoint component
\[ \deloop^2\Omega^2 X \to \deloop \Omega X \] as a universal cover of a connected space; in particular, the fiber is $\pi_1 X $. More generally, the composable chain of maps
\[ \cdots \to \deloop^{n+1} \Omega^{n+1} X \to \deloop^n \Omega^n X\to \cdots \deloop \Omega X \to X \]
are simply the “upside-down” Postnikov tower, or Moore-Postnikov tower of the basepoint inclusion. At each stage we have a fiber sequence
\[ \deloop^{n+1}\Omega^{n+1}X\to \deloop^n \Omega^n X \to \deloop^{n} \pi_n (X) .\]

One other handy bit of niftiness, the fiber of a product map is the product of the several fibers: if $f_b : F_b \to G_b \to E_b $ are a continuous family of fiber sequences (meaning that the spaces $G_b$ and $E_b$ are continously chosen, and the maps $G_b\to E_b$ and the basepoints $*\to E_b$ as well) then so is
\[ \prod_b F_b \to \prod_b G_b \to \prod_b E_b \]
a fiber sequence, at the basepoint section. This was already used implicitly in the recent note on special local systems, to obtain a long exact sequence of relative depedent homology groups.

Now, the title of the present note mentions “Leray-Serre spectral sequences”, which are about the relations between the cohomologies of $F,X,B$ for a fiber sequence $F \to X\overset{f}{\to} B$. When $B$ has several components it makes more sense to talk of the relations among $X$,$B$ and *all the fibers*, but what comes down to the same thing, we migh as well suppose $B$ is connected. So, fix a representable cohomology theory $E$, and consider all the long fiber sequences of dependent spectra over $B$:
\[ \deloop \map(F_b,E_n) \to \map(F_b,E_{n+1}) \to H^{n+1}(F_b,E) \to \deloop^2 \map(F_b,E_n) \to \deloop (F_b,E_{n+1}) \to \cdots \]
By the product fiber, this leads to a long fiber sequence
\[ \prod_b \deloop \map(F_b,E_n)\to \prod_b \map(F_b,E_{n+1})\to \prod_b H^{n+1} (F_b,E)\to \cdots \]
\[ \prod_b \deloop^k\map(F_b,E_{n+1}) \to \prod_b \deloop^k H^{n+1} (F_b,E) \to \prod_b \deloop^{k+2} \map(F_b,E_n) \]
The long exact sequences of component groups now gives a spectral sequence whose first discernible page consists of (finite!) chain complexes
\[ H^0(B,\mathcal{H}^{n+1}(F_b,E)) \to H^2(B,\mathcal{H}^n(F_b,E)) \to \cdots \]
\[ H^1(B,\mathcal{H}^{n+1}(F_b,E))\to H^3(B,\mathcal{H}^n(F_b,E))\to \cdots \]
and converging to the graded complex of $ H^n(X,E) $ associated to space-of-origin filtraiton in the sequence
\[ \prod_b \deloop^n \map(F_b,E_0)\to \prod_b \deloop^{n-1} \map(F_b,E_1) \to \cdots \]