On the Leray spectral sequence and sheaf cohomology I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two topological spaces (say as nice as we want) spaces, I would like to understand how to construct the $E_2$ term $$E_2^{pq} = H^*(Y,\mathcal{H}^*),$$ where $\mathcal{H}^*$ is the pre-sheaf $U \mapsto H^*(f^{-1}(U))$. I know a bit about sheaf cohomology, but honestly I don't really understand how this pre-sheaf appears, or why/if we need to look at its sheafification. 
I have several general questions regarding all this, some being even more general:


*

*Why are we always talking about the $E_2$-term and not the $E_1$-term of a spectral sequence?

*From where to I need to start in order to construct this $E_2$-term ?

*Suppose that there exists an open cover $\{ U_i \}$ of $Y$ such that $\mathcal{H}^*(U_i) = \mathcal{G}^*(U_i)$, where $\mathcal{G}^*$ is another pre-sheaf (for instance the constant one). Under what condition could we have: $$H^*(Y,\mathcal{H}^*) = H^*(Y, \mathcal{G}^*) ?$$


Does anyone have a nice reference to these concepts (important to note that I'm no specialist in algebraic geometry :))
Thanks a lot for your help !
 A: Edited Here's a  quick slightly expanded answer to the first couple of questions. First, pick a coefficient sheaf $F$, e.g. a constant sheaf for what you seem to want. To compute sheaf cohomology $H^*(X,F)$ choose a $\Gamma$-acyclic (e.g. injective, flasque, or, in good cases, fine) resolution $I^\bullet$ of $F$, then
$$H^i(X,F) = H^i(\Gamma(X, I^\bullet))$$
Alternatively, resolve $f_*I^\bullet$ on $Y$ by acyclics to get a double complex $J^{\bullet\bullet}$. (You can either use a Cartan-Eilenberg resolution, or  Godement's canonical flasque resolution; see Weibel's Homological Algebra for the first, or Godement's Theorie de Faisceaux for the second.) Then apply $\Gamma(Y,-)$. One of the two spectral sequences for this double complex is
$$E_1= H^{p+q}(\Gamma(J^{p\bullet}))\Rightarrow H^{p+q}(Tot(\Gamma(J)))= H^{p+q}(X, F)$$
This can be shown to give the the Leray spectral sequence from the $E_2$ page. So there is an $E_1$ page but as you can see it is complicated and  not so useful. 
Additional Notes:  


*

*I'm really just repeating what TKe said in somewhat more explicit fashion.

*Under appropriate conditions (e.g. if X & Y are paracompact), you can do this with Cech cohomology, but it's going to be messy since you need to choose a cover on $Y$ and then refine the preimage.

*Your question 3 has a negative answer: let $Y$ be a wedge of two  circles, $f$ be the identity, $\{U_i\}= \{Y\}$, $F=\mathbb{Z}$, $G= F\oplus L$, where $L$ is a  locally constant sheaf with nontrivial irreducible monodromy. Then $F$ and $G$ have the same sections on the cover but the $H^1$'s differ.

