What Information/Advantage do we Gain by Substituting a Continuous Map by a Fibration?

I'm trying to understand the usefulness of "substituting" a continuous map f , by a fibration F. By substituting, I mean there is the result that given a continuous map $f:X \rightarrow Y$ , for any topological spaces $X,Y$, one can define a new space $X'$, and a fibration $F : X' \rightarrow Y$ , so that that the maps $f, F$ (together with the identity map $id: Y \rightarrow Y$) satisfy the commutative square $$\newcommand{\ra}{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} X' & \ra{F} & Y \\ \da{d} & & \da{=} \\ X & \ras{f} & Y \\ \end{array}$$ I know that once we have a fibration $F: X \rightarrow Y$ , we have two new conditions satisfied by $F$ that are not satisfied by $f$:

i) The fibers of $F$ are homotopically-equivalent to each other, and,

ii) A fibration $F$ gives rise to a long-exact sequence of homotopy groups.

So we may gain some information about homotopy groups from ii). Still: what can I really learn about $f: X \rightarrow Y$ from this associated fibration ? How is this associated fibration useful in general?

• Well, a minor point is that it is part of one of the necessary axioms in order for $\mathbf{Top}$ to be a model category en.wikipedia.org/wiki/Model_category. Jan 30 '14 at 0:39
• I hope you don't mind, I added in the commutative square you mentioned (and called the canonical deformation retraction $d$). Jan 30 '14 at 0:47
• No problem, Daniel Rust; thanks, actually. Jan 30 '14 at 0:57
• @HansLundmark: Thanks ;). Feb 27 '14 at 0:46

2) The compulsion to replace a map by a fibration (or cofibration) is explained well on: http://math.uoregon.edu/~ddugger/hocolim.pdf page 6 where Dugger explains the more general situation of homotopy colimits. This is related because, basically, one way to compute homotopy limits is to replace maps by fibrations (actually he discusses homotopy colimits but the theory is dual) first in order to get a limit that is homotopy invariant and tractable" via the methods of homotopy theory.
3) Point ii) really buys us quite a lot: say I want to show that two CW complexes $E$ and $E'$ are homotopy equivalent. Then I am reduced to showing that they have isomorphic homotopy groups by Whitehead's theorem, and then I can pray that there is a comparison map between the two spaces and that they each sit in fiber sequences --- standard homological algebra techniques then apply.