**Part I**

Let $f: X\to Y$ be an arbitrary map between topological spaces/simplicial sets $X$ and $Y$. Then we can associate with $f$ a tower of fibrations $$ \dots \to F_i\to\dots \to F_1\to F_0=Y $$

constructed in a following way:

0.let $F_0 = Y$ and we take $C_0$ as homotopy cofiber of $f$

1.define $F_1$ as homotopy fiber of $Y\to C_0$ and since composition $X\to Y\to C_0$ is null-homotopic we have a map $X\to F_1$. $C_1$ is a homotopy cofiber of this map.

2.$F_2$ is a homotopy fiber of $Y\to C_1$ and so on.

Therefore we got a tower of principle (under some assumptions on $\pi_1$-action I believe) fibrations $$ \Omega C_i\to F_{i+1}\to F_i $$ together with collection of maps $X\to F_i$. Many questions about this construction can be asked. What does this tower tell us about $f$? About $X$ ? Under what conditions on $f$ $X\to \lim F_i$ is an equivalence ? etc

**Q1:** I'm sure such construction have been studied before, where I can find any information about it?

**Part II**

Actually, Postnikov tower for a space $X$ is constructed in a very similar way, with one more additional step: instead of taking homotopy fiber straight away, we kill homotopy groups of cofiber using fundamental class.

In particular, we start with $X\to K(\pi_1 X,1)$, take its homotopy cofiber $C_0$ and define next step of a tower as a homotopy fiber of composition $K(\pi_1 X, 1)\to C_0\to K(\pi_3 C_0, 3)$. Here $C_0$ is $2$-connected and $\pi_3 C_0=\pi_2 X$.

**Q2** What is the ``phylosophical'' meaning of taking this additional step $C_n\to K(\pi_{n+2} X,n+3)$ ? What will happen with Postnikov tower if we will not kill homotopy groups of cofiber, apart from losing nice description of $k$-invariants as cohomology classes ?

P.S. In general, I would like to compare two constructions (without killing homotopy groups of cofiber and with it) and to understand, which one is more ``canonical''. Conjecturally, taking different morphisms $f$ (for example $X\to \Omega^{\infty}\Sigma^{\infty} X$) we will get towers, that a related to already known constructions (such as Goodwillie tower, for example).

**EDIT**

Actually (if I'm not wrong), if $X$ is simple, we can start the Postnikov tower for $X$ straight away from a map $X\to *$. Fundamental class of $\Sigma X=\mathrm{hocofib\{X\to *\}}$ is then represented by $\Sigma X\to K(\pi_1 X, 2)$ and $X_1=\mathrm{hofib}\{*\to K(\pi_1 X,2)\}=K(\pi_1 X,1)$. If we apply my construction to a map $X\to *$, then as $F_1$ we will have $\Omega\Sigma X$, instead of $X_1=K(\pi_1 X,1)$ in a usual Postnikov tower.

In general, comparing two constructions, we have a map $F_i\to X_i$, in the example above in the level one this is $\Omega\Sigma X\to K(\pi_1 X,1)$, which is an isomorphism on $\pi_1$.