**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 $\pi\_1$.



Postnikov-type tower for a map between spaces
Here is an answer to your "Part I."


The construction you outlined (and its dual) was considered in detail in the papers of Ganea. In particular:


Ganea, T.
Induced fibrations and cofibrations. 
Trans. Amer. Math. Soc. 127 1967 442–459. 


(See Section 3 of the above for your case.)


For the dual question see my answer to: [Fibrations and Cofibrations of spectra are "the same"](https://mathoverflow.net/questions/177241/fibrations-and-cofibrations-of-spectra-are-the-same/177332?s=9|0.0833#177332)


