orthogonality of $n$-connected and $n$-truncated maps I am currently learning about factorization systems. One seminal example of such a FS in the category of spaces is the pair of ($n$-connected, $n$-truncated) maps. Here, a map of spaces $f:X \to Y$ is said to be $n$-connected (resp. $n$-truncated) if its (homotopy) fiber $F_f$ is an $n$-connected (rep. $n$-truncated) space, ie. $\pi_k F_f \cong 0$ for $k<n$ (resp. $k>n$).
My problem is that I cannot convince myself that this is indeed a FS.
The following assertions should be classical facts in homotopy theory, yet I am not able to prove any of them :

*

*The two sets of maps should be (strongly) orthogonal, ie. in any commutative square

$$
\begin{array}{ccc} 
A & \xrightarrow{} & B \\\
\downarrow{f} & & \downarrow{g} \\\
D & \xrightarrow{} & E 
\end{array}  
$$
where $f$ is an $n$-connected map and $g$ an $n$-truncated map, there should exist a lift $D \to B$ rendering the usual diagram commutative. This lift should furthermore be unique up to homotopy.
In fact, this property should completely characterise $n$-connected and $n$-truncated maps : $f$ should be $n$-connected iff the lift exists for any $n$-truncated $g$ (dually $g$ should be $n$-truncated iff the lift exists for any $n$-connected $f$).


*Any map $f:X \to Y$ should factor (again, uniquely up to homotopy) as $$
X \to |f|_n \to Y
$$
where $X \to |f|_n$ is an $n$-connected map and $|f|_n \to Y$ is an $n$-truncated map. Here I suppose one sould try to build the factorization using both the Postnikov section $X \to X_n$ and the $n$-connected cover $Y\langle n \rangle \to Y$ ...

Any clarification on these facts would be of great help !
 A: For the lifting properties, I think you can get some mileage by adjusting the definitions.  Warning: I always get the index $n$ in $n$-connected and $n$-truncated wrong, so be mindful of possible off-by-one errors.
First, some generalities.  Let $I$ be a class of maps in the category of topological spaces.  Define:

*

*$I$-inj is the class of maps with the right lifting property with respect to all maps in $I$.

*$I$-proj is the class of maps with the left lifting property with respect to all maps in $I$.

*$I$-cof is the class ($I$-inj)-proj.

*$I$-cell is the class of maps that can be obtained as a (transfinite) composition of pushouts of $I$.  One can show that $I$-cell $\subseteq$ $I$-cof, and that every map in $I$-cof is a retract of a map in $I$-cell.

In our situation, let $I = \{S^{k+1} \to D^{k+2}\}_{k \geq n}$.  Then $I$-inj is precisely the class of $n$-truncated maps. It remains to show that $I$-cof is the class of $n$-connected maps.  To see this, note that $I$-cell consists of those generalized CW-pairs where only cells of dimension $> n+1$ are allowed, so they are $n$-connected.  $I$-cof consists of retracts of such maps, so a map in $I$-cof is $n$-connected as well.
Moreover, here is a trick to show every solution to the lifting problem $$\begin{array}{ccc} A & \rightarrow & Y \\ \downarrow &  & \downarrow \\ X & \rightarrow & B, \end{array}$$ where the left vertical arrow is $n$-connected and the right vertical arrow is $n$-truncated, are automatically homotopic.   Given two such lifts, the existence of a homotopy corresponds to a solution for another lifting problem $$\begin{array}{ccc} A \times I \cup X \times \partial I & \rightarrow & Y \\ \downarrow & & \downarrow \\ X \times I & \rightarrow & B. \end{array}$$  Since $A \times I \cup X \times \partial I \to X \times I$ is $n$-connected if $A \to X$ is, and $Y \to B$ is $n$-truncated, a lift exists.
