Suppose you have a pair of orthogonal factorization systems, $(E\_0, M\_0), (E\_1, M\_1)$ in a category $\cal C$ such that $M\_0\subseteq M\_1$; this [entails](https://ncatlab.org/nlab/show/ternary+factorization+system) that there is a ternary factorization
$$
X\xrightarrow{e\_1} A\xrightarrow{e\_0 m\_1} B\xrightarrow{m\_0} Y
$$
where each arrow is labeled according to the class it belongs to.


Suppose now to have *another* OFS $(L,R)$ on $\cal C$, and to factor the middle arrow $A\to B$ into $A\xrightarrow{l} S\xrightarrow{r} B$; this gives a factorization $X\to S\to B$.



> 
> Does this define a third factorization system $(E\_{01}\wr L, M\_{01}\wr R)$?
> 
> 
> 



Factorization system "tilted" by $(L,R)$
Yes.


First note that Galois connections are far more common than field theorists would have you believe: *any* binary relation gives rise to one and the two fixed sets a closed under any appropriate algebraic structure.


Write


* $P\perp Q$ for "every member of $P$ is orthogonal (in the sense of factorisation systems) to every member of $Q$";
* $P;Q$ for the set of all composites of members of $P$ followed by members of $Q$;
* $L'=L\cap E\_0\cap M\_1$ and $R'=R\cap E\_0\cap M\_1$ (to avoid any ambiguity in this argument); and
* $E=E\_1;L'$ and $M=R';M\_0$.


Then


* $E\_1\perp M\_1\supset R'$,
* $E\_1\perp M\_1\supset M\_0$,
* $L'\subset L\perp R\supset R'$ and
* $L'\subset E\_0\perp M\_0$.


Since orthogonality respects composition, it follows that $E\perp M$, whilst by construction $E;M$ is the entire hom-class of the category.


Therefore $(E,M)$ is a factorisation system.


The things that I have assumed are all proved in [my book](http://paultaylor.eu/prafm) but I don't have a copy to hand to look up the references.


In the question as stated, it is given that $M\_0\subset M\_1$, but there is no order-relationship between these and $R$. This is why it was necessary to introduce $R'$. The construction gives a way of handling expressions in the lattice of factorisation systems on a category, so it would be an interesting exercise in lattice theory to find out whether this is modular or even distributive.


