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Under what assumptions can one prove existence of weak factorization systems (co)generated by a class of morphisms ? Are there counterexamples ? I am interested both in assumptions on the class of morphisms, and I am happy to assume Vopenka principles. I am mostly interested in the category of topological spaces or its variations.

For a class $P$ of morphisms of a category $C$, let $P^l$, resp. $P^r$, denote the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class $C$. Let $P^{lr}:=(P^l)^r$, etc.

Under what assumptions on $P$ is $(P^l,P^{lr})$ a weak factorisation system, i.e.
each morphism $f$ in $C$ decomposes as $f=f_{lr}\circ f_{l}$ where $f_l\in P^l$ and $f_{lr}\in P^{lr}$ ?

Dually, under what assumptions on $P$ is $(P^{rl},P^{r})$ a weak factorisation system, i.e. each morphism $f$ in $C$ decomposes as $f=f_{r}\circ f_{rl}$ where $f_r\in P^r$ and $f_{rl}\in P^{rl}$ ?

For example, can one prove this for $P$ consisting of a single morphism of finite topological spaces ?

That is, does the following hold in the category of topological spaces (or one of its variations):

Question. For each map $g$ of finite topological spaces, are both $(\{g\}^l,\{g\}^{lr})$ and $(\{g\}^{rl},\{g\}^{r})$ weak factorisation systems ?

(T. Beke, Sheafifiable homotopy model categories, Math. Proc. Cambr. Phil. Soc. 1239 (2000), 447–475.). Proposition 1.3. proves that for a locally presentable category $C$ and a set $P$ of morphisms, $(P^{rl}, P^{r})$ is a weak factorisation system. Unfortunately, this does not apply to the category of topological spaces.

Question. Are there natural examples of classes $P$ and morphisms which fail to decompose as required above ? Or perhaps when existence of a decomposition implies large cardinals.

I am also interested in the set theoretic aspects involved.

A recent paper (Sean Cox, Jiří Rosický. Fibrantly generated weak factorization systems.) poses the problem whether $(P^{l}, P^{lr})$ is a weak factorisation system for each set $P$, perhaps under assumption that the category is locally presentable.

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