I'm interested in instances of the following data:
$C$ is a (possibly higher) category;
$(L,M)$ is a weak factorization system (wfs) on $C$;
$(M,R)$ is a unique factorization system (fs) on $C$.
Definition: I'll call such data $(L,M,R)$ interlocking factorization systems on $C$ because the right half of the wfs coincides with the left half of the fs.
Examples:
On the 1-category $Set$ we have the $(Mono,Epi)$ wfs and the $(Epi,Mono)$ fs.
On the $\infty$-category $Spaces$ we have (as discussed here) the $((n+1)\text{-skel_r}, n\text{-conn})$ wfs and the $(n\text{-conn},n\text{-trunc})$ fs [1], for any $n \in \mathbb Z_{\geq -2}$.
On the 1-category $Ch_{\geq 0}(R\text{-Mod})$, we have the $(\text{acyclic mono with projective cokernel}, \text{epi})$ wfs and the $(\text{epi}, \text{mono})$ fs, for any ring $R$.
Question 1: What are some other examples of the above data of "interlocking factorization systems"?
Observations:
If $(L,M,R)$ forms interlocking factorization systems on $C$, then $(M,R)$ is a modality (i.e. a fs with basechange-stable left class). Moreover, $M$ is closed under co-transfinite-composition. In fact, we might think of $(L,M,R)$ as being fundamentally a fs $(M,R)$ where $M$ satsifies some further closure conditions like these (though it's not clear that there's actually an equivalent formulation along these lines).
Dually, we might think of an interlocking factorization system $(L,M,R)$ on $C$ as being fundamentally a wfs $(L,M)$ where $M$ satisfies some further closure conditions (namely, cobase-change and colimits in the arrow category). If $C$ is locally presentable, then modulo checking that $M$ is accessibly embedded, this is actually an equivalent formulation.
Question 2: Given a class of morphisms $M$ in a ($\infty$-)category $C$ (perhaps assumed to be locally presentable),
does knowing that $M$ is the left half of a fs $(M,R)$ in any way "simplify" the task of checking whether $M$ is the right half of a wfs $(L,M)$?
dually, does knowing that $M$ is the right half of a wfs $(L,M)$ imply anything interesting about whether $M$ is also the left half of a fs $(M,R)$?
Question 3: Are interlocking factorization systems just a curiosity, or is there anything special you can do with them? For instance, do they lead to some kind of obstruction theory?
[1] Here,
- $(n+1)\text{-skel_r}$ denotes the retracts of relative $(n+1)$-dimensional CW complexes
- $n\text{-conn}$ denotes the $n$-connected maps (= maps with $n$-connected fibers, off by 1 from the most classical convention)
- $n\text{-trunc}$ denotes the $n$-truncated maps (= maps with $n$-truncated fibers, which again may be off by 1 from your favorite convention)
