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I'm interested in instances of the following data:

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)
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    $\begingroup$ One example worth understanding could be the following: it is known that t-structures on the homotopy category of a given $(\infty,1)$-category are in bijection with a suitable class of factorization systems (called normal torsion theories). It could be interesting to study which $t$-structures correspond to "normal interlocking torsion theories"... e.g., are they the TTFs studied in the PhD thesis of Pedro Nicolas (arxiv.org/pdf/0801.0507.pdf) or do we need to impose some more conditions to get a bijection? $\endgroup$ Aug 15, 2020 at 22:55
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    $\begingroup$ This isn't quite the same phenomenon, but ternary factorisation systems share similar structure in being overlapping factorisation systems in some sense, though they're stronger than the notion you're interested in (at least when $R_1 = L_2$). $\endgroup$
    – varkor
    Aug 15, 2020 at 23:29
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    $\begingroup$ Long ago I asked what's up with those FS such that $(E,M)$ is a OFS and $(M,E)$ a WFS; this doesn't seem to be understood. (I would call these particular interlocking FS Frobenius, for obvious reasons). Fiorenza, Marchetti and I studied the notion of "stable n-ary OFS" to describe Postnikov towers in arxiv.org/abs/1501.04658: not that this was really unknown. It just come out better when done $\infty$. Hopefully you can adapt that technology? $\endgroup$
    – fosco
    Aug 16, 2020 at 9:09
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    $\begingroup$ Tim Campion. Your last observation can be extended. Dual of an interlocking factorization system is $(R,M,L)$ where $(R,M)$ is a factorization system and $(M,L)$ is a weak factorization system. I met this situation in my arXiv:1702.08684 paper. Examples are on Boolean algebras, Banach spaces, or commutative $C^\ast$-algebras. The last example yields an interlocking factorization system on compact Hausdorff spaces. $\endgroup$ Aug 18, 2020 at 14:13
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    $\begingroup$ Trivial observation about question 2: at least it's automatically closed under retracts... (-:O $\endgroup$ Aug 21, 2020 at 2:20

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