# Interlocking (weak) factorization systems

MathOverflow Asked by Tim Campion on November 9, 2020

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 $$((n+1)text{-skel_r}, ntext{-conn})$$ wfs and the $$(ntext{-conn},ntext{-trunc})$$ fs , for any $$n in mathbb Z_{geq -2}$$.

• On the 1-category $$Ch_{geq 0}(Rtext{-Mod})$$, 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?

 Here,

• $$(n+1)text{-skel_r}$$ denotes the retracts of relative $$(n+1)$$-dimensional CW complexes
• $$ntext{-conn}$$ denotes the $$n$$-connected maps (= maps with $$n$$-connected fibers, off by 1 from the most classical convention)
• $$ntext{-trunc}$$ 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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