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Morgan Rogers (he/him) (Nov 14 2023 at 15:48):It depends on additional hypotheses on the factorization system (see for example "algebraic factorization system"). Often or will be universal in some way (so might be a kernel/cokernel, say).
Mike Shulman (Nov 14 2023 at 16:58):It's always a reflection from the slice category into the subcategory ...
Mike Shulman (Nov 14 2023 at 16:59):(Assuming by "factorization system" you mean an "orthogonal/unique" factorization system.)
fosco (Nov 14 2023 at 20:02):Sometimes is a pullback, and arises as the induced map into it; google "simple factorization system", and in particular a paper from Cassidy, Hebert and Kelly. Or read my PhD thesis, which is an account of torsion theories (particularly well-behaved FS, covered in non-abelian setting also in Borceux, handbook tome 1) in triangulated/stable-homotopic setting.
fosco (Nov 14 2023 at 20:04):As Morgan says, AFS are another source of factorization obtained from universal properties; Garner's paper is the best reading.
In general, very little can be said, without knowing what are the specifics of the FS you have to handle. For example, is it right proper (=every is a monic)? Extremal-right proper (=every is an extremal monic)?
Matteo Capucci (he/him) (Nov 15 2023 at 06:53):Mike Shulman said:
(Assuming by "factorization system" you mean an "orthogonal/unique" factorization system.)
Yeah I do!
Matteo Capucci (he/him) (Nov 15 2023 at 06:54):Mike Shulman said:
It's always a reflection from the slice category $\mathcal{C}/c'$ into the subcategory $\mathcal{M}/c'$...
This sounds like what I need! Thanks Mike