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In Garner's Understanding the Small Object Argument he assumes the classes of a WFS are closed under retracts, while at [[weak factorization systems]], or in other sources I consulted, this isn't assumed. Is it just a terminological divergence or is it because someone later found out that assumption isn't needed?
Uhm ok the nLab proves that indeed one gets closure under retracts for free. Unless I misunderstood something?
You're right, and one possible reference is the joyal catlab https://ncatlab.org/joyalscatlab/published/Weak+factorisation+systems#closureofcomplementsfirst and https://ncatlab.org/joyalscatlab/show/Cartesian+squares#retractquasicart
Garner uses an equivalent definition of wfs, where instead of requiring that and , he simply asks (i.e. any lifting problem of a morphism in against a morphism in has a solution). If you add that and are closed under retracts, then you can recover back that the two classes and determine each other
See for instance Riehl Definition 2.2 in comparison with Lemma 2.3
Oooh I see! Thanks