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Stream: theory: category theory

Topic: closure under retracts of weak factorization systems

Matteo Capucci (he/him) (Feb 17 2025 at 16:29):

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?

Matteo Capucci (he/him) (Feb 17 2025 at 16:32):

Uhm ok the nLab proves that indeed one gets closure under retracts for free. Unless I misunderstood something?

fosco (Feb 17 2025 at 17:23):

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

Clémence Chanavat (Feb 17 2025 at 17:48):

Garner uses an equivalent definition of wfs, where instead of requiring that $L = llp(R)$ and $R = rlp(L)$, he simply asks $L \pitchfork R$ (i.e. any lifting problem of a morphism in $L$ against a morphism in $R$ has a solution). If you add that $L$ and $R$ are closed under retracts, then you can recover back that the two classes $L$ and $R$ determine each other

Clémence Chanavat (Feb 17 2025 at 17:49):

See for instance Riehl Definition 2.2 in comparison with Lemma 2.3

Matteo Capucci (he/him) (Feb 17 2025 at 21:28):

Oooh I see! Thanks