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I gave functorial factorization its own little entry, for ease of pointing to the precise definition.
This is for the moment just copied over from the corresponding paragraph at weak factorization system (where I have re-organized the sectzion outline slightly, for clarity). Also I added cross-links with some relevant entries.
In a simplicial model category with small object argument. Can functorial factorization be simplicially functorial?
Yes, because every simplicial set is cofibrant. See 4.3.8 in Hirschhorn’s book or 24.2 here.
Thanks! I have added these pointers here.
added pointer to:
Daniel Dugger, Prop. 7.1 in: Combinatorial model categories have presentations, Advances in Mathematics 164 1 (2001) 177-201 [arXiv:math/0007068, doi:10.1006/aima.2001.2015]
Clark Barwick, Prop. 2.5 in: On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 2 (2010) 245–320 [doi:10.4310/hha.2010.v12.n2.a9, euclid:1296223884, subsuming: arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]
as references for functorial factorization in combinatorial model categories, and made this an Example-item (here)
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