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I am reading about the Reedy model structure from Hovey's book and

I was wondering if the Reedy model structure on $\mathcal{M}^{\Delta}$ is cofibrantly generated by a small set of arrows and permits the small object argument.

I see that in theorem 5.2.5 he proves the existence of a functorial factorization by transfinite induction assuming functorial factorization on $\mathcal{M},$ so there would not be any need to apply the SOA, but I am still left wondering if the Reedy model structure can be seen as cofibrantly generated. Is there any reference for this?

Thanks in advance.

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  • $\begingroup$ At minimum, I guess you're assuming that $\mathcal M$ itself is cofibrantly generated as a hypothesis? $\endgroup$
    – Tim Campion
    Dec 29, 2021 at 17:42
  • $\begingroup$ Yes. I thought that if $\mathcal{M}$ is cofibrantly generated by $\mathcal{J}$ then $\mathcal{M}^{\Delta}$ would be cofibrantly generated by the set of maps $X\to Y$ such that $X_i \bigsqcup_{L_iX}L_iY \to Y_i \in \mathcal{J}$ for every $i \in \Delta$ but I am not sure how to prove this. $\endgroup$ Dec 29, 2021 at 17:49
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    $\begingroup$ The answer is yes: see Thm 15.6.27 in Hirschhorn's book Model categories and their localizations. $\endgroup$ Dec 29, 2021 at 19:42
  • $\begingroup$ perfect! thanks very much for the precise reference. $\endgroup$ Dec 30, 2021 at 16:47
  • $\begingroup$ @D.-C.Cisinski Maybe post that as an answer? $\endgroup$ Jan 2, 2022 at 6:10

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