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By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's principle).

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  • $\begingroup$ I am very confused by the question, if the underlying category is locally presentable, then it is accessible in first place. $\endgroup$ – Ivan Di Liberti Mar 26 at 14:45
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    $\begingroup$ @IvanDiLiberti Accessible as a model category, with accessible weak factorization systems. $\endgroup$ – Philippe Gaucher Mar 26 at 14:48
  • $\begingroup$ Oh, thanks for the clarification. $\endgroup$ – Ivan Di Liberti Mar 26 at 14:48
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    $\begingroup$ @IvanDiLiberti I put a link towards the nLab for the definition. $\endgroup$ – Philippe Gaucher Mar 26 at 14:50
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I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky Locally presentable and accessible categories the locally presentable category $\bf Gra$ of graphs has a reflective subcategory that is not accessible, and by Proposition 3.5 of Salch The Bousfield localizations and colocalizations of the discrete model structure this reflector is the fibrant replacement functor of a model structure on $\bf Gra$.

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An example which does not depend on set theory is the equivariant model structure on the category of maps of spaces by Emmanuel Farjoun.

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