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 LibertiCommented Mar 26, 2019 at 14:45
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2$\begingroup$ @IvanDiLiberti Accessible as a model category, with accessible weak factorization systems. $\endgroup$– Philippe GaucherCommented Mar 26, 2019 at 14:48
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$\begingroup$ Oh, thanks for the clarification. $\endgroup$– Ivan Di LibertiCommented Mar 26, 2019 at 14:48
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3$\begingroup$ @IvanDiLiberti I put a link towards the nLab for the definition. $\endgroup$– Philippe GaucherCommented Mar 26, 2019 at 14:50
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2 Answers
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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 $\bf Gra$.
answered Mar 27, 2019 at 4:03
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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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1$\begingroup$ Are you sure it is not accessible ? it is not combinatorial of couse. but I think it is accessible. Well to be honest I don't quite know about the version using actual space but then the underlying category is not locally presentable. I claim that a similar construction using simplicial sets instead would be accessible however. $\endgroup$ Commented May 8, 2020 at 18:56
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$\begingroup$ I was trying to write a rigorous proof recently, but did not succeed so far. I still believe that the main idea is correct and hope to finish the argument eventually. Frankly, I would be happier if the equivariant model structure turns out to be accessible, so if you have a proof (for simplicial sets, of course), please tell me. $\endgroup$ Commented May 10, 2020 at 8:51