# Example of non accessible model categories

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).

• I am very confused by the question, if the underlying category is locally presentable, then it is accessible in first place. – Ivan Di Liberti Mar 26 at 14:45
• @IvanDiLiberti Accessible as a model category, with accessible weak factorization systems. – Philippe Gaucher Mar 26 at 14:48
• Oh, thanks for the clarification. – Ivan Di Liberti Mar 26 at 14:48
• @IvanDiLiberti I put a link towards the nLab for the definition. – Philippe Gaucher Mar 26 at 14:50

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 $$\bf Gra$$.