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Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category such that there exists a set of generating cofibrations $I$ and such that all objects are fibrant. Then it is combinatorial. Pro …

Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable categories. $\mathcal{A}$ reflective and closed under filtered colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let …

I understand where is the mistake. $r$ does preserve finite presentability (the proof is straightforward and it is due to the fact that it is a left adjoint of a functor preserving filtered colimits). …

This question is a follow-up of Extending functors defined on dense subcategories. Let $\mathcal{K}$ be a locally presentable category. An object $X$ of $\mathcal{K}$ is called a Yoneda object i …

I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mat …

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)

The category of topological spaces is the category of models of a relational universal strict Horn theory $T$ without equality, i.e. the axioms are of the form $(\forall x)(\phi(x) \rightarrow \psi(x) …

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 princ …