TITLE: Cofibrant replacements of a given object in a combinatorial model category
QUESTION [7 upvotes]: In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-filtered colimit of a diagram of cofibrant replacements of a given object $X$ is a cofibrant replacement of $X$. Does the contrary hold, i.e. is the full subcategory of cofibrant replacements of a given object accessible ?
EDIT : the class of cofibrations is accessible so for $\lambda$ regular big enough, a $\lambda$-filtered colimit of cofibrations $\varnothing \to X_i$ is a cofibration ; the class of weak equivalences is accessible so for $\lambda$- regular big enough, a $\lambda$-filtered colimit of weak equivalences $X_i\to X$ is a weak equivalence ; so for $\lambda$ regular big enough, a $\lambda$-filtered colimit of cofibrant replacements of $X$ is a cofibrant replacement of $X$.

REPLY [3 votes]: (sorry I have troubles with comments, I post here even if it is not an answer) I have a new information. In On a fat small object argument, it is proved that in a λ-combinatorial model category, every cofibrant object is a λ-filtered colimit of λ-presentable cofibrant objects, which is close to what I wanted.