# Existence of a $\lambda$-generated Model Category Structure

Apologies if this is a stupid question: Let $C$ be a cofibrantly generated model category ($\mathbf{Edit}$: Combinatorial) and let $[X,C]$ be functor category equipped with the projective model structure.

Recall that for an infinite cardinal $\kappa$, a model structure on a category $D$ is $\kappa$-generated if every (trivial) cofibration in $D$ is a transfinite composition of pushouts of coproducts of (trivial) cofibrations between $\kappa$-small objects.

Are there any conditions on $C$ (or something else) that would ensure that the model structure on $[X,C]$ is $\kappa$-generated?

It seems like a bit of a pain to manually check if the $\kappa$-generated condition holds.

• $C$ being $\kappa$-generated suffices. I mean the existen of generating (trivial) cofibrations with $\kappa$-small source and target. – Fernando Muro Jul 21 '16 at 18:38
• As long as X is nonempty, a necessary and sufficient condition for [X,C] to be κ-generated is that C itself is κ-generated. The generating acyclic cofibrations for the projective model structure have the form R⊗j, where R is representable and j is a generating acyclic cofibration in C. The (co)domains of R⊗j are κ-small if the (co)domains of j are. – Dmitri Pavlov Jul 21 '16 at 18:40