TITLE: Looking for generalization of projective model structure
QUESTION [8 upvotes]: If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences and fibrations are transformations that are such on each object of $\mathcal{C}$. I recall seeing a generalization in which we take a subset $D$ of objects of $\mathcal{C}$ and say that a transformation $f$ is a weak equivalence or fibration if $f(d)$ is one for every $d\in D$ (and require nothing at the other objects). Am I remembering correctly that this works in this generality, and does anyone have a reference for where this is written down? I'd like to use this but see no need to reinvent the wheel if I can just refer to it.

REPLY [4 votes]: You might be thinking of the paper "On Modified Reedy and Modified Projective Model Structures" by Mark Johnson. It does exactly what you asked in your question, in Proposition 6.4. Funny you should mention reinventing the wheel; I reproved this result a few years ago in a mathoverflow answer, before I knew about Mark's paper.