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Oct 3, 2013 at 15:22 comment added David White @ZhenLin: Thanks for your close reading of the answer. Actually, when I wrote this I was thinking a bit about your recent paper on universes. I think the notion in (3) is a bit more general than your comment suggests, because I'm not assuming $M$ is generated under filtered colimits by a set as you'd need for combinatorial (nor under a class as Chorny requires for class-combinatorial). I really just mean an option between Quillen's original definition (finite (co)limits) and the one which appears in Hovey's book (all (co)limits). I don't want to restrict to combinatorial because the OP didn't.
Oct 3, 2013 at 14:32 comment added Zhen Lin Option 3 is not really a "new notion" – Quillen's original definition only required finite limits and colimits, and did not require functorial factorisation. In option 2, if you restrict to $\kappa$-accessible functors between combinatorial model categories that are locally $\kappa$-presentable, then the category you get is locally $\kappa$-presentable and has a cofibrantly-generated projective model structure (Kan et al.) as well as an injective model structure (Lurie). In particular it is locally small.
Oct 3, 2013 at 12:23 history answered David White CC BY-SA 3.0