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Jan 12, 2019 at 0:39 vote accept Tim Campion
Jan 12, 2019 at 0:39 comment added Tim Campion Ah, this is a great example -- left horn inclusions are a pseudo-generating set for the Quillen model structure but not a generating set! So this definitively answers Question B with a "no". And yes, injective model structures seem to answer Question A with a "no" as well. This doesn't answer Question C, but I think Question C is not really so interesting given that the answer to Question B is "no". Thanks!
Jan 12, 2019 at 0:29 history answered D.-C. Cisinski CC BY-SA 4.0