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Brendan Murphy (Oct 29 2023 at 23:43):What do Quillen adjunctions give you over "derivable adjunctions" as defined in Shulman's paper on comparing composites of left and right adjoints? The conditions in that paper (a left derivable functor must preserve trivial cofibrations between cofibrant objects and preserve cofibrant objects themselves, dually for right/fibrant) make sense to me, we want derived functors to exist and we want to be able to compose derivable functors. But why the stronger requirement of preserving trivial (co)fibrations between all objects, vs just between the objects necessary to derive the functor? I would appreciate a straightforward example of something that needs the stronger hypothesis on the functors
Mike Shulman (Oct 30 2023 at 00:51):One thing that Quillen adjunctions give you is preservation under more constructions. As a simple example, if is a Quillen adjunction and is an object, the induced adjunction between coslice categories is also Quillen. But if the original adjunction is only derivable, you can't say that: the cofibrant objects in are the cofibrations in , so unless you know that preserves cofibrations you can't conclude that preserves cofibrant objects.
Mike Shulman (Oct 30 2023 at 00:52):Similar observations apply to model structures on functor categories, etc.