@PiotrPstrągowski Adams covers are what you get when you build minimal Adams resolutions. The $E_2$-term of the ASS of the $n$th Adams cover of $X$ looks like that of $X$ with the lowest $n$ lines removed.
@DenisNardin Finite limits commute with filtered colimits in any compactly-generated category, so yes. But I'm confused what argument you had in mind about spaces -- I don't think that $Cat_\infty$ is generated under colimits by spaces -- e.g. I don't see how to construct the arrow category as a colimit of spaces.
Oh I see
yeah, check it in the category of spaces, not because of any generation property.
@TimCampion My argument is that there is a conservative collection of limit-preserving filtered-colimit preserving functors F_i:C→Space (this is Map(c_i, -) for c_i a generator)
@PiotrPstrągowski I don't know if there's a standard term for a Grothendieck category admitting a conservative functor to $Ab$, but note that this is pretty close to the condition of having a generator. And if I remember right, any Grothendieck category with a generator is locally presentable -- similar to how any elementary topos with coproducts and a generator is a Grothendieck topos, and hence locally presentable.
Oh wait-- having a generator is part of the condition of being Grothedieck. So the strength of the condition is in being exact and cocontinuous...
Note, though, that if you have a conservative, cocontinuous, exact functor to $C \to Ab$, then that functor is comonadic. I don't know much about exact comonads on $Ab$, though.
