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Reid Barton (May 09 2020 at 16:33):What are the appropriate morphisms of quasitopoi? The Elephant casually suggests using the same definition as for geometric morphisms of topoi, but in a situation where "concrete sheaves" has a meaning, I'd imagine one would want the inclusion of concrete sheaves in all sheaves to be a quasitopos morphism, while it couldn't be a geometric morphism by the ordinary definition (unless the concrete sheaves happen to form a topos), unless I am quite confused.
Reid Barton (May 09 2020 at 16:39):https://arxiv.org/abs/1106.5331 has a description of the localizations of presheaf categories which result in quasitopoi, but it's not obvious to me how to extract a good general notion of morphism from this (though I've put very little thought into it so far).
Morgan Rogers (he/him) (May 09 2020 at 17:21):That's a solid argument for not using geometric morphisms. iirc the reflector does preserve some limits (maybe finite products?); whatever it is, that's the quickest fix.
The motivation for using geometric morphisms in the first place is that they correspond to continuous maps between (sober) topological spaces. So you should look first at a notion that fits well with how you're presenting/constructing your quasitoposes.
Reid Barton (May 09 2020 at 17:30):Let's say I am presenting them as suitable reflective subcategories of a category of presheaves, or as the separated sheaves on a bisite.
Reid Barton (May 09 2020 at 17:31):I guess I could try to imagine what the morphisms of bisites ought to be, though I suspect this is harder than the original question...
Morgan Rogers (he/him) (May 09 2020 at 17:48):Reid Barton said:
Let's say I am presenting them as suitable reflective subcategories of a category of presheaves, or as the separated sheaves on a bisite.
Comorphisms of (bi)sites are also a reasonable starting point here, and in that case the resulting "inverse image functor" preserves exactly as much limit structure as the for sheafification functor .
Mike Shulman (May 10 2020 at 15:21):An ordinary morphism of sites is required to be sufficiently flat that the induced functor on sheaves preserves finite limits. Is there a standard notion of "morphism of bisites" that weakens this somehow? It doesn't seem offhand to be an easier problem than the original question.
Reid Barton (May 10 2020 at 15:44):I should probably add that this is mostly curiosity at this point, and I was really just hoping someone would come along with a strong opinion about the answer (that's different than the one for topoi).
Reid Barton (May 10 2020 at 15:45):Now that @Mike Shulman has commented, most of that hope is gone :upside_down:
Morgan Rogers (he/him) (May 10 2020 at 16:56):So, looking before Theorem 4.4.8 in Sketches of an Elephant, the reflector preserves monomorphisms (I had misremembered) for a universal closure operator on a quasitopos . Theorem 4.4.8 itself is of interest: the inclusion of a quasitopos into a topos has cartesian reflector if and only if the quasitopos is a topos. So yes, geometric morphisms are too strong, but preserving monomorphisms (or at least cocovers) seems like it should be a feature of the chosen morphisms.