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sarahzrf (Mar 28 2020 at 22:10):so sheafification is the left adjoint to the inclusion (or more generally, the inclusion of the subcategory of sheaves for some lawvere-tierney topology on an elementary topos). but as far as i can tell, nlab seems to be making reference to indeed being a subtopos of , which means that there needs to be a right adjoint too, right...?
sarahzrf (Mar 28 2020 at 22:10):what's that?
Morgan Rogers (he/him) (Mar 28 2020 at 22:13):No, there isn't a right adjoint (and examples where a right adjoints exist are rather rare).
Morgan Rogers (he/him) (Mar 28 2020 at 22:17):It's one of those cases where the name came from convenience rather than following the usual most general convention for subobjects. Inclusions are monos in the category of toposes and geometric morphisms, but they don't constitute all monomorphisms. However, they fit into a nice orthogonal (up to equivalence) factorization system with surjections forming the right-hand class, so they're a very nice class of monos to work with, and we take subtopos to mean "domain of an inclusion into a given topos"
sarahzrf (Mar 28 2020 at 23:36):ah... nlab's page "subtopos" defines it as having a geometric embedding >.<
sarahzrf (Mar 28 2020 at 23:36):perhaps it's inconsistent usage between the pages
sarahzrf (Mar 28 2020 at 23:38):actually no wait wtf it's on the same page
sarahzrf (Mar 28 2020 at 23:38):https://ncatlab.org/nlab/show/subtopos#sheaves_localization_closure_and_reflection
sarahzrf (Mar 28 2020 at 23:39):i guess it doesn't say that the subtopoi are in fact the topoi of sheaves, but
sarahzrf (Mar 28 2020 at 23:39):i thought that was a pretty reasonable interpretation of what it's saying >.>
vikraman (Mar 28 2020 at 23:59):Having a right adjoint would mean the inclusion preserves colimits, which is definitely not true!
sarahzrf (Mar 29 2020 at 00:01):yeah, i think that occurred to me, and then i confused myself :thinking:
sarahzrf (Mar 29 2020 at 00:01):so what the heck is it talking about?
Reid Barton (Mar 29 2020 at 03:16):The inclusion is the right adjoint. Its left adjoint is sheafification, which preserves finite limits, so the inclusion is a geometric morphism. And it's a geometric embedding because it's fully faithful.
sarahzrf (Mar 29 2020 at 05:38):the direction of the geometric morphism is the direction of the left adjoint, though
sarahzrf (Mar 29 2020 at 05:38):...is what i would say if i couldnt fuckin read!!!
sarahzrf (Mar 29 2020 at 05:38)::face_palm:
Morgan Rogers (he/him) (Mar 30 2020 at 09:37):sarahzrf said:
ah... nlab's page "subtopos" defines it as having a geometric embedding >.<
In case it's a further source of confusion it should be noted that (geometric) "inclusion" and "embedding" are names for the same concept