Timeline for Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint?
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18 events
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| Aug 10, 2021 at 0:04 | comment | added | Jonas Frey | @DmitriPavlov: I agree with you that admitting only invertible 2-cells is "the minimal required modification" and the best way of looking at the question. But the point I was trying to make is in the setting with non-invertible 2-cells the decomposition Top --> Loc --> Topos has to be replaced by Top --> Loc_0 --> Loc --> Topos, where Loc_0 has disctrete homs, and Loc has ordered ones. Then the first functor has a right adjoint and the last one has a left adjoint, and I don't know about the middle one. | |
| Aug 9, 2021 at 23:56 | comment | added | Dmitri Pavlov | @JonasFrey: I see, you were talking about reflection for noninvertible 2-morphisms of toposes. Originally, I had in mind the minimal required modification of OP's statement, which only adds invertible 2-morphisms, but the arguments continue to make sense for noninvertible 2-morphisms. | |
| Aug 9, 2021 at 23:45 | comment | added | Jonas Frey | I think if we say that Loc is reflective in the (2,2)-category Topos, then we're speaking about the locally ordered category of locales? | |
| Aug 9, 2021 at 23:20 | comment | added | Dmitri Pavlov | @JonasFrey: For the locally ordered version of Loc, which comparison result do you have in mind? | |
| Aug 9, 2021 at 19:28 | comment | added | Jonas Frey | I think to stay as close as possible to the spirit of the question makes the most sense to view the involved categories as (2,1)-categories. Otherwise there arises the question whether Loc should be viewed as locally ordered or locally discrete. One is most natural when comparing with spaces, the other when comparing with toposes. | |
| Aug 6, 2021 at 21:47 | vote | accept | user333306 | ||
| Aug 6, 2021 at 21:47 | comment | added | user333306 | Ah, of course! :-) | |
| Aug 6, 2021 at 21:37 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Aug 6, 2021 at 21:31 | comment | added | Dmitri Pavlov | @user333306: The localic reflection is the left adjoint of the inclusion Loc→Topos. See, for example, the nLab: ncatlab.org/nlab/show/locale#RelationToToposes | |
| Aug 6, 2021 at 21:30 | comment | added | user333306 | @DmitriPavlov Ah, thanks. But what about the localic reflection Benjamin Steinberg mentioned. Isn't it a right adjoint of Loc -> Topos? | |
| Aug 6, 2021 at 21:13 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Aug 6, 2021 at 21:12 | comment | added | Dmitri Pavlov | @user333306: It does answer your question, since the example in the last paragraph is a counterexample to the claim that Top→Topos has a right adjoint. A left adjoint must necessarily preserve the indicated homotopy colimit, which is not the case with Top→Topos. | |
| Aug 6, 2021 at 21:11 | comment | added | Dmitri Pavlov | @TimCampion: Everything is treated bicategorically here, since Topos is a bicategory. The 1-category of toposes is not a well-defined notion until you pick a specific model (and different models need not be equivalent as 1-categories). | |
| Aug 6, 2021 at 19:39 | comment | added | Benjamin Steinberg | If I recall you take subobjects of the terminal object as a frame. This may require the topos to be Grothendieck but this is stuff I learned by osmosis so might be off. | |
| Aug 6, 2021 at 19:37 | comment | added | Benjamin Steinberg | Isn’t the location reflection, described here mathoverflow.net/questions/271096/… a right adjoint? I’m not an expert | |
| Aug 6, 2021 at 18:59 | comment | added | user333306 | @Dmitri: Thanks. But this doesn't answer whether Top -> Topos has a right adjoint. | |
| Aug 6, 2021 at 18:36 | comment | added | Tim Campion | Because $Loc \to Topos$ is fully faithful, it reflects limits (as well as colimits). So your first point -- that $Top \to Loc$ does not preserve finite products -- implies that $Top \to Topos$ also does not preserve finite products. So $Top \to Topos$ does not have a left adjoint. Your second point starts to make it matter exactly what is meant by "category" here -- you seem to be treating $Topos$ as at least a $(2,1)$-category. | |
| Aug 6, 2021 at 18:27 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |