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Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let $$\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$$ be the functor sending a space $X$ to the topos of sheaves on $X$. Does this functor have a left or a right adjoint?

Of course, $\mathrm{Sh}$ factorizes as $$\mathbf{Top}\to\mathbf{Loc}\to\mathbf{Topos},$$ since the sheaves on $X$ are the sheaves on the locale of open subsets of $X$.

It is well-known that $\mathbf{Top}\to\mathbf{Loc}$ has a right adjoint and $\mathbf{Loc}\to\mathbf{Topos}$ has a left adjoint ($\mathbf{Loc}$ is reflective in $\mathbf{Topos}$). So the question whether $\mathrm{Sh}$ has a left adjoint reduces to the question whether $\mathbf{Top}\to\mathbf{Loc}$ has a left adjoint and the question whether $\mathrm{Sh}$ has a right adjoint reduces to the question whether $\mathbf{Loc}\to\mathbf{Topos}$ has a right adjoint.

Remark. I already asked this on math.SE but I didn't get an answer.

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  • $\begingroup$ Are your toposes Grothendieck toposes? $\endgroup$
    – Tim Campion
    Aug 6, 2021 at 18:28
  • $\begingroup$ Are the reductions you give meant to be "if and only if" reductions? Clearly if $Loc \to Topos$ and $Top \to Loc$ had these additional adjoints then so would $Top \to Topos$, but the converse is not clear (to me) (though see my comment on Dmitri Pavlov's answer below). $\endgroup$
    – Tim Campion
    Aug 6, 2021 at 18:41
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    $\begingroup$ "I already asked this on math.SE but I didn't get an answer" Well, I left a comment there with which you could have find an answer yourself. (math.stackexchange.com/questions/4217568) Notice that Dmitri's answer exactly follows that suggestion. $\endgroup$ Aug 7, 2021 at 2:23

1 Answer 1

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In this answer, Topos is interpreted as a 2-category. (As a side remark, the 1-category of toposes does not make sense until one picks a specific model for toposes and geometric morphisms, and different models need not be equivalent as 1-categories. For the 1-categorical framework to make sense, at the very least one needs to organize toposes into a relative category, so that different models can be shown to be Dwyer–Kan equivalent as relative categories.)

whether Top→Loc has a left adjoint

The functor Top→Loc does not have a left adjoint because it does not preserve finite products. For example, the product of rational numbers with themselves as locales and as topological spaces produces nonisomorphic locales. In particular, rational numbers form a topological group, but not a localic group.

whether Loc→Topos has a right adjoint

Loc→Topos does not have a right adjoint because it does not preserve homotopy colimits.

For example, suppose G is a discrete group acting on a point. Then the colimit of this action in locales is again a point. But the homotopy colimit of this action in toposes is the delooping of G, which is not equivalent to a point.

Thus, both Loc→Topos and Top→Topos do not have a right adjoint functor.

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  • $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Aug 6, 2021 at 18:36
  • $\begingroup$ @Dmitri: Thanks. But this doesn't answer whether Top -> Topos has a right adjoint. $\endgroup$
    – user333306
    Aug 6, 2021 at 18:59
  • $\begingroup$ Isn’t the location reflection, described here mathoverflow.net/questions/271096/… a right adjoint? I’m not an expert $\endgroup$ Aug 6, 2021 at 19:37
  • $\begingroup$ 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. $\endgroup$ Aug 6, 2021 at 19:39
  • $\begingroup$ @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). $\endgroup$ Aug 6, 2021 at 21:11

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