Does the Zariski spectrum of a ring arise formally from the inclusion of the big Zariski topos into the classifying topos for rings?

Question

Let $\iota_\ast : \mathcal A \to \mathcal B$ be a geometric morphism. I'm looking for some functor

$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$

which can be defined from the data of $\iota$. When $\mathcal B = \mathrm{BRing} = \operatorname{Fun}(\mathrm{Ring}^\text{fp}, \mathrm{Set})$ is the classifying topos for (commutative) rings, so that $\mathrm{Topos}_{//\mathcal B}$ is the category of ringed topoi, I'd like the following to be true when the input $\langle R \rangle_\ast : \mathrm{Set} \to \mathrm{BRing}$ classifies an ordinary ring $R$:

1. $F_{\mathrm{Zar} \to \mathrm{BRing}}(R) = \operatorname{Spec}^{\mathrm{Zar}}(R) \xrightarrow{\langle \mathcal O_R\rangle_\ast} \mathrm{Zar}$.

2. $F_{\mathrm{Et} \to \mathrm{BRing}}(R) = \operatorname{Spec}^\text{ét}(R) \xrightarrow{\langle \mathcal O_R\rangle_\ast} \mathrm{Ét}$.

That is, when $\mathcal A$ is the classifying topos for local rings, I want to carry a ring to its Zariski spectrum (with the usual structure sheaf of local rings), and when $\mathcal A$ is the classifying topos for strictly henselian local rings, I want to carry a ring to its étale spectrum.

Question: Does such a general construction $\iota \mapsto F_\iota$ exist?

Notes:

• This should be related to Cole's theory of spectrum, or maybe to Lurie's theory of spectrum, but in these cases it seems the input data is a bit different?

• Of course, I'd expect this functor to carry morphisms to morphisms of locally ringed topoi in case (1), which are not the same as the morphisms in $\mathrm{Topos}_{/{\mathrm{Zar}}}$, but what I'm asking for should still be perfectly well-defined, even if one can say more. An answer shedding light on what more can be said about the output morphisms would be nice, but not necessary for me.

• By $\mathrm{Topos}_{//\mathcal B}$ I mean the lax slice category (which the nlab calls the "slice 2-category") of the 2-category $\mathrm{Topos}$. (Note that if we used $\mathrm{Topos}_{/\mathcal B}$ instead, then we would not have a copy of the category of rings sitting inside when $\mathcal B = \mathrm{BRing}$, just its underlying groupoid.)

Answer 1

This is really the point of Cole and Dubuc approach, and it really requires also to conveniently chose some factorization data to work.

I - In Cole formalism, suppose you have a geometric embedding $ \iota : Set[\mathbb{T}_2] \hookrightarrow Set[\mathbb{T}_1]$ with $ \mathbb{T}_1 $. You can still turn a $ \mathbb{T}_1$ modelled topos $F : \mathcal{F} \rightarrow Set[\mathbb{T}_1]$ into a $ \mathbb{T}_2$-modelled one by simply taking either the bipullback or the bicomma of $F$ along $ \iota$, but in both cases the universal property is too rigid for this construction to define a right 2-adjoint to the inclusion $ Topos//Set[\mathbb{T}_2] \hookrightarrow Topos//Set[\mathbb{T}_1]$. Bipullbacks provide a right adjoint at the level of pseudoslices (in other words, if your morphisms of modelled topoi only allow restrictions of models along geometric morphisms), while the bicomma construction still implies to be restricted to the pseudoslice over $Set[\mathbb{T}_2]$, because the residual 2-cell in the property of the bicomma have to be invertible. Yet the bicomma $ F \downarrow \iota$ can be seen as a "gros spectrum", classifying all morphisms from $F$ toward a $\mathbb{T}_2$-model.

One has to specify suited factorization data in order to allow for more 2-cells over $ Set[\mathbb{T}_2]$. In most cases, it will not be possible to have an adjunction over the whole oplax slice, but at least on a large enough class where the 2-cells will lie in some right-like class for a factorization system (for instance, in morphisms of locally ringed topoi, the underlying morphism of models is required to have a conservative morphism of local rings as inverse image).

If $\mathbb{T}_1$ is a cartesian theory and $(\mathcal{L}, \mathcal{R}) $ a "left generated" factorization system on $\mathbb{T}_1[Set]$ (for instance the (localization, conservative) factorization for rings homomorphisms), then any category of models $ \mathbb{T}_1[\mathcal{E}]$ inherits such a factorization system, and in particular the universal 2-cell $ \mu_{F,\iota}$ of the bicomma object itself admits a factorization. If now your factorization system satisfies a special admissibility condition relative to $ \mathbb{T}_2$-models stipulating that the middle term of the factorization of an arrow $ f : F \rightarrow E$ with $E$ a $\mathbb{T}_2$-model is itself a $\mathbb{T}_2$-model, then taking the biinverter of the right part of the factorization provides a good definition of the spectrum $$ Spec(F) = Inv( r_{\mu_{F,\iota}})$$ It classifies left maps from (inverse image of) $F$ toward $ \mathbb{T}_2$-models, with the projection $$ \widetilde{F} : Spec(F) \rightarrow Set[\mathbb{T}_2]$$ being the "structure sheaf", in some sense, the "free" $\mathbb{T}_2$-model under $F$. The admissibility condition ensures that this construction $Spec(-)$ really defines a right bi-adjoint to the inclusion $ Topos//Set[\mathbb{T}_2]^{R} \hookrightarrow Topos//Set[\mathbb{T}_1] $ where the domain 2-category allows now for oplax 2-cells over $ Set[\mathbb{T}_2]$ that have an underlying morphism of models in $R$. Moreover the underlying morphism of model $h_F : \eta_{F} ^*F \Rightarrow \tilde{F}$ in the counit $(Spec(F), \widetilde{F}) \rightarrow (\mathcal{F}, F)$ always is a left map (morally, it is the universal left maps toward $ \mathbb{T}_2$-models). Note that here the condition that $ \iota$ is a geometric embedding is important to establish the equivalence of homcategories in the bi-adjunction.

One cannot avoid in general to put a condition on the morphisms you allow between locally modelled topoi. Indeed, allowing all maps to be "local" (morally, to be in $\mathcal{R}$) would in counterpart implies of left maps to be trivial, but as the counit is a left map, one would not have an adjunction unless all $ \mathbb{T}_1$-models are already $ \mathbb{T}_2$-model so everything gets trivialized. The opposite situation is when all maps are left, so only iso are right, and the bicomma is a spectrum, but as told above this get something very rigid with only restrictions maps as morphisms of locally modelled topoi... Choosing a non trivial suited factorization system is a compromise between those two.

II - Dubuc approach relaxes put no condition on $\mathbb{T}_1$ nor $\iota$ and processes through bicolimits of topoi rather than limits. In his formalism, he replaces the factorization data with a more invariant object, an etale class $ \mathscr{H}$ in $Set[\mathbb{T}_2]$, but this plays essentially the same role. For a topos $\mathcal{E}$ equipped with an etale class $ \mathscr{H}$, one can construct the etale topos $ \mathcal{E}_\mathscr{H}$ associated to the etale class $\mathscr{H}$ (mode of those objects whose terminal map is etale). This requires a condition ("ETC"), essentially equivalent to admissibility, relating the etale class with $ \iota$, to ensure that the category of etale objects is a Grothendieck topos. But when it is so, it appears actually as a coinverter of some universal right maps.

Then in a context as considered by Dubuc, one has a geometric morphism $ \iota : Set[\mathbb{T}_2] \rightarrow \mathcal{S}[\mathbb{T}_1]$ (which is not anymore required to be an embedding) and an etale class in $\mathcal{S}[\mathbb{T}_1]$ satisfying ETC relative to $ \iota$. Then one construct again the "gros spectrum" of $F$ as the bicomma $ F \downarrow \iota$, but now the spectrum is obtained as the etale topos for some induced etale class in the bicomma, and it is related through a local geometric morphism $ F \downarrow \iota \rightarrow Spec(F)$ in a "gros vs petit" topos manner, the spectrum being the petit topos. Beware again the spectrum only defines a right adjoint for the restricted oplax slice over $\mathbb{T}_2$.

You can find all details on those either in Cole or Dubuc, or in the chapter 5 of my thesis.