The 'gros' functor from schemes into (strictly) locally ringed topoi

Question

Consider the category of finitely presented commutative rings, and equip its opposite category with either the Étale or Zariski topology. These give rise to topoi $\operatorname{Sh}(\mathbf{Et})$ and $\operatorname{Sh}(\mathbf{Zar})$ by taking sheaves. By the descent theorem for schemes, the functors of points of schemes are all sheaves in either of these topoi. In what follows, we will consider the Étale case:

Given a scheme $S$ viewed as an object of $\operatorname{Sh}(\mathbf{Et})$, we can form the slice topos $\operatorname{Sh}(\mathbf{Et})_{/S},$ and by the Yoneda lemma, we see that the functor taking a scheme to this slice topos is fully faithful into the category of categories over $\operatorname{Sh}(\mathbf{Et})$. However, given a map $f:S\to T$ of schemes, the induced functor $$f_!:\operatorname{Sh}(\mathbf{Et})_{/S}\to \operatorname{Sh}(\mathbf{Et})_{/T},$$ is not a geometric morphism. It is however a morphism in $\mathbf{Cat}_{/\operatorname{Sh}(\mathbf{Et})}$ with respect to the projection functors.

Since all of the functors in this triangle admit right adjoints that themselves admit right adjoints, the calculus of mates tells us that we have a triangle of essential (in fact Étale) geometric morphisms:

$$\begin{matrix} \operatorname{Sh}(\mathbf{Et})_{/S} & \xrightarrow{f_*} & \operatorname{Sh}(\mathbf{Et})_{/T}\\ S_* \searrow&\overset{\alpha}{\Leftarrow}&\swarrow T_*\\ &\operatorname{Sh}(\mathbf{Et}) \end{matrix},$$

where the natural transformation $\alpha$ is the mate of the mate of the commuting triangle arising from the Yoneda embedding:

$$\begin{matrix} \operatorname{Sh}(\mathbf{Et})_{/S} & \xrightarrow{f_!} & \operatorname{Sh}(\mathbf{Et})_{/T}\\ S_!\searrow&=&\swarrow T_!\\ &\operatorname{Sh}(\mathbf{Et}) \end{matrix},$$

In particular, since $\operatorname{Sh}(\mathbf{Et})$ is the classifying topos of strictly local rings (also called strictly henselian rings), the geometric morphism $f_*$ together with the 2-cell $\alpha$ determines a map of strictly locally ringed topoi. Since everything is natural and functorial, this defines a functor from schemes (in fact from $\operatorname{Sh}(\mathbf{Et}$)) to the category of strictly locally ringed topoi.

Question: Is this assignment fully faithful in the category of strictly locally ringed topoi?

Answer 1

So the final answer, is 'no', but there is still something interesting to say:

Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ (where $Top$ is the 2-category of toposes) of toposes $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

As the category of scheme indentifies in the way described in the question with a full subcategory $Sh(Et)$ this shows that morphisms of scheme $X \rightarrow Y$ corresponds exactly to morphisms between their (gros) étale topos, "over the étale topos of Spec $\mathbb{Z}$, i.e. geometric morphisms $f : Et_X \rightarrow Et_Y$ endowed with an isomorphism $f^* \mathcal{O}_Y \simeq \mathcal{O}_X$ between their structural sheaves (as described in the question) as sheaves of rings, so they corresponds to a special kind of morphisms of locally ringed toposes.

One might wonder whether all morphisms of locally ringed toposes between étale toposes of scheme are actually of this form, but unfortunately, the answer is no. I'm quite far from algebraic geometry so I could be completely off, but I believe the following gives a counter example:

Consider $X = Spec (\mathbb{Z}/p \mathbb{Z} )$ and $\mathcal{T}$ its gros étale topos.

Its structural sheaf is a sheaf of strict local $\mathbb{Z}/p\mathbb{Z}$ algebra, in fact the universal one.

I claim the Frobenius endomorphism of this sheaf (which is always a local morphism) provides a non-trivial non-invertible endormorphism of this sheaf: if it were trivial or invertible, then by universality, it would mean that for any sheaf of strictly local $\mathbb{Z}/p\mathbb{Z}$-algebra on any topos the Forbenius endomorphism would be trivial/invertible.

Hence the identity of the topos together with the Frobenius endomorphism provides a morphism of locally ringed topos that is not of the form above and hence do not corresponds to a morphism of scheme.