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Let $\mathcal{Z}$ be the Zariski topos, in the sense of the classifying topos for local rings. I was wondering whether there might be any connection between $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$ and $\mathcal{Z}$. Certainly, there is a geometric morphism $\mathcal{Z} \to \mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$, and there’s also a geometric inclusion $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z}) \to \mathcal{Z}$. On the other hand, there’s no chance of $\mathcal{Z}$ itself being localic, since it has a proper class of (isomorphism classes of) points. Let’s write $L \mathcal{Z}$ for the localic reflection of $\mathcal{Z}$; the first geometric morphism I mentioned then corresponds to a locale map $L \mathcal{Z} \to \operatorname{Spec} \mathbb{Z}$. But what is $L \mathcal{Z}$ itself?
The open objects in $\mathcal{Z}$ can be identified with certain saturated cosieves on $\mathcal{Z}$ in the category of finitely-presented commutative rings, and so may be identified with certain sets of isomorphism classes of finitely-presented commutative rings. If I’m not mistaken, every finitely-presented commutative ring gives rise to an open object in $\mathcal{Z}$. This suggests that $L \mathcal{Z}$ might be some kind of (non-spatial) union of all isomorphism classes of affine schemes of finite type over $\mathbb{Z}$, which is an incredibly mind-boggling thing to think about. It’s not clear to me whether other kinds of open objects exist. For example, does every not-necessarily-affine open subset of $\operatorname{Spec} A$, for every finitely-presented ring $A$, also show up…?
I haven’t really thought about this at all yet, but the general kind of question is interesting, I think.
Just briefly, before I spend more thoughts on it: are you sure you are looking for the localic reflection? That would be sheaves on the subobject lattice of the terminal object in $\mathcal{Z}$. Isn’t that trivial?
You might be thinking of the hyperconnected / localic -factorization of $\mathcal{Z} \to Sh(Spec \mathbb{Z})$ instead?
Let me know I am mixed up here. I don’t really have the leisure to think about this right now. Hopefully later.
But, incidentally, I was thinking just recently of a kind of question that is at least vaguely similar: some logic colleagues of mine have figured out that what is called the topos of types in logic is such a hyperconnected/localic factorization for “big topos“es not over Lawvere theories (for rings, as in your case), but over coherent theories.
There one stars with a coherent category $C$ (think of this as analogous to the Lawvere theory of rings in the following), considers a full subcategory $\mathcal{K}$ of its models (think of this as analogous to $Aff^{op}$, the full subcategory of finitely presented rings), then observes that there is a canonical geometric morphism
$PSh(\mathcal{K}^{op}) \to Sh(C)$(which, accordingly, we think of as analogous to “$PSh(Aff) \to Sh(ThRing)$”) and finds that its hyperconnected/localic factorization is precisely that through the “topos of types” (which thereby, in the analogy, is the localic image of the “category of all (pre-)schemes” in $Sh(C)$).
This arises in the literature in a purely logic context. I was wondering just recently if something could be gained from looking at this from the geometric perspective, which very much reminds me of what you are asking here.
No, I really do mean the localic reflection: I was hoping to find a logical approach to explaining how the $\operatorname{Spec}$ construction is inevitable as soon as you decide that rings should be algebras of functions over locally ringed spaces. My first hope was that $\mathcal{Z}$ itself would turn out to be equivalent to $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$ but perhaps that was a little too naïve of me. Then my next thought was to look at the localic reflection of $\mathcal{Z}$.
I don’t think the subterminals of $\mathcal{Z}$ are trivial – surely it’s a very non-trivial lattice? Or maybe I’ve erroneously concluded that the principal cosieve generated by a ring is automatically a saturated cosieve and so is a non-trivial subsheaf of the terminal sheaf… I guess what I’m really looking for is an explicit description of $L \mathcal{Z}$.
I don’t think the subterminals of $\mathbb{Z}$ are trivial
They are rings that receive an epimorphism from $\mathbb{Z}$.
hoping to find a logical approach to explaining how the $Spec$ construction is inevitable as soon as you decide that rings should be algebras of functions over locally ringed spaces.
The $Spec$ construction can be nicely understood as being the left adjoint to the forgetful functor from locally ringed spaces to ringed spaces.
This is nicely discussed in section 2 of Structured Spaces.
I don’t think the subterminals of $\mathbb{Z}$ are trivial
They are rings that receive an epimorphism from $\mathbb{Z}$.
Is that all? That’s a little disappointing…
The $Spec$ construction can be nicely understood as being the left adjoint to the forgetful functor from locally ringed spaces to ringed spaces.
This is nicely discussed in section 2 of Structured Spaces.
Shouldn’t it be a right adjoint? I suppose it makes perfect sense given the property $Hom(X, Spec A) \cong Hom(A, \Gamma (X, \mathcal{O}_X))$… should have thought of that first!
Peter Arndt suggested another nice way of recovering Spec: apparently $\mathbf{Sh}(Spec A)$ is the “pullback” of the Zariski topos along the classifying morphism $\mathbf{Set} \to [\mathbf{Alg}^{fp}_{\mathbb{Z}}, \mathbf{Set}]$ of $A$. This makes some sense, since it suggests that what Spec is doing is finding the “smallest” topos in which $A$ becomes a local ring. I should probably first go find out what “pullbacks” of toposes are though…
I don’t think the subterminals of ℤ are trivial
They are rings that receive an epimorphism from ℤ.
Is that all?
Let’s check: the terminal object is $Spec \mathbb{Z}$. By the universal property of the spectrum, morphisms $X \to Spec \mathbb{Z}$ are in bijection with ring homomorphisms $\mathbb{Z} \to \mathcal{O}(X)$. Hence for the former to be a mono, this needs to be an epi.
That’s a little disappointing…
But it should be intuitively clear: $\mathcal{Z}$ is a big topos defining a geometry. These should themselves look like big fat points, and not having any inner structure beyond that universal geometric structure that they define. This is as for cohesive toposes (only that $\mathcal{Z}$ is not quite cohesive, I think, but the general idea is the same.)
Shouldn’t it be a right adjoint?
Sure, depending on which choice of variance you start with.
I should probably first go find out what “pullbacks” of toposes are though…
Ah, did Arndt mean pullback in the (2, 1)-categorical sense? I was having trouble reconciling his claim with Johnstone’s characterisation of pullbacks of subtoposes (Example A4.5.14(e)): because the pullback of a subtopos is supposed to be a subtopos, and there are no interesting subtoposes of $\mathbf{Set}$…
Actually, even the obvious “commutative” square only commutes up to a non-invertible (!!) geometric transformation $A \Gamma \Rightarrow i \mathcal{O}$, where $A : \mathbf{Set} \to [\mathbf{Alg}^{fp}_{\mathbb{Z}}, \mathbf{Set}]$ is the classifying morphism of the ring $A$ and $\mathcal{O} : \mathbf{Sh}(Spec A) \to \mathcal{Z}$ is the classifying morphism of the structure sheaf of $Spec A$, so I guess Arndt must have meant a lax pullback of some kind. This makes some sense, since morphisms of locally ringed spaces (should?) correspond to diagrams of geometric morphisms over $\mathcal{Z}$ commuting up to a non-invertible geometric transformation…
Right, I need to think more about that statement that you say Peter Arndt has made. I haven’t thought much about lax pullbacks of toposes, to be frank. (And probably what he has in mind is what more precisely is called the comma object, not the lax pullback.)
I’m getting more confused the more I think about it. For one thing, suppose we only test the universal property of the comma object over localic toposes. Then, that would mean that we are looking for a bijection between ringed locale morphisms $(X, \mathcal{O}_X) \to Spec A$ and ringed locale morphisms $(X, \mathcal{O}_X) \to (*, A)$, where $X$ is locally ringed. But Hartshorne gives an example [Ch. II, Example 2.3.2] of a ringed space morphism $Spec K \to Spec A$ which is not induced by a ringed space homomorphism $Spec K \to (*, A)$ – for concreteness, we could have $A = \mathbb{Z}_p$ and $K = \mathbb{Q}_p$. If I understand correctly there is in fact a two-to-one map because $Spec K$ can be mapped to either the open point or the closed point, and in either case we have a full $Hom (A, K)$ worth of morphisms. So in the end it looks like the only way to recover $Spec A$ correctly is via the adjunction between ringed spaces and locally ringed spaces… how disappointing.
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