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.

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.)

]]>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…

]]>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…

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 $\mathbb{Z}$ 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*.

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 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.

]]>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…?

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