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    • CommentRowNumber1.
    • CommentAuthorZhen Lin
    • CommentTimeMar 17th 2012

    Let 𝒵\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 Sh(Spec)\mathbf{Sh}(\operatorname{Spec} \mathbb{Z}) and 𝒵\mathcal{Z}. Certainly, there is a geometric morphism 𝒵Sh(Spec)\mathcal{Z} \to \mathbf{Sh}(\operatorname{Spec} \mathbb{Z}), and there’s also a geometric inclusion Sh(Spec)𝒵\mathbf{Sh}(\operatorname{Spec} \mathbb{Z}) \to \mathcal{Z}. On the other hand, there’s no chance of 𝒵\mathcal{Z} itself being localic, since it has a proper class of (isomorphism classes of) points. Let’s write L𝒵L \mathcal{Z} for the localic reflection of 𝒵\mathcal{Z}; the first geometric morphism I mentioned then corresponds to a locale map L𝒵SpecL \mathcal{Z} \to \operatorname{Spec} \mathbb{Z}. But what is L𝒵L \mathcal{Z} itself?

    The open objects in 𝒵\mathcal{Z} can be identified with certain saturated cosieves on 𝒵\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 𝒵\mathcal{Z}. This suggests that L𝒵L \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 SpecA\operatorname{Spec} A, for every finitely-presented ring AA, also show up…?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2012
    • (edited Mar 17th 2012)

    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 𝒵\mathcal{Z}. Isn’t that trivial?

    You might be thinking of the hyperconnected / localic -factorization of 𝒵Sh(Spec)\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 CC (think of this as analogous to the Lawvere theory of rings in the following), considers a full subcategory 𝒦\mathcal{K} of its models (think of this as analogous to Aff opAff^{op}, the full subcategory of finitely presented rings), then observes that there is a canonical geometric morphism

    PSh(𝒦 op)Sh(C) PSh(\mathcal{K}^{op}) \to Sh(C)

    (which, accordingly, we think of as analogous to “PSh(Aff)Sh(ThRing)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)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.

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeMar 17th 2012
    • (edited Mar 17th 2012)

    No, I really do mean the localic reflection: I was hoping to find a logical approach to explaining how the Spec\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 𝒵\mathcal{Z} itself would turn out to be equivalent to Sh(Spec)\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 𝒵\mathcal{Z}.

    I don’t think the subterminals of 𝒵\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 L𝒵L \mathcal{Z}.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2012

    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 SpecSpec construction is inevitable as soon as you decide that rings should be algebras of functions over locally ringed spaces.

    The SpecSpec 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.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeMar 19th 2012

    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 SpecSpec 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,SpecA)Hom(A,Γ(X,𝒪 X))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 Sh(SpecA)\mathbf{Sh}(Spec A) is the “pullback” of the Zariski topos along the classifying morphism Set[Alg fp,Set]\mathbf{Set} \to [\mathbf{Alg}^{fp}_{\mathbb{Z}}, \mathbf{Set}] of AA. This makes some sense, since it suggests that what Spec is doing is finding the “smallest” topos in which AA becomes a local ring. I should probably first go find out what “pullbacks” of toposes are though…

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2012
    • (edited Mar 19th 2012)

    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 SpecSpec \mathbb{Z}. By the universal property of the spectrum, morphisms XSpecX \to Spec \mathbb{Z} are in bijection with ring homomorphisms 𝒪(X)\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: 𝒵\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 𝒵\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…

    See at Topos the section Limits (scroll down a bit).

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeMar 19th 2012

    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 Set\mathbf{Set}

    Actually, even the obvious “commutative” square only commutes up to a non-invertible (!!) geometric transformation AΓi𝒪A \Gamma \Rightarrow i \mathcal{O}, where A:Set[Alg fp,Set]A : \mathbf{Set} \to [\mathbf{Alg}^{fp}_{\mathbb{Z}}, \mathbf{Set}] is the classifying morphism of the ring AA and 𝒪:Sh(SpecA)𝒵\mathcal{O} : \mathbf{Sh}(Spec A) \to \mathcal{Z} is the classifying morphism of the structure sheaf of SpecASpec 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 𝒵\mathcal{Z} commuting up to a non-invertible geometric transformation…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2012

    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.)

    • CommentRowNumber9.
    • CommentAuthorZhen Lin
    • CommentTimeMar 20th 2012
    • (edited Mar 20th 2012)

    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,𝒪 X)SpecA(X, \mathcal{O}_X) \to Spec A and ringed locale morphisms (X,𝒪 X)(*,A)(X, \mathcal{O}_X) \to (*, A), where XX is locally ringed. But Hartshorne gives an example [Ch. II, Example 2.3.2] of a ringed space morphism SpecKSpecASpec K \to Spec A which is not induced by a ringed space homomorphism SpecK(*,A)Spec K \to (*, A) – for concreteness, we could have A= pA = \mathbb{Z}_p and K= pK = \mathbb{Q}_p. If I understand correctly there is in fact a two-to-one map because SpecKSpec K can be mapped to either the open point or the closed point, and in either case we have a full Hom(A,K)Hom (A, K) worth of morphisms. So in the end it looks like the only way to recover SpecASpec A correctly is via the adjunction between ringed spaces and locally ringed spaces… how disappointing.

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