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I made some edits at classifying topos to correct what I thought were some inaccuracies. One is that simplicial sets classify interval objects, but offhand I didn’t see the exact notion of linear interval over at interval object that would make this a correct sentence. In any event, I went ahead and defined the notion of linear interval as a model of a specified geometric theory.
The other is for local rings. I think when algebraic geometers refer to a sheaf of local rings, they refer to a sheaf of rings over a (sober) space all of whose stalks are local. I wasn’t sure that description would be kosher for a general (Grothendieck) topos $E$ since there may not be any “stalks” (i.e., points $Set \to E$) to refer to. In any case, it seems to me safer to give the geometric theory directly.
Thanks! That laziness about intervals bothered me too, but I was too lazy to unlazify it. I think Joyal (?) sometimes calls these “strict intervals”, to indicate that top and bottom are distinct.
I also agree that it’s certainly not kosher to define anything stalkwise over a topos without enough points. Perhaps the geometric theory should also be mentioned at locally ringed topos?
Thanks! I added the remark about strict intervals (a little rashly now that I look back, since you put a question mark by Joyal), and mentioned that the non-strict ones are classified by presheaves on the augmented simplex category. On your recommendation, I also added a geometric theory-style definition to locally ringed topos.
Notice that we do have an entry interval, which does describe the poset-intervals needed here.
It appears at interval object only under “related concepts”, though. It would be good to somehow clarify the relation between these entries further, or else otherwise harmonize them.
Urs, your first sentence is true, but the strictness assumption is missing (and of course there is a linearity assumption as well). Because there are classically equivalent formulations which become inequivalent when interpreted in a general topos, I think it’s best not to leave anything to chance, and spell everything out carefully at classifying topos.
Sure, I didn’t imply anything else.
I didn’t know that fact about non-strict intervals and the augmented simplex category. Where is a reference?
If it’s not in Mac Lane and Moerdijk (and I don’t know, because my copy went AWOL), then I don’t know where. Actually, I said it a bit offhandedly – let me see if it’s actually true! The idea of course is that the schizophrenic or ambimorphic adjunction induced by the 2-element set in different guises, which is familiar in the form of an adjoint equivalence
$\Delta^{op} \simeq StrictFinInterval$extends to an equivalence $\Delta_{a}^{op} \simeq FinInterval$. Then, for a topos $E$, a general (non-strict) interval corresponds to a filtered colimit of representables of finite intervals,
$FinInterval^{op} \stackrel{\hom(-, [n])}{\to} Set \stackrel{\Delta}{\to} E$(again, just as in the strict interval case), which corresponds to a left exact left adjoint
$Set^{\Delta_{a}^{op}} \simeq Set^{FinInterval} \to E$as required.
The proof in ML&M is very concrete and not very conceptual. If you know a nice conceptual proof, it would be a service to have it on the nLab! My intuition is certainly something along the lines of the fact that diagrams on the category of finitely presented models of a theory classify models of that theory, but that precise theorem is only true for finite-limit theories, which doesn’t include (even non-strict) intervals. Is the central fact that makes it still work the one you mentioned above, that a general interval is still a filtered colimit of finite(ly presented) ones? That seems offhand like it should be true for models of all finitary theories, since the category of such models ought to be finitely accessible. Maybe I’m just confused….
Mike, maybe I’m confused myself. In fact I found myself getting confused trying to understand the scope of ’finitary theory’ here, and reading Makkai-Paré’s book wasn’t helping me out of this confusion… on page 58 they write, “Let us call a category elementary if it is equivalent to $Mod(\mathcal{S})$ for a coherent sketch $\mathcal{S}$ (see Section 3.1). The term “elementary” is suggested by the established term “elementary class” in model theory. An elementary class is the class of all models of a small set of axioms in first order logic [my emphasis]. By what we said in the previous sections, the elementary classes are precisely the same as the object-classes of categories of the form $Mod(\mathcal{S})$ for coherent sketches $\mathcal{S}$.”
What confuses me is this unrestricted “small set of axioms in first order logic”, whereas coherent logic puts restrictions on the formulas. I wasn’t sure what sorts of logical restrictions you had in mind, if any.
Anyway, let me step back and say what I was probably unconsciously thinking in #8, which was something of a sledgehammer approach. A coherent theory (like the theory of linear intervals or of strict linear intervals) should be uniquely determined from its models in $Set$, by Deligne’s theorem. Therefore (I want to conclude), to check that $Set^{C^{op}}$ is a classifying topos for a coherent theory $T$, it’s enough to test this in $Set$, i.e., establish that $T$-models in $Set$ are equivalent to flat functors $C \to Set$ – this special case would ensure the general case for any topos in place of $Set$.
Assuming this meta-principle, here’s roughly how I’d argue that augmented simplicial sets classify (non-strict) linear intervals. Given a filtered diagram of representables
$J \stackrel{F}{\to} FinInterval \stackrel{yoneda}{\to} Set^{\Delta_a}$let $I$ be the filtered colimit of $J \stackrel{F}{\to} FinInterval \stackrel{i}{\hookrightarrow} Interval$. Then I want to claim that $Interval(i-, I)$ is the colimit of the first filtered diagram. But that’s obvious, because finite intervals $[n]$ are finitely presentable in $Interval$, so
$Interval(i [n], I) = Interval(i [n], colim i \circ F) \cong colim Interval(i [n], i \circ F) \cong colim FinInterval([n], F)$naturally in $[n]$, in other words $Interval(i-, I) \cong colim (yoneda \circ F)$. Then, because finite intervals are dense in all intervals, we conclude that the category of intervals is equivalent to the category of flat functors $\Delta_a \to Set$.
This sounds just like what you were suggesting when you mentioned the category of models being finitely accessible.
When I wrote “finitary” I giess I was really thinking of “coherent”. But I think what Makkai and Pare have in mind is their Prop. 3.2.8 which shows that any first-order theory is equivalent over classical logic to a coherent one. In the Elephant (D1.5.13) this is called Morleyization.
Let me think a bit about the rest of your comment.
Mike #11: ohhh!!! I think I remember seeing it in the Elephant before, and thinking to myself, “so this is the precise technical sense in which people say Deligne’s theorem is equivalent to the Gödel completeness theorem for first-order logic.” And then forgot about it, or at least the result didn’t hit me with force until now.
We should record that somewhere on the nLab.
Todd #12, Mike #13: I’ve heard that analogy from my lecturer several times… I wish I knew what it meant! (If I had to guess, “enough points” means that truth can be checked at the level of stalks…?)
(If I had to guess, “enough points” means that truth can be checked at the level of stalks…?
I haven’t been following closely, so if your question is about something more arcane than then notion of a topos with enough points, then please ignore the next sentence.
But if you are asking for what it means for a topos to have enough points, see def. 2 at point of a topos.
Zhen, the result I was actually hoping for in #10, and which may be only loosely associated with “enough points”, goes like this: suppose $T$ is a coherent theory, and that $E$ is a Grothendieck topos equipped with a $T$-model, such that the canonical map
$Topos(Set, E) \to Model_T(Set)$is an equivalence. Then $E$ is the classifying topos for $T$. In other words, I was hoping for a meta-principle that says, at least for some class of theories or toposes, that as long as you get the right answer in $Set$, you have the correct classifying topos. But something smells pretty fishy about the blunt generality in which I’ve stated it here, and now I only hope that at least something along these lines is true, but I don’t want to guess what the something might be until I think it over more.
Let’s see. Suppose $E$ is as you suggest, and let $Set[T]$ denote the actual classifying topos of $T$. Then the given $T$-model in $E$ gives us a geometric morphism $E\to Set[T]$, and the hypothesis tells us that $Set$ sees this morphism as an equivalence. On the other hand, the hypothesis that $E$ has enough points tells us that the collection of geometric morphisms $Set\to E$ is jointly co-faithful and co-conservative. We’d like to conclude from these facts that $E\to Set[T]$ is an equivalence, but offhand I don’t see how.
And actually, aren’t there examples of constructively inequivalent coherent theories that are classically equivalent?
Todd #16: Ah. That definitely looks like a completeness theorem to me, thanks! Now that I have an idea of what’s being discussed, I think this is the relevant claim in Mac Lane and Moerdijk [Ch. X, §7, Cor. 2]:
Let $T$ be a [coherent] theory. A formula $\forall x (\phi (x) \to \psi (x))$ as above holds in all models of $T$ in any topos, iff it holds in all models of $T$ in the topos $\mathbf{Sets}$.
(I have replaced their use of “geometric” with “coherent”, in line with current practice.) The proof explicitly makes use of Deligne’s theorem that the classifying topos of $T$ has enough points because it is a topos over a site of finite type.
It appears to me that your principle should follow if one can show that Set is a separator (in the appropriate 2-categorical sense) in the category of toposes with enough points: that is, if we have two non-isomorphic geometric morphisms $f, g : E \to F$ with $f \ncong g$, then there should be a geometric morphism $p : \mathbf{Set} \to E$ such that $f \circ p \ncong g \circ p$. Regrettably I am insufficiently practised in 2-category theory to see how this might be proven…
Zhen: how would that imply Todd’s principle? If Set were a strong generator, then I would see it.
Agh, I must have confused the fact that $1$ separates in $\mathbf{Set}$ with the fact that $1$ represents the forgetful/identity functor, of which the 2-analogue is what we actually want to prove. Sorry!
I asked Olivia Caramello about this after seminar today and she said that if a geometric morphism $E \to F$ induces an equivalence $\mathrm{Geom}(\mathbf{Set}, E) \simeq \mathrm{Geom}(\mathbf{Set}, F)$, and $E$ and $F$ are coherent toposes, then $E$ and $F$ are indeed equivalent; apparently it’s a corollary of the conceptual completeness theorem of Makkai and Reyes.
Thank you, Zhen! I’d like to think that over a bit before getting back to you.
Oh! It’s even stated explicitly as D3.5.9 in the Elephant. “Let S and T be coherent theories, and suppose we are given an interpretation of S in T for which the induced functor T-Mod(Set) $\to$ S-Mod(Set) is an equivalence. Then S and T are Morita-equivalent [i.e. their classifying toposes are equivalent]”.
Now I actually believe that we can prove the theorem about simplicial sets in this way. But I still don’t quite follow the argument in #10 — I think I understand up until the sentence “Then, because finite intervals are dense in all intervals, we conclude that the category of intervals is equivalent to the category of flat functors $\Delta_a \to Set$.” Can you explain how that sentence follows from the preceeding?
Mike, let me back up a bit. I first argued that a flat functor $\Delta_a \to Set$, in other words a functor which arises as a colimit of a filtered diagram
$J \stackrel{F}{\to} FinInterval \stackrel{yoneda}{\to} Set^{\Delta_a},$is isomorphic to a functor of the form $Interval(i-, I)$ for some interval $I$. So $Flat(\Delta_a, Set)$ is equivalent to the full subcategory $\mathcal{F}$ of $Set^{\Delta_a}$ consisting of functors of the form $Interval(i-, I)$. Now the functor
$Interval \to \mathcal{F}: I \mapsto \hom(i-, I)$is essentially surjective, and full and faithful precisely because $i: FinInterval \hookrightarrow Interval$ is a dense inclusion. Therefore the composite
$Interval \to \mathcal{F} \simeq Flat(\Delta_a, Set)$is an equivalence.
Ah, okay, I think I follow. I was forgetting that a functor into Set is flat iff it is a filtered colimit of representables. This seems to work, but I’m still worried because it doesn’t seem like you’ve used anything other than that (1) the theory of intervals is coherent and (2) the category of intervals in Set is finitely accessible. Is is true that the classifying topos of any coherent theory whose category of Set-models is finitely accessible is a presheaf category on its category of finitely presented Set-models?
I recalled that not every coherent theory has a finitely accessible category of models, and put a counterexample at coherent logic, but this would still be a (to me) surprisingly widely applicable result.
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