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edited classifying topos and added three bits to it. They are each marked with a comment "check the following".
This is in reaction to a discussion Mike and I are having with Richard Williamson by email.
Okay, I put that in now.
The first two were essentially right, I reordered a bit to make it I think more clear. I don't know enough oo-theory yet to answer the third one.
thanks
in the part on sites, you mention the "universal object". What is that, actually, in general?
@ Tim #2
When I read this, I thought ‘Isn't a theory of theories a doctrine?’. Then I saw that this isn't quite what Tim meant, but I added it to the page anyway.
The "universal object" meant the "generic model of the theory," i.e. the canonical functor from the syntactic category of the theory (in this case, the site) into its classifying topos. It's not in general a single object, of course, but for many theories, say the theory of a ring, we think of it as a single object (the ring) with structure.
the canonical functor from the syntactic category of the theory (in this case, the site) into its classifying topos.
Okay, so just to be sure: for our "theories of limits and cover-preserving functor" or whatever its called, this canonical functor is just Yoneda embedding followed by sheafificaton, as in Lurie's structured Spaces 1.4, right?
Yes.
In ’The idea’, the classifying topos $B G$ for $G$-torsors is being likened to the classifying space $\mathcal{B} G$. Is there an analogue on the classifying topos side for the total space, $E G$, of the universal $G$-bundle?
Interesting question, David. My off-the-cuff guess would be $E G = Set^G/G$, which is equivalent to $Set$. (Alternatively, you can think of $Set^G/G$ as presheaves over the action groupoid $G/G$ of $G$ regarded as $G$-set.) The covering projection would be
$\Pi_G: Set^G/G \to Set^G$
which takes an object $p: X \to G$ to the object of (equivariant) sections of $p$. I’m thinking that given a geometric morphism $\phi: E \to Set^G$ corresponding to a torsor $T$ of $\Delta G$ in $E$, the pullback $\phi^* E G$ is $E/T$, again equipped with the sections functor to $E$.
The big question seems to be: to which extent does the functor
$PSh_{(\infty,1)} : \infty Grpd \to \infty Toposes$and/or
$Sh_{(\infty,1)} : Top \to \infty Toposes$preserve limits and colimits.
That looks promising. A natural next question is whether other classifying toposes have some cover over them, but then it says at the entry “every topos F is the classifying topos for something”, which means I’d be asking “Does every topos has a cover?”. I don’t suppose the Freyd cover fits the bill. Now what is the Freyd cover? Will start a stub, and new latest changes discussion.
Ah, not the Freyd cover. We have at localic topos that $Set^G/G$ is a localic slice covering $Set^G$.
for any Grothendieck topos $E$, there exists an open surjection $F \to E$ where $F$ is localic.
which means I’d be asking “Does every topos has a cover?”.
Every (Grothendieck)-topos is equivalent to one of the form $Sh(N \mathcal{G}_{loc})$ for $\mathcal{G}_{loc}$ a localic groupoid. Suppose for simplicity this comes from a localic group $G_{loc}$, hence is $\mathcal{G}_{loc = }\mathbf{B} G_{loc}$. Then there is the corresponding $\mathbf{E} G_{loc} = G_{loc}/G_{loc}$. Then I’d think $Sh(N (G_{loc}/G_{loc})$ should be what you are looking for.
Give me a minute to collect some references and put this into the nLab…
I would be more inclined to say that the analogue of EG in the world of classifying topoi is the generic G-torsor, which is an internal G-torsor in the topos $Set^G$. The important aspect of the space EG is that as a principal G-bundle over BG, it is a universal element, i.e. the natural transformation $Hom(X,BG) \to G Bdl(X)$ that it induces (by the Yoneda lemma) is the isomorphism which exhibits BG as the object representing the functor $X\mapsto G Bdl(X)$. For the same Yoneda reasons, the classifying topos $Sh(C_T)$ of any geometric theory T comes with a generic T-model, which is a T-model in $Sh(C_T)$ which represents the functor $E\mapsto T Mod(E)$ in the same way. For T = the theory of G-torsors, this generic model is the generic G-torsor.
Now, because (as we’ve discussed elsewhere) an object X of a topos E can equivalently be identified with the “local homeomorphism of topoi” $E/X \to E$, and as an object of $Set^G$ the generic G-torsor is just G itself regarded as a G-set, for the case of G-torsors the point of view I espoused above gives the same answer as Todd. However, for a general geometric theory, the generic model can’t be as simply described as a single topos covering the classifying topos. In fact, for a given Grothendieck topos, the generic model that we get living in it will look quite different depending on what (presentation of a) geometric theory we choose to consider it as classifying. (Along the same lines, a given topos admits many different presentations in terms of sheaves on a localic groupoid.)
Is this worth noting at classifying topos, Mike?
Probably. (-: But I don’t have time right now to add it. (It takes a bit more effort for me to incorporate something like that into the flow of an existing page than to write it as a standalone forum post.)
I opted for the easiest incorporation, pasting in at the end of the page here. It needs working on.
Actually, I need to work through the page to see what’s going on. In the universal bundle case, $E G \to B G$ is a map of spaces. Pulling back can happen in some category of spaces. Given a geometric theory $T$, we’re saying that the equivalent is $U: C_T \to S[T]$ as in the definition section? In which case, maybe this analogy should be mentioned there. You have $C_T \to E$ factoring through $S[T]$. Is there not a fourth corner of the square, pulling back $U: C_T \to S[T]$ along $E \to S[T]$. Is there a problem that $C_T$ is not a topos?
I’d think we can follow the general logic of generalized universal bundle and realize the topos incarnation of the universal $G$-bundle as the (topos incarnation of the) point mapping into the (topos incarnation of) the classifying space.
So in $Grpd$ the universal $G$-bundle is simply
$\array{ * \\ \downarrow \\ \mathbf{B}G }$in that every $G$-principal bundle classified by a map $X \to \mathbf{B}G$ is the weak pullback of this morphism.
We hit this with $PSh(-) : Grpd \to Topos$ to get
$\array{ PSh(*) = Set \\ \downarrow \\ PSh(\mathbf{B}G) = Set^G } \,.$This should be the universal $G$-bundle in the world of toposes, in that for $Sh(X) \to PSh(\mathbf{B}G)$ any geometric morphism, the corresponding $G$-principal bundle over $X$ is in its topos-incarnation the weak pullback
$\array{ \mathcal{P} &\to& PSh(*) \\ \downarrow &\swArrow& \downarrow \\ Sh(X) &\to& PSh(\mathbf{B}G) } \,.$Now, of course the geometric morphism $PSh(*) \to PSh(\mathbf{B}G)$ picks a single object in $PSh(\mathbf{B}G)$, which is the universal $G$-torsor as in Mike’s message: that morphism sends a set $S$ to the $G$-set $S \times G$ equipped with the $G$-action on itself.
If we are not talking about group-principal bundles but about groupoid principal bundles, then of coruse this story becomes more involved, and I’d tink that this is what Mike refers to when he says that we don not in general have a covering space.
@Urs: If by “weak pullback” you mean pullback up to isomorphism, then yes, I think that’s what I was getting at with the remark about objects of a topos as local homeomorphisms over it. I really prefer “2-pullback” or “pseudo-pullback,” since “weak pullback” is also used as a case of a weak limit. I also get worried when you write a $\swArrow$ in the diagram without any isomorphism sign, since you might mean a comma object.
@David: yes, $C_T \to S[T]$ is the analogue of $EG \to BG$. But it should be regarded as a map into $S[T]$ instead of a space over $BG$, which is why we compose with $f^*\colon S[T] \to E$ to get the model $C_T \to E$ classified by $f$, instead of pulling back. Or, if you like, the notion of “pulling back” is already incorporated into the inverse image part $f^*$ of the geometric morphism $f$, which for instance can be realized as actual pullback of sheaves regarded as local homeomorphisms.
Yes, I mean, ahm “homotopy pullback” :-) I suggest we call that $(2,1)$-pullback, being a special case of an (infinity,1)-pullback.
So I get back to my remark I made elsewhere: there is good evidence that the $(\infty,1)$-functors
$PSh_{(\infty,1)} : \infty Grpd \to (\infty,1)Toposes$and
$Sh_{(\infty,1)} : Top \to (\infty,1)Toposes$preserve $(\infty,1)$-colimits and $(\infty,1)$-pullbacks. At least those involved in principal $\infty$-bundle theory (namely the colimits defining effective epimorphisms and the pullbacks defining homotopy fibers).
But I don’t know how to attack this statement beyond that low dimensional evidence.
added a section Universal bundle topos.
But needs to be expanded and harmonized with the paragraph on universal elements.
Why (2,1)-pullback? The categories in question are 2-categories, not (2,1)-categories.
Why (2,1)-pullback? The categories in question are 2-categories, not (2,1)-categories.
Yes, but we want the pullback with an invertible 2-cell.
A 2-pullback already has an invertible 2-cell. If the 2-cell isn’t invertible, it’s not a pullback, it’s a comma object.
It’s not common usage, but I am suggesting that a good terminology would be or would have been:
“2-pullback” or $(2,2)$-pullback for lax pullback
“(2,1)-pullback” for the notion where the 2-cells are required isos, i.e. for the pullback in the maximal sub-$(2,1)$-category of the given $(2,2)$-category.
And then (2,0)-pullback would be strict pullback, I presume?
Given an $n$-category $C$ I thought it would make good sense to speak of an $(n,r)$-pullback as a (fully general) pullback in the $(n,r)$-core of $C$, i.e. the maximal sub $(n,r)$-category.
This is for instance what we do when we speak of the (infinity,1)-pullbacks in (infinity,1)Cat (which is really an $(\infty,2)$-category) which are computed by homotopy pullbacks in the Joyal model structure.
With that convention an $(n,0)$-limit would be an limit in the core. Not so interesting.
Well, I disagree, as I’ve said before. I don’t think comma objects should be called any kind of pullback; they’re really a quite different beast to my mind. More specifically, since a “2-limit” is a limit up to isomorphism, so should a “2-pullback” be, or people will get pretty confused.
By the way, a pseudo-pullback in a 2-category is not exactly the same as a comma object in the (2,1)-core. Any one of the former is also one of the latter, but not conversely; the former has a stronger universal property that also says something about noninvertible 2-cells factoring through it uniquely. I presume that the (∞,1)-limits in the (∞,1)-category (∞,1)-Cat actually do possess this stronger (∞,2)-dimensional universal property, if one took the time to write it down in (∞,2)-categorical terms.
At least in 2-categorical situation I agree with Mike on this issue (comma vs. pseudopullbacks discussion). Though I tend to be sloppy in these issues.
But I am not talking about comma objects. (?)
You said you would have wanted to use “2-pullback” to mean “lax pullback.” I assumed that by “lax pullback” you meant a comma object; did you actually mean what I would call a lax pullback, namely the lax limit of a cospan? I think those are even less like ordinary pullbacks than comma objects are….
Urs, you will agree that if you are talking on comma objects or on pullbacks depends on what you are trying to do. I did not follow the discussion carefully but if you are talking about some sort of classifying object construction for bundles and using steps like Grothendieck construction, then in low dimensions, in my unreliable memory one does deal with the approapriate versions of comma objects. No ?
You said you would have wanted to use “2-pullback” to mean “lax pullback.” I assumed that by “lax pullback” you meant a comma object; did you actually mean what I would call a lax pullback, namely the lax limit of a cospan?
Yes. I am imagining that there is an evident definition of lax $(\infty,n)$-limit for all $n$: imagine you have a simplicial set incarnation of $(\infty,n)$-categories (say along the lines of Verity), then use Joyal’s simplicial formula for the simplicial set of cones over a diagram and find the terminal object in there. In other words, proceed verbatim as for $(\infty,1)$-limits modeled in quasi-categories. Just have more relaxed assumptions on what the simplicial set has to satisfy.
I did not follow the discussion carefully but if you are talking about some sort of classifying object construction for bundles and using steps like Grothendieck construction, then in low dimensions, in my unreliable memory one does deal with the approapriate versions of comma objects. No ?
The Grothendieck construction is a comma object, but the pullback that defines a principal bundle is a homotopy pullback. I am thinking this should remain true if the principal bundle is realized in its topos incarnation.
It’s possible that a simplicial approach would work to define higher-lax sorts of limits, although I’m a little skeptical that it would get the directions of everything right; simplicial nerves start to get really weird when the 2- and higher cells are noninvertible. However, I haven’t thought very much about what n-lax things would mean for n any bigger than the smallest possible (wherever the numbering should start), or what they would be good for; maybe you’re right. But my main point remains: lax pullbacks should not be called 2-pullbacks. If you want a number indexing the level of laxness, I would put a prefix on the word “lax” instead.
I have added to classifying topos a section Geometric morphisms equivalent to morphisms of sites containing the crucial lemma that explains “why classifying toposes work”.
have added a section for local algebras to classifying topos, with the 1-categorical analog of structured (infinity,1)-topos.
It is amazing that this 1-categorical analog has not been discussed or at least not discussed prominently before. Looks to me like a major omission. Lurie of course discusses it in the $\infty$-categorical context, but somebody should write it all out just for 1-categories.
I am going to make a puny start. Have been thinking about better terminology, suitably generalizing “locally ringed topos” but being more descriptive than structured topos.
How about
locally algebra-ed topos
??
There are people (Benno vdB, in the context of Bohrification) who speak of “C-stared toposes” to mean ringed toposes whose ring object is in fact a $C^*$-algebra object. So generally “algebra-ed toposes” seems to be a notion to go for. I am just not sure how exactly to spell it. ;-)
I have added to classifying topos a section For inhabited linear orders with statement and one half of the proof that $Set^{\Delta } = Sh(\Delta^{op})$ classifies those simplicial objects which are nerves of posets that are inhabited linear orders.
I prefer to say a “space with a structure sheaf (of rings)” than a strange abbreviation “ringed space”, which I always found misleading and when I was a student it confused me. Even worse with “locally ringed space” as if it were ringed locally, what is not true; it is so much easier to say with a structure local ring and everybody understands. So I like also “topos with a structure ring” and alike phrases which are understandable to people outside the subject unlike the strange abbreviation “ringed topos” (worse with “lined topos” where I still forget what is exactly meant). In fact there is no need to devise words which are unlikely to get acquired by all interested mathematicians Thus monoided and algebred and so on topoi and spaces makes me quite uneasy, it is inventing for no need; it looks also like it is an operation on topoi and not a structure. If it is a structure, one says with such structure. It is friendly, and it is in general a problem with category theory to be unfriendly to outsiders with lots of terminology and conventions. If it were needed than I would agree with the terminology, but rather lets not without need proliferate unclear abbreviations. We should rather save readers from confusion rather than few letters of ink.
You are replying to #38 from last year, right?
In principle I very much agree with what you say. Only that since the terminology “ringed space” etc. is entirely standard, it is not clear to me if saying “locally algebra-ed space” is more unfriendly to outsiders than changing their standard term and saying “space with structure ring”. But I agree that in principle the latter woule be better.
Ringed space is not any more standard than a “space with a structure ring” (or sheaf of rings). BOTH are standard, and many references introduce both terms simultaneously just the full version does not lead to confusion as often as the first (outside of the algebraic geometry community). Also the longer term is more often used when speaking to people of wider community (say in a colloqium talk, as opposed to an expert community). Plus in many languages “ringed space” does not translate, so one has to stick with the literal translation of the long version. And as I said, “locally” is far more confusing, as well as “lined topos”, for which I still tend to forget what is meant. The worst combination is monoided space where one can be mislead to having a space in some monoidal category or something like that (such notions exist). It is like an invitation for a confusion. A similar thing is with “simplicial category” which is ambigous abbreviation for either simplicial object in Cat or a simplicially enriched category. At least when mentioning it the first time within a context/paper/talk one should say the full name and not appeal to the jargon of his microcommunity. (The fact that the first can be viewed in a canonical way as a special case of the second does not make itan excuse, but rather it makes it worse.)
It is so easy to type several more characters and being unambiguous (surely, I myself used the locally ringed spaces in some papers of mine, but I am not happy about that decision, as it was intended to a wider community than alg. geom.). Of course in a context when one repeats often, and uses one and the same version of structure, one can simply make a local convention and say “space” for gadgets within his category of whatevered spaces.
I noticed that at classifying topos there was no pointer back to forcing. So I have now added in the Idea-section right after the mentioning of the universal model the following paragraph:
The fact that a classifying topos is like the ambient set theory but equipped with that universal model is essentially the notion of forcing in logic: the passage to the internal logic of the classifying topos forces the universal model to exist.
Please feel free to improve as need be.
I’m not sure what you mean by “spooky” or “sophistry” there. An empty assignment is of course just an empty function.
Well, I don’t doubt that it can be made precise, actually it works pretty well as stated. What I am a bit uneasy about is for one the codomain class of this empty function, then the empty collection of maps needs a composition defined in order to give a category of models and the idea that any model arises as $f^*(U)$ from the generic model seems to be lost. For a signature with sorts one thinks of a model rather in terms of the objects in the image of the assigment and for me this gives the empty assigment the feeling of being a model in name only since the concrete object under it is missing. But feel free to improve or correct the section! I never felt very happy with it and hesitated quite some time before adding it, since it is a bit too long and too syntax leaning in comparison with the rest of the examples, and probably too confusing by bringing in the idea of classification relative to a signature. On the other hand, $Set$ and $1$ are the most basic examples, so they should be mentioned somewhere and then I didn’t want to give impression that $Set$ only classifies the empty theory over the empty signature.
the codomain class of this empty function
Well, of course there are multiple empty functions: one from the empty set of sorts to the set of objects of the topos, another from the empty set of function symbols to the set of morphisms of the topos, etc.
the empty collection of maps needs a composition defined in order to give a category of models
Of course there is a unique empty function $\emptyset\times\emptyset\to\emptyset$, defining the empty category.
the idea that any model arises as $f^*(U)$ from the generic model seems to be lost
Not at all: there is exactly one model of the empty theory in any topos, including exactly one (generic) model in $Set$, and $f^*$ indeed maps the latter to the former. There’s no trouble defining $f^*$ on models of the empty theory.
For a signature with sorts one thinks of a model rather in terms of the objects in the image of the assigment
I’m not sure exactly what that means, but if it produces such confusion one should probably stop thinking that way. (-: You certainly can’t in general think of a model as a subset of the set of objects of the topos equipped with structure, if that’s what you mean, since some object might have to be used for more than one sort (e.g. a model of any multi-sorted theory in the one-object trivial topos).
I think the intent of that paragraph was to illustrate the fact the empty structure isn’t the only thing $Set$ classifies. E.g. it classifies natural number objects, and for any small set $S$, it classifies “objects that are $S$-indexed disjoint unions of the terminal object”.
The concrete example is to show $Set$ classifies initial objects, but this idea is being framed as formulating a theory with a single sort.
@49. In nutshell, my intuitive idea of a model seems to correspond basically to the situation prevalent in classical model theory: a carrier set plus structure. Since it is rather commonplace to blur the distinction between carrier and model, one is then tempted to read (the induced model) $f^*(U)$ as arising from the application of $f^*$ to the object $U$ carrying the generic model (I guess the leading examples as well as the rather casual introduction of the notation in the MacLane-Moerdijk book reinforce this intuition). A nice and useful picture as far as it goes but unfortunately inadequate to handle cases when more than one sort is around as you point out, as well as in the cases that cause my phantom pain here, when there is less than one sort around, creating a strange nostalghia for a concrete carrier object where none is to be found.Thanks for setting me straight!
Put a pointer to Diaconescu’s theorem that justifies passing to the presheaf topos in the case of the finite limit theory for groups.
I know it’s mentioned later in the article, but (IMO) the theory of groups seems to be the reference example and the one that’s most fleshed out, so this last detail should be included there.
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