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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 27th 2010
    • (edited Jan 14th 2013)

    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.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeJan 27th 2010
    Your question as to finding a new phrase to replace theory theory has an `obvious' answer `the theory of theories'.
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 27th 2010

    Okay, I put that in now.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJan 27th 2010

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 27th 2010
    • (edited Jan 27th 2010)

    thanks

    in the part on sites, you mention the "universal object". What is that, actually, in general?

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeJan 27th 2010

    @ 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.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 28th 2010

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2010

    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?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJan 28th 2010

    Yes.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 12th 2010

    In ’The idea’, the classifying topos BGB G for GG-torsors is being likened to the classifying space G\mathcal{B} G. Is there an analogue on the classifying topos side for the total space, EGE G, of the universal GG-bundle?

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 12th 2010
    • (edited Jul 12th 2010)

    Interesting question, David. My off-the-cuff guess would be EG=Set G/GE G = Set^G/G, which is equivalent to SetSet. (Alternatively, you can think of Set G/GSet^G/G as presheaves over the action groupoid G/GG/G of GG regarded as GG-set.) The covering projection would be

    Π G:Set G/GSet G\Pi_G: Set^G/G \to Set^G

    which takes an object p:XGp: X \to G to the object of (equivariant) sections of pp. I’m thinking that given a geometric morphism ϕ:ESet G\phi: E \to Set^G corresponding to a torsor TT of ΔG\Delta G in EE, the pullback ϕ *EG\phi^* E G is E/TE/T, again equipped with the sections functor to EE.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2010

    The big question seems to be: to which extent does the functor

    PSh (,1):GrpdToposes PSh_{(\infty,1)} : \infty Grpd \to \infty Toposes

    and/or

    Sh (,1):TopToposes Sh_{(\infty,1)} : Top \to \infty Toposes

    preserve limits and colimits.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 12th 2010

    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.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 12th 2010

    Ah, not the Freyd cover. We have at localic topos that Set G/GSet^G/G is a localic slice covering Set GSet^G.

    for any Grothendieck topos EE, there exists an open surjection FEF \to E where FF is localic.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2010
    • (edited Jul 12th 2010)

    which means I’d be asking “Does every topos has a cover?”.

    Every (Grothendieck)-topos is equivalent to one of the form Sh(N𝒢 loc)Sh(N \mathcal{G}_{loc}) for 𝒢 loc\mathcal{G}_{loc} a localic groupoid. Suppose for simplicity this comes from a localic group G locG_{loc}, hence is 𝒢 loc=BG loc\mathcal{G}_{loc = }\mathbf{B} G_{loc}. Then there is the corresponding EG loc=G loc/G loc\mathbf{E} G_{loc} = G_{loc}/G_{loc}. Then I’d think Sh(N(G loc/G loc)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…

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2010

    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 GSet^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)GBdl(X)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 XGBdl(X)X\mapsto G Bdl(X). For the same Yoneda reasons, the classifying topos Sh(C T)Sh(C_T) of any geometric theory T comes with a generic T-model, which is a T-model in Sh(C T)Sh(C_T) which represents the functor ETMod(E)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/XEE/X \to E, and as an object of Set GSet^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.)

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 13th 2010

    Is this worth noting at classifying topos, Mike?

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2010

    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.)

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2010

    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, EGBGE G \to B G is a map of spaces. Pulling back can happen in some category of spaces. Given a geometric theory TT, we’re saying that the equivalent is U:C TS[T]U: C_T \to S[T] as in the definition section? In which case, maybe this analogy should be mentioned there. You have C TEC_T \to E factoring through S[T]S[T]. Is there not a fourth corner of the square, pulling back U:C TS[T]U: C_T \to S[T] along ES[T]E \to S[T]. Is there a problem that C TC_T is not a topos?

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010
    • (edited Jul 14th 2010)

    I’d think we can follow the general logic of generalized universal bundle and realize the topos incarnation of the universal GG-bundle as the (topos incarnation of the) point mapping into the (topos incarnation of) the classifying space.

    So in GrpdGrpd the universal GG-bundle is simply

    * BG \array{ * \\ \downarrow \\ \mathbf{B}G }

    in that every GG-principal bundle classified by a map XBGX \to \mathbf{B}G is the weak pullback of this morphism.

    We hit this with PSh():GrpdToposPSh(-) : Grpd \to Topos to get

    PSh(*)=Set PSh(BG)=Set G. \array{ PSh(*) = Set \\ \downarrow \\ PSh(\mathbf{B}G) = Set^G } \,.

    This should be the universal GG-bundle in the world of toposes, in that for Sh(X)PSh(BG)Sh(X) \to PSh(\mathbf{B}G) any geometric morphism, the corresponding GG-principal bundle over XX is in its topos-incarnation the weak pullback

    𝒫 PSh(*) Sh(X) PSh(BG). \array{ \mathcal{P} &\to& PSh(*) \\ \downarrow &\swArrow& \downarrow \\ Sh(X) &\to& PSh(\mathbf{B}G) } \,.

    Now, of course the geometric morphism PSh(*)PSh(BG)PSh(*) \to PSh(\mathbf{B}G) picks a single object in PSh(BG)PSh(\mathbf{B}G), which is the universal GG-torsor as in Mike’s message: that morphism sends a set SS to the GG-set S×GS \times G equipped with the GG-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.

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeJul 14th 2010

    @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 TS[T]C_T \to S[T] is the analogue of EGBGEG \to BG. But it should be regarded as a map into S[T]S[T] instead of a space over BGBG, which is why we compose with f *:S[T]Ef^*\colon S[T] \to E to get the model C TEC_T \to E classified by ff, instead of pulling back. Or, if you like, the notion of “pulling back” is already incorporated into the inverse image part f *f^* of the geometric morphism ff, which for instance can be realized as actual pullback of sheaves regarded as local homeomorphisms.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010
    • (edited Jul 14th 2010)

    Yes, I mean, ahm “homotopy pullback” :-) I suggest we call that (2,1)(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 (,1)(\infty,1)-functors

    PSh (,1):Grpd(,1)Toposes PSh_{(\infty,1)} : \infty Grpd \to (\infty,1)Toposes

    and

    Sh (,1):Top(,1)Toposes Sh_{(\infty,1)} : Top \to (\infty,1)Toposes

    preserve (,1)(\infty,1)-colimits and (,1)(\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.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    added a section Universal bundle topos.

    But needs to be expanded and harmonized with the paragraph on universal elements.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeJul 14th 2010

    Why (2,1)-pullback? The categories in question are 2-categories, not (2,1)-categories.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    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.

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2010

    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.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010

    It’s not common usage, but I am suggesting that a good terminology would be or would have been:

    • “2-pullback” or (2,2)(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)(2,1)-category of the given (2,2)(2,2)-category.

    • CommentRowNumber28.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 15th 2010

    And then (2,0)-pullback would be strict pullback, I presume?

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    Given an nn-category CC I thought it would make good sense to speak of an (n,r)(n,r)-pullback as a (fully general) pullback in the (n,r)(n,r)-core of CC, i.e. the maximal sub (n,r)(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 (,2)(\infty,2)-category) which are computed by homotopy pullbacks in the Joyal model structure.

    With that convention an (n,0)(n,0)-limit would be an limit in the core. Not so interesting.

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2010

    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.

    • CommentRowNumber31.
    • CommentAuthorzskoda
    • CommentTimeJul 15th 2010

    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.

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010

    But I am not talking about comma objects. (?)

    • CommentRowNumber33.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2010

    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….

    • CommentRowNumber34.
    • CommentAuthorzskoda
    • CommentTimeJul 15th 2010

    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 ?

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010

    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 (,n)(\infty,n)-limit for all nn: imagine you have a simplicial set incarnation of (,n)(\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 (,1)(\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.

    • CommentRowNumber36.
    • CommentAuthorMike Shulman
    • CommentTimeJul 16th 2010

    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.

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2011

    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”.

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2011
    • (edited Apr 27th 2011)

    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 *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. ;-)

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2012
    • (edited Apr 16th 2012)

    I have added to classifying topos a section For inhabited linear orders with statement and one half of the proof that Set Δ=Sh(Δ op)Set^{\Delta } = Sh(\Delta^{op}) classifies those simplicial objects which are nerves of posets that are inhabited linear orders.

    • CommentRowNumber40.
    • CommentAuthorzskoda
    • CommentTimeApr 17th 2012
    • (edited Apr 17th 2012)

    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.

    • CommentRowNumber41.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2012

    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.

    • CommentRowNumber42.
    • CommentAuthorzskoda
    • CommentTimeApr 17th 2012
    • (edited Apr 17th 2012)

    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.

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2013
    • (edited Jan 14th 2013)

    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.

    • CommentRowNumber44.
    • CommentAuthorThomas Holder
    • CommentTimeAug 26th 2018

    What started as typo fixing ended by throwing in some remarks and references on the relation between NNOs and classifying toposes.

    diff, v64, current

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