Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

Atrium > Mathematics, Physics & Philosophy: A course on topos theory

Bottom of Page

1 to 30 of 30

  1.  
    • CommentRowNumber1.
    • CommentAuthorDavidCarchedi
    • CommentTimeFeb 13th 2013

    Hello fellow category enthusiasts. I am giving serious consideration into teaching a course for master students on topos theory at the University of Bonn this coming German semester (starting in April). Does anyone have any helpful suggestions of topics to include, good lecture notes and/or references, or just other general advice? Does anyone have experience in teaching such a course? Much obliged, thank you!

    • CommentRowNumber2.
    • CommentAuthoradeelkh
    • CommentTimeFeb 13th 2013

    This may be somewhat relevant: Olivia Caramello recently gave a two-week course "Les topos de Grothendieck comme ponts unificateurs des mathématiques" in Paris, and the videos are available at:

    https://sites.google.com/site/logiquecategorique/cours/topos_caramello

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 14th 2013

    I have a copy of some lecture notes from when Peter Johnstone lectured the course in Cambridge some years ago, which I can send by email if you like. I haven’t had a look at them but they are probably almost entirely unlike the course Olivia Caramello gave last year.

    • CommentRowNumber4.
    • CommentAuthorDavidCarchedi
    • CommentTimeFeb 14th 2013

    @adeelkh, thanks for the link!

    @Zhen Lin: Yes, please email me those, that would be great! My email is: my_user_name_here@gmail.com. Thanks!

    • CommentRowNumber5.
    • CommentAuthorDavidCarchedi
    • CommentTimeFeb 15th 2013

    By the way, I am going to concentrate primarily on Grothendieck topoi in my course.

  6. Hello everyone,

    I confirm what Zhen Lin said as to the fact that the lecture notes from Peter Johnstone's course are "probably almost entirely unlike the course I gave last year". In fact, even if I studied in Cambridge (with Johnstone himself as a supervisor), the view of Grothendieck toposes as unifying 'bridges' which I have elaborated since the beginning of my Ph.D. studies is radically different from the 'elementary' approach to toposes which Johnstone has always favoured in his research.

    To say it briefly, the systematic study of classifying toposes, Morita-equivalences and (bijective) site characterizations for topos-theoretic invariants, which constitutes the 'core' of my theory, has been almost completely neglected by the categorical community in the past thirty years, with the unpleasant result that topos theory found only a very limited number of applications into 'concrete' mathematical areas; a situation to which I have been trying to remedy by concentrating on Grothendieck toposes (rather than on the class of elementary toposes). To 'taste the fruits' of this novel approach, I suggest visiting my website www.oliviacaramello.com , which contains general explanations of the abstract theory as well as links to my papers.
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013
    • (edited Feb 25th 2013)

    To say it briefly, the systematic study of classifying toposes, Morita-equivalences and (bijective) site characterizations for topos-theoretic invariants, which constitutes the ’core’ of my theory, has been almost completely neglected by the categorical community in the past thirty years,

    Could you briefly say what you mean by “(bijective) site characterizations for topos-theoretic invariants”?

  8. I mean characterizations of invariant properties of toposes (or constructions on them) direcly in terms of the sites of definitions for the toposes, that is logical biimplications of the kind: "a topos of sheaves on a site (C, J) satisfies an invariant I if and only if the site (C, J) satisfies a certain property explicitly written in the 'language' of the site (C, J)" (holding for 'large' classes of sites if not for any site). Various examples of such characterizations are given in my TAC paper "Site characterizations for geometric invariants of toposes" : http://www.tac.mta.ca/tac/volumes/26/25/26-25.pdf . As explained there, such characterizations are useful in connection with the view 'toposes as bridges' as they allow to transfer information between two distinct sites of definitions for the same topos.

    Throughout the past years several characterizations for topos-theoretic invariants had been established (see the 'Elephant', for instance); however, most of them had the form "A Grothendieck topos E satisfies an invariant I if and only if *there exists* a site of definition for E satisfying a certain property" and as such they could not be used for transferring information between different sites of definition for the same topos (in other words, they only allowed to 'enter' a given 'bridge' and not to 'exit' from it).

    A more comprehensive explanation of the concept of site characterization is available from my website at the address: http://www.oliviacaramello.com/Unification/TechnicalExplanation.html .
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013
    • (edited Feb 25th 2013)

    Thanks, I see.

    Now I also remember that I once looked at this article of yours and added it to the references at locally connected topos.

    Together with the characterization of sites for local toposes this gives almost a site characterization for cohesive toposes. I had been looking into finding decent site characterizations such that also the (infinity,1)-topos over the site is cohesive.

    You don’t happen to have an extension of your tools to higher toposes, by any chance?

  10. In principle, the tools that I have developed in the classical setting should be extendable to the higher-dimensional one; indeed, there is no reason why this should not be possible as my techniques are based on fundamental topos-theoretic notions which should admit - if they do not already - natural analogues in the higher-dimensional context. The problem in practice is simply that many of such analogues have not been technically developed yet: for instance, one still lacks a theory of higher classifying toposes as well as a good invariant theory for higher toposes. I am confident that, once the theory of higher toposes will be fully developed, it will be easy and natural to import the unifying techniques into it.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013
    • (edited Feb 25th 2013)

    Okay, I see. Could you point me to where in your article you lay out the general tools which you use in order to find “site characterizations”?

    In “Site characterizations for geometric invariants of toposes” I see a list of examples but no general mechanism (I haven’t yet read the article line-by-line though, I have to admit). But I gather you are saying that you produce these examples from some general machinery, some algorithm maybe? What is that general machinery and where is it discussed?

  12. The general method is explained in section 2 of the paper and formalized by Theorem 2.1: this metatheorem ensures the existence of bijective site characterizations for a large class of 'geometric' invariants of toposes. All the examples in the paper arise as applications of this general result to particular invariants.
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013
    • (edited Feb 25th 2013)

    Oh: I was looking at the arXiv version of your article which has no theorem 2.1.

    But now I went to the TAC webpage for your article, downloaded your published article from there and find that this now does have an expanded section 2 and does have a theorem 2.1 in it! :-)

    • CommentRowNumber14.
    • CommentAuthorDavidCarchedi
    • CommentTimeFeb 26th 2013

    @OliviaCaramello:

    Thanks for your explanations. Perhaps you would have an idea how to answer the following then:

    If (C,J)(C,J) is a Grothendieck site, when is the topos Sh(C,J)Sh(C,J) Hausdorff (i.e. has proper diagonal)?

    Also- any advice about giving a course on (Grothendieck) topoi?

    Thanks!

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 27th 2013

    @Olivia

    I like the idea of talking about toposes Sh(C,J)Sh(C,J) with an acknowledgement they have a specified site of definition. Allow me to rehash the famous quote:

    A lady (or gentleman) never chooses a site of definition.

    But if it has a canonical one, she (or he) should make use of it!

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2013

    David,

    a site for a topos is just like a basis for a vector space. Ladies and gentleman choose such all the time and for good reason. The problem is not in choosing it, but in forgetting that it is a choice.

    • CommentRowNumber17.
    • CommentAuthorDavidCarchedi
    • CommentTimeFeb 27th 2013

    @DavidRoberts: Also, there is always a canonical choice: the topos itself with the canonical topology. But at any rate, I agree with Urs- there are lots of good reasons to pick a site.

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 27th 2013

    @Urs, I know, hence the quote. But I think Olivia’s philosophy fits in with the riposte to the quote, that sometimes you don’t need to choose, as one comes for free, and it shouldn’t be completely forgotten.

    @DavidC - but if you want a small site… and universes are in scare supply… :-)

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2013
    • (edited Feb 27th 2013)

    @DavidR:

    not sure if we should further dwell on this, but I don’t agree with that quote (who are you quoting?) and I don’t see why this is “Olivia’s philosophy”. If you want to construct a topos with some properties and know how to generate it from a site, then you want to know which properties the site needs to have in order for the topos to have these properties. This is something one does all the time when working with toposes.

    As far as I understand, Olivia’s central point is rather that its worthwhile to look at this situation when one has two different sites for the same topos, since the resulting “Morita equivalence” of sites can be non-obvious from the point of view of just the sites.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 27th 2013
    • (edited Feb 28th 2013)

    That’s ok. It was only meant to be a light-hearted observation, nothing deep. (Edit: I should clarify I think the quote itself, which I merely constructed myself from the usual quote about vector spaces, doesn’t fit with the philosophy)

    • CommentRowNumber21.
    • CommentAuthorOliviaCaramello
    • CommentTimeFeb 28th 2013
    • (edited Feb 28th 2013)

    @DavidCarchedi

    Here are a few ideas on how to obtain a site characterization for the property of a topos to be Hausdorff. One could use the following well-known criterion for a geometric morphism to be proper: f:F→Ef:F\to E is proper if and only if the internal locale f *(Ω F)f_*(\Omega_{F}) in EE is compact. Notice that in our case (i.e., ff equal to the diagonal morphism on F=Sh(C,J)F=Sh(C, J), regarded as a topos over sets), f *f_* and Ω F\Omega_{F} admit explicit descriptions in terms of the site (C,J)(C, J). To unravel what the condition “f *(Ω F)f_*(\Omega_{F}) is compact” means in terms of the site (C,J)(C, J), it is necessary to consider internal directed posets in F× SetFF\times_{Set} F and subterminal objects in categories of internal (co)presheaves; internal directed posets should admit an explicit description in terms of the site (C,J)(C, J), while to obtain an explicit description of the subterminal objects one could use the “externalization” technique for categories of internal sheaves (see Proposition C2.5.4 in the “Elephant”) and then apply the characterization of the subterminal objects in a topos Sh(D,K)Sh(D, K) as the KK-ideals on DD.

    Concerning the topos theory course, I can send you by e-mail the syllabus of the course that I gave in Cambridge in 2011 and 2012 if you like; it was less advanced than the series of lectures that I have recently given in Paris (which were mostly about my own research), so it might be more suitable for beginners. It is probably a good idea to decide what to cover in the course also on the basis of the level of mathematical sophistication of the audience and the specific interests of the participants. For instance, the course that I gave in Cambridge was examinable and primarily addressed to students; accordingly, I explained the technical steps in more detail and did not treat research-level topics at length.

    • CommentRowNumber22.
    • CommentAuthorDavidCarchedi
    • CommentTimeFeb 28th 2013

    @Olivia:

    Thanks for the pointers about Hausdorffness. I will see if I can unravel them. I have an interesting application for having such a description.

    As far as the course, sure, send me the syllabus for your course, that could be helpful, thanks! (my email is my username here @gmail).

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2013
    • (edited Mar 3rd 2013)

    re #21 @Olivia: Is that reply you now gave an application of your theorem 2.1?

    I am still a bit unsure as to what’s going on. Somehow it seemed to me that there maybe was a claim that questions like in Dave’s #14 would have an entirely systematic “non-creative” answer, where we just apply some algorithm. Is that so?

    re #20 @David R.: Ah, now I finally found the source of that quote. Hadn’t heard of that before.

    • CommentRowNumber24.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 3rd 2013

    (I thought of that quote too, but thought it in turn might have a still older, more popularly based source.)

  25. @Urs

    My reply does not represent an application of Theorem 2.1. Indeed, the property of Hausdorffness does not seem to admit a characterization in terms of the existence of a class of objects and arrows in the topos satisfying some (invariant) property as in the hypotheses of the theorem; a specific argument is thus needed in this case. On the other hand, for invariants satisfying the hypotheses of the theorem, the process of obtaining site characterizations for them is essentially canonical and 'mechanical', as you expect.

    @DavidR

    I basically agree with the claim that one should ideally try to prove things about a certain mathematical object without using 'non-canonical' presentations for it; anyway, my primary aim is not to investigate toposes by using sites, but rather to investigate sites (and hence geometric theories) by using toposes. The focus is on getting concrete results and establishing fruitful relationships between specific mathematical theories by using toposes as 'bridge objects' connecting them; the toposes at the end of the process completely disappear, they are just instrumental for performing the 'translations' of properties and constructions between the theories as well as for 'calculating' on them.
  26. @DavidCarchedi

    I have just uploaded to my website the slides which I used as a support for the lectures of the topos theory course that I gave in Cambridge in 2012, together with the example sheets for the same course: http://www.oliviacaramello.com/Teaching/Teaching.htm

    I hope this is helpful :)
    • CommentRowNumber27.
    • CommentAuthorDavidCarchedi
    • CommentTimeApr 9th 2013

    In case anyone is interested, I have made a website for my course, and I also have (prematurely) posted the first homework assignment. Any comments are welcome! Thanks.

    http://people.mpim-bonn.mpg.de/carchedi/topos.html

    • CommentRowNumber28.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 9th 2013

    A typo on the page:

    David Carched, Max Planck Institute for Mathematics, office: 314

    • CommentRowNumber29.
    • CommentAuthorDavidCarchedi
    • CommentTimeApr 9th 2013

    @DavidR: Thanks!

    • CommentRowNumber30.
    • CommentAuthoradeelkh
    • CommentTimeApr 9th 2013

    You should make the text lighter!

1 to 30 of 30

  1.