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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!
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
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.
@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!
By the way, I am going to concentrate primarily on Grothendieck topoi in my course.
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”?
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?
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?
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! :-)
@OliviaCaramello:
Thanks for your explanations. Perhaps you would have an idea how to answer the following then:
If $(C,J)$ is a Grothendieck site, when is the topos $Sh(C,J)$ Hausdorff (i.e. has proper diagonal)?
Also- any advice about giving a course on (Grothendieck) topoi?
Thanks!
@Olivia
I like the idea of talking about toposes $Sh(C,J)$ 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!
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.
@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.
@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… :-)
@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.
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)
@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: $f:F\to E$ is proper if and only if the internal locale $f_*(\Omega_{F})$ in $E$ is compact. Notice that in our case (i.e., $f$ equal to the diagonal morphism on $F=Sh(C, J)$, regarded as a topos over sets), $f_*$ and $\Omega_{F}$ admit explicit descriptions in terms of the site $(C, J)$. To unravel what the condition “$f_*(\Omega_{F})$ is compact” means in terms of the site $(C, J)$, it is necessary to consider internal directed posets in $F\times_{Set} F$ and subterminal objects in categories of internal (co)presheaves; internal directed posets should admit an explicit description in terms of the site $(C, J)$, 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 $Sh(D, K)$ as the $K$-ideals on $D$.
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.
@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).
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.
(I thought of that quote too, but thought it in turn might have a still older, more popularly based source.)
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
A typo on the page:
David Carched, Max Planck Institute for Mathematics, office: 314
@DavidR: Thanks!
You should make the text lighter!
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