Timeline for The unification of Mathematics via Topos Theory
Current License: CC BY-SA 2.5
20 events
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| Mar 18, 2013 at 14:24 | comment | added | Chandan Singh Dalawat | En tout cas, les mathématiques contemporaines fournissent un exemple d'équivalence extraordinairement profonde et très etudiée qui, sans avoir jusqu'à présent été formulée comme une équivalence de Morita, semble tout de même très proche du cadre général de la théorie de Caramello : c'est la correspondance de Langlands. $$ $$ Laurent Lafforgue (ihes.fr/~lafforgue/math/TheorieCaramello.pdf) | |
| Jul 16, 2010 at 21:42 | comment | added | Marco Benini | Incidentally, you should comment on the crucial features of Olivia's methods as well as on the other examples I provided and those contained in her paper before reaching a final judgement on the overall sense of her claims. It seems unfair to me to judge a work from a single example; instead I would be happy to discuss about the meaning of the whole theory, as the original Question asks for. And I would like to encourage all the mathematicians who are reading us to familiarize themselves with the methods in the paper so to be able to generate themselves their favourite applications. | |
| Jul 16, 2010 at 21:42 | comment | added | Marco Benini | The fact that, besides generating new results, one sometimes recovers classical theorems by applying Olivia's theory is a clear indication of the centrality of these methods and should not be interpreted as a proof of the fact that the theory lacks substance, as you seem to suggest. It is precisely the fact that one can arrive at interesting results by means of completely different and abstract techniques that should prompt the serious consideration of any mathematician. | |
| Jul 16, 2010 at 21:41 | comment | added | Marco Benini | While we are on the topic of fields, another less trivial (although well-known) result that can be read straight-off from the description of the Booleanization of the classifying topos for fields given in "De Morgan's law and the theory of fields" is the fact that any two countable algebraically closed fields of a given finite characteristic in which every element is algebraic over the prime field are isomorphic. | |
| Jul 16, 2010 at 21:40 | comment | added | Marco Benini | Dear BCnrd, I have to say that I'm quite surprised by your answer. I see now that the example we discussed is "easy" in the theory of fields, but the point I wanted to make is that, once you have a description of the classifying topos, this result as well as many others (e.g., the ones I previously reported) flow naturally and immediately from it (and I really mean by a one-line yet entirely rigorous proof) - note that to prove this result by standard methods, a basic knowledge of the theory of field extensions is needed, while the topos-theoretic proof follwos a completely different path. | |
| Jul 12, 2010 at 12:36 | comment | added | BCnrd | @Marco: Thanks very much for the follow-up! Now that I see the definitions of terms I didn't know, the assertion with finite fields is very well-known, easy to prove directly (not deep), and has been used for a long time. If someone told you it's hard to see with standard methods, they're wrong. It supports my original "concern": after unraveling the logic terminology, the "transferred" result in the other area of math will be obvious if no creative effort was required. It's up to Olivia to justify her claim of interesting/deep applications to other areas of math; I'll leave it to her. | |
| Jul 11, 2010 at 15:16 | comment | added | Marco Benini | If you (or other interested people) want further clarification on this or other details of Olivia’s work, I would suggest to contact her directly by e-mail; she confirmed to me that she would be entirely happy to explain/discuss any aspect of her work with you. And I must also confess that I'm unable to answer your last question since my training and intuition in pure algebra is limited - I'm mainly a logician. | |
| Jul 11, 2010 at 15:11 | comment | added | Marco Benini | The free filtered-colimit completion D of a category C is characterized by the following universal property: for any category V with filtered colimits, any functor F:C --> V extends, uniquely up to canonical isomorphism, to a filtered-colimit-preserving functor D --> V (the fact that every field which is algebraic over its prime field is a filtered colimit of finite fields then follows immediately from this characterization). Anyway, I think that the discussion is getting too specialized and far from the original question to be appropriate for this website and for my own expertise. | |
| Jul 8, 2010 at 2:51 | comment | added | BCnrd | Marco, thanks for the reply. Unfortunately I don't know enough about the part of topos theory which you use (e.g., I've never heard of "flat functors" or "filtered colimit completion", etc.). So let's focus on your first claim, to unravel what it is saying. Can you describe in concrete terms what it means that the category of fields algebraic over a finite field is the "free filtered-colimit completion" of the category of finite fields? Presumably this means something more than the assertion that such fields are directed unions of their finite subfields. Can you say what it means? | |
| Jul 5, 2010 at 8:10 | comment | added | Marco Benini | As I have already said, due to the novelty of these methods, not many applications outside logic have been obtained so far, but this should not distract us from the potential of the theory and its range of applicability. Which, by the way, is exactly what the original question asks for. | |
| Jul 5, 2010 at 8:08 | comment | added | Marco Benini | In my opinion, what should be especially noted in all of this is not just the concrete interest of these results but the nature of the methods, which - as it can be understood even from a non-technical reading of Olivia's work - are completely general and extremely flexible (starting from any Morita-equivalence, a great number of results can be extracted/transferred by using topos-theoretic invariants). | |
| Jul 5, 2010 at 8:00 | comment | added | Marco Benini | As an example, by transferring the invariant property of the classifying topos to be two-valued (respectively Boolean, De Morgan) across the two different sites one gets that the category of finitely presentable models of the theory is strongly connected (respectively is a groupoid, satisfies the dual of the amalgamation property) if and only if the theory satisfies a logical condition such as (geometric) completeness (respectively the property of being Boolean, the property of being De Morgan). | |
| Jul 5, 2010 at 7:51 | comment | added | Marco Benini | More in general, whenever one has a theory of presheaf type T (Olivia gives several concrete examples of theories of this kind belonging to different fields of mathematics in her papers/PhD thesis), one can transfer results between the theory (in terms of the syntactic site of definition of its classifying topos) and the theory of flat functors on the opposite of the category of finitely presentable models of T. Easy properties of one theory may translate into 'deep' properties of the other. | |
| Jul 5, 2010 at 7:49 | comment | added | Marco Benini | Then we can use invariants, as suggested by Olivia in her paper, to transfer results between the two theories. In this way one derives, for example, the fact that the category of fields of finite characteristic which are algebraic over their prime fields is the free filtered-colimit completion of the category of finite fields. As far as I know (and, as said before, I'm not an expert), results like these are of interest in Algebra and hard to obtain using standard techniques. In this case, the result is just "information-extraction" from the classifying topos via a suitable invariant. | |
| Jul 5, 2010 at 7:44 | comment | added | Marco Benini | Even though the paper "De Morgan's law and the theory of fields" is entirely written in topos-theoretic language, one can easily derive from it interesting insights in the theory of fields. One main point of the paper is the explicit description of the classifying topos of the theory of fields of finite characteristic which are algebraic over their prime fields as the category of functors from the category of finite fields to the category of sets. This implies that we have a Morita-equivalence between this theory and the theory of flat functors on the opposite of the category of finite fields. | |
| Jul 5, 2010 at 7:42 | comment | added | Marco Benini | @BCnrd: Sorry for the delay in replying, but I was away attending a lecture course given by Olivia on the contents of her paper "The unification of Mathematics via Topos Theory". I also discussed your comments with Olivia: in the following, I will try to report the answers, as far as I have understood them. Since I'm a logician and not an expert in the theory of fields, I'll apologize in advance for not going into the details of an algebraic treatment. | |
| Jun 30, 2010 at 4:34 | comment | added | BCnrd | Dear Marco: what is interesting (let alone deep) from the viewpoint of the theory of fields in the link you give? (I don't think I have ever encountered a situation where the DeMorgan issue is relevant, and I know the theory of fields extremely well and have used it in many many ways.) In particular I still don't see what justifies the claim that her works allows one to translate something simple in one part of math into something deep in another. As far as I can tell, the grand sweep of what Olivia claims is not appropriate at the present time. | |
| Jun 29, 2010 at 14:31 | comment | added | Marco Benini | @BCnrd: a partial answer to your question can be find in arxiv.org/abs/0808.1972 which shows a non-trivial result obtained in Algebra, specifically in the theory of fields. On the other side, one should consider that Olivia's theory has been conceived in 2008. It has reached its full expression at the end of 2009, and in the last months it has been refined and extended, as one can check in the Math Archive. It is not strange that there is a limited number of examples outside logic: If one has to develop a new theory, s/he would first focus on the field where it naturally lives. | |
| Jun 29, 2010 at 3:40 | comment | added | BCnrd | @Marco: It will help to clarify matters if you can provide one example of a deep result in some part of mathematics apart from logic or set theory which is deduced from a simple result in another theory by means of topoi. (Note: results which are routine applications of Zorn's Lemma and/or easily equivalent to the Axiom of Choice do not count as deep.) | |
| Jun 28, 2010 at 20:11 | history | answered | Marco Benini | CC BY-SA 2.5 |