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+Olivia Caramello#toposIHES -

#OliviaCaramelloHere is a text recently posted by Olivia Caramello on her website:

http://www.oliviacaramello.com/Unification/InitiativeOfClarificationResults.html#UpdatesIn spite of the implicit recognition of the importance of my research programme made in his letter, A. Joyal has not stopped showing signs of disdain and hostility towards my work and my person in the last months. His behavior at the recent conference “Topos à l’IHÉS” that I co-organized is exemplary of the apparent impossibility for (some of) the category theorists of the old generation to understand and accept my ideas. I had proposed that he be invited to give one of the two tutorials on topos theory taking place on the first two days of the conference (the other one being given by me). He happily accepted the invitation formulating at the same time the unusual request to give an additional separate lecture at the conference as well. A few weeks before the beginning of the conference, I contacted him to agree on the contents of our respective courses, so to offer the audience a coherent picture of the subject without significant overlaps. He tried to press me to devote a substantial part of my course to elementary toposes in place of the ‘bridge’ technique, which nonetheless was not supposed to be mentioned in his own course. We finally agreed on giving two independent courses. According to the abstract that he had submitted (available here), he would introduce the fundamental notion of site and that of a category of sheaves on a site in the first part of his course; I could then build on these notions to present the theory of classifying toposes and the ‘bridge’ technique, in which sites play a central role. Contrary to what he had announced, Joyal did not introduce the notion of site in the first two lectures (so I had to define it in mines), which caused the complaints of a number of people. The problem was not remedied in the following lectures, where neither the notion of category of sheaves on a site nor the fundamental concept of flat functor were defined. These choices made it crystal clear that sites and presentations are essentially irrelevant in his “big picture” of topos theory. This greatly contrasts with the essential role that these concepts play in the ‘bridge’ technique, as the carriers of ‘concrete’ mathematical information that make it possible to apply topos theory in a variety of different mathematical situations and use classifying toposes as tools for unifying different mathematical theories with each other.

As it was already explained here, the strict and blind adherence to the Lawverian ‘site-free’ ideology, and the resulting mathematics done on one level rather than two, is the main mathematical reason behind the lack of understanding of my work that I have experienced from some senior category theorists since the times of my Ph.D. thesis. I was therefore not overly surprised that Joyal did not attend the third and fourth lectures of my course, where I presented the theory of topos-theoretic ‘bridges’ and its applications (I should add that I attended instead all of his lectures, and he attended all those of the other speakers!). Once again, an opportunity of scientific dialogue and confrontation in light of the mathematical fruits brought by this novel approach has been deliberately missed.

I should say that I find Joyal’s statement from his letter that my methodology ‘toposes as bridges’ is a vast extension of Klein’s Erlangen Program correct both technically and conceptually. Indeed, every group gives rise to a topos (namely, the category of its actions on sets), but the notion of topos is much more general. As Klein classified geometries by means of their automorphism groups, so we can study first-order geometric theories by studying the associated classifying toposes; as Klein established surprising connections between very different-looking geometries through the study of the algebraic properties of the associated automorphism groups, so the methodology ‘toposes as bridges’ allows to discover non-trivial connections between properties, concepts and results pertaining to different mathematical theories through the study of the categorical invariants of their classifying toposes.

Nonetheless, the few exchanges that I had with Joyal following his letter have left me with the impression that he has not actually understood the sense of my work (nor its technical aspects) and that he is not interested in trying to read it at all. This is all the more paradoxical since in the seventies Joyal was one of the inventors of the theory of classifying toposes of geometric theories, together with Makkai, Reyes and others. On the other hand, it might be precisely the lack of understanding from that generation of scholars of the deep meaning and immense applicative potential of the notion of classifying topos (which is revealed by the methodology ‘toposes as bridges’), and the resulting decision of not pursuing the development of that theory, the real reason behind their hostility towards my work. It is quite clear to me that most of these people, unlike Grothendieck (and me), have always conceived toposes as special kinds of abstract ‘algebraic’ structures to be studied essentially for their own sake rather than as meta-mathematical tools that can be used for the investigation and solution of concrete mathematical problems.

In this respect, the Introduction of Johnstone’s “Topos Theory” (1977), whose main focus and inspiration is the theory of elementary toposes, is particularly illuminating: in it, Johnstone notably talks about the “fundamental uselessness” of the general existence theorem for classifying toposes (!), complains that “the full import of the dictum that “the topos is more important than the site” seems never to have been appreciated by the Grothendieck school” and concludes that, unlike Grothendieck, he does not “view topos theory as a machine for the demolition of unsolved problems in algebraic geometry or anywhere else”. This was in 1977, but he has not changed his mind ever since. My work, which is Grothendieckian in spirit and actually allows to vindicate Grothendieck’s intuition of (Grothendieck) toposes as unifying spaces across different mathematical theories, is thus unbearable for him (despite the fact that I was his own Ph.D. student!). I have been regularly told (by different category theorists) things such as “you are not one of us” (referring to my different way of doing mathematics), “you have made me and Lawvere seem idiots” (referring to my address at the CT 2010) and “it is not that we do not understand, it is that we do not want to understand”. This is an attitude that one might label as “mathematical fanaticism”. I therefore no longer consider it my problem to make my word heard by people that do not want to hear.