Olivia Caramello's website

Unifying theory

Controversy with category theorists

Overview and results of the initiative of clarification

Introduction

Reasons why I have undertaken this initiative of clarification

The denigratory campaign

Reasons behind the denigratory campaign

A bit of history

A few selected episodes

André Joyal’s debate and concluding letter

Responses to the initiative of clarification

Some updates

Introduction

On January 1 2015 I sent a letter to some of the most prominent members of the categorical community containing six questions to which they were kindly invited to publicly answer in order to clarify their positions with regard to a number of persistent accusations, of both scientific and personal nature, that I had received throughout the past years. Specifically, whilst there have never been public criticisms as to the scientific solidity of my research program (or to the soundness of my technical papers), I have been repeatedly accused by some of these people of "over-selling" my research, or of proving results that were already known (but admittedly never written down or stated in public or recorded occasions).

The experts whom I contacted were: Steve Awodey, Michael Barr, Francis Borceux, Marta Bunge, Eduardo Dubuc, Martin Hyland, Peter Johnstone, André Joyal, F. William Lawvere, Michael Makkai, Ieke Moerdijk, Gonzalo Reyes, Philip Scott and Ross Street. The questions that I had posed were the following:

(1) Do you think that I made any mistakes, of either scientific or ethical nature, in presenting the research program of toposes as 'bridges' at the CT2010 (or in this paper), such as to justify the resulting persistent hostility from influential members of the category theory community? If yes, please explain.

(2) Do you think that it is right to comment, in the context of peer-review (or with colleagues in a more informal way), about an extended piece of research work in terms such as "I have known most, if not all, of this for decades but I have not written it down" or "all these things are well-known even though there are no traces of them in the literature or in recorded talks" to the point of recommending its rejection? I have had various experiences of this kind related to my research work throughout the past years, and I am not the only young researcher in category theory to have suffered from receiving ungrounded evaluations of this kind or pressures to present one's own results as "well-known by the experts" in spite of the lack of any written reference.

(3) If you think that (some of) my results are "well-known", can you give a precise reference of a theorem (or more if applicable) that I attributed to myself when in fact it was proved by someone else before me?

(4) Do you think that the transfer of topos-theoretic invariants across different representations of a given topos (such as different sites of definitions for it) constitutes a form of unification across theories naturally attached to such representations ? If not, please explain why.

(5) Do you have any objections to the statement that there is an element of canonicity (or, in more computational terms, automatism) in the generation of results through the theory of topos-theoretic ‘bridges’, to the point that many non-trivial (although not necessarily 'interesting' in the traditional sense of the term) insights in different mathematical contexts can be obtained in an essentially mechanical way, as argued in this paper? If not, please explain why.

(6) Do you think that it is important to devote research efforts to the development of the program of toposes as unifying 'bridges' ? If not, do you think that there are any preferable alternative methodologies for investigating (first-order) mathematical theories in relation to each other and enabling an effective transfer of concepts, properties and results across them (when the theories have an equivalent or strictly related semantic content)?

I specifically informed all the contacted experts that their answers would be published on my website (unless requested otherwise). A couple of experts sent private replies which, in accordance to their will, have not been made public.

I would like to thank all the experts who have responded to my initiative of clarification by providing answers to my six questions or general replies. I would also like to encourage those who have not replied so far to do so; their contributions will be very appreciated and published here. I report here all the responses that I have received, without any moderation, together with some commentaries to them, to which the involved experts are kindly invited to further respond if they wish.

Before proceeding further, I should remind the reasons for which I had undertaken this quite unusual initiative of clarification with some of the main experts of the category theory community.

Reasons why I have undertaken this initiative of clarification

Throughout the past years, and precisely starting from my invited lecture at the International Category Theory Conference 2010 presenting the preprint “The unification of Mathematics via Topos Theory”, I have been repeatedly accused by some of the most influential category theorists of "over-selling" my research, or of proving results that were already known (but admittedly never written down or stated in public or recorded occasions). Another recurrent accusation has been that of being arrogant or disrespectful of the experts of the old generation. These accusations have led to a widespread attitude of suspect and denigration surrounding my work, which materialized in a number of difficulties in getting my papers published throughout the past years and in unfair treatments in the context of my applications for academic positions. Most importantly, this attitude has prevented, or at least strongly discouraged, many young people from studying a promising subject and hence contributing to the development of a research direction in topos theory which has already proved to be very fruitful. The serious problems in the attitude of a specific mathematical community towards the work of a young researcher documented on this page are unfortunately not unique and are apparently becoming more common these days, thus affecting more and more young researchers in different areas of mathematics and natural sciences. It is a responsibility of the leading specialists of a given field to encourage and promote the development of a new theory which promises to bring many novel insights and applications. Not only this has not happened to any extent in this case, but some of the leading category theorists have pretended to completely ignore the theory of ‘topos-theoretic bridges’ introduced in the above-mentioned preprint, labelling it, depending on the person, as "absurd", "uninteresting", "irrelevant", or "well-known", and to utter personal attacks against me (such as the generic accusations of being “full of myself”, “arrogant” or “disrespectful” of the main experts of the field) so to discourage everyone from pursuing any closer investigation. What is even more unfortunate is that, as the development of the theory progressed and more applications were obtained, this aprioristic attitude of hostility did not decrease, and even amplified in some cases. I have had therefore no other possibility, after five years of silent suffering from these ungrounded accusations, to organize a public debate in order to promote a return to scientific objectivity and a serious ethical conduct.

The denigratory campaign

The main accusations against me that have been spread out through the categorical community in the past five years are the following: that I would prove “well-known/folklore” (though unpublished nor publicly communicated) results (my “duality theorem” has been taken by some experts as the ‘representative example’ of this), that I would “oversell” my research (the accusation of “absurdity” or “exaggeration” with regard to my statement that many results can be obtained by applying the theory of ‘topos-theoretic bridges’ in an essentially automatic way is an exemple) and that I would be arrogant and unrespectful towards the experts of the old generation.

As I explain below, and as it clearly emerges from the results of this initiative of clarification reported on these pages, all these accusations are unfair and completely ungrounded.

The main experts whom, on the basis of the episodes described here as well as of their responses to my initiative of clarification (or lack thereof), appear to have held forms of scientific hostility towards my work, are William Lawvere, Peter Johnstone, Martin Hyland, Ieke Moerdijk and Steve Awodey. There is in particular no doubt as to the crucial role played by Johnstone and Hyland in the context of the accusations that I have received (see this section ); in fact, the denigratory campaign could have not been effective, to the point of misleading even a specialist such as André Joyal (see this section ), had it not been ‘invested’ of the authority of the professors of the research group in which I got my Ph.D. and to which I still belonged in 2010. Whilst Johnstone and Awodey have responded to my letter thereby revealing their positions, Lawvere has not replied, whilst Hyland and Moerdijk have explicitly refused to answer. Of course, any late response from them which clarifies their positions will always be welcome, and published here. The responses received so far are available here.

The accusation of proving “folklore theorems”

From the reactions of the contacted experts, it has emerged that indeed the word had been spread around that I prove “well-known” or “folklore” results (with all the imaginable negative consequences that this naturally entails), without nevertheless there being any proof whatsoever supporting such claims. Indeed, none of the contacted experts was able to provide a single reference containing a proof or a statement of a result that I attributed to myself but which had been proved before, nor anyone showed that any of my results could be deduced from previously existing results in an essentially straightforward way.

It is clearly unfair to pretend that a young researcher presents his/her results, which he/she had discovered on his/her own without being told about them by anyone, as non-original on the grounds that “experts knew them but never wrote them down (or publicly communicated them)”.

Sadly enough, Peter Johnstone, my former Ph.D. supervisor, revealed himself in the context of this initiative of clarification as one of the primary sources of the accusations of non-originality of my results that I have received on multiple occasions, as well as of the defamation consisting in asserting, in a completely ungrounded way, that I would be a sort of ‘exalted’ person that considers herself superior to everybody else and does not want to give due credit to the experts of the old generation (see this section and his answers to my six questions). Whilst when I was a Ph.D. student under his supervision he had confirmed to me the originality of all the results in my thesis, he subsequently started to oppose me and began to spread the word among colleagues that some of these results were not new in the sense of being “folklore theorems” known by all the experts since a long time (though admittedly never written down). A result of which I have been accused in particular to claim credit for is my “duality theorem”, which appears nowhere in the literature (neither the statement nor the proof) and for which in fact there is even a considerable amount of evidence that it was not known by some of the main experts in topos theory; indeed, Lawvere extensively argued on a public occasion with me that the theorem was false (see this section), while Moerdijk could have easily solved his own 20 year old conjecture by applying it if he had known it (as I did in my paper “A characterization theorem for geometric logic”). Joyal was misled to believe that this result was folklore but after analyzing the matter more closely he concluded that it was not, and publicly declared it in his letter.

The possibility of ‘automatically’ generating results through the theory of topos-theoretic bridges

Another element of controversy, which was the subject of question n. 5, concerned the possibility of ‘automatically’ generating new results through the theory of topos-theoretic ‘bridges’ to the point that many non-trivial (although not necessarily 'interesting' in the traditional sense of the term) insights in different mathematical contexts can be obtained in an essentially mechanical way. This statement, which is certainly very strong but adequately justified (it comes at the end of a long technical and methodological development in the paper “The unification of Mathematics via Topos Theory”), requires a working familiarity and in-depth knowledge of the technique ‘toposes as bridges’ to be properly understood. As a matter of fact, none of the contacted experts has invested enough so far, on a technical level, in learning about this methodology so to be able to make an informed judgement about this claim (their short and vague answers received in this respect witness to this); unfortunately, the lack of an adequate in-depth knowledge has not prevented some of them from brutally discarding it as ‘overselling’, thus raising doubts as to my scientific seriousness (the title “Absurdities” referred to this statement by Bob Walters in a post on his blog - which was subsequently removed upon my request to scientifically justify it, or to let me respond - is a notable example). In order to further clarify this point, I have written a technical justification for this statement in the context of my response to Marta Bunge’s answers.

The attitude of the two Cambridge professors

As mentioned above, Johnstone played a crucial role in the diffusion of negative opinions on my work across the community. Several are the actions that he has taken in order to delegitimize me, even in the context of this initiative of clarification.

As it can be seen here, he responded to the question n. 3 (about the references of a theorem - or more if applicable - that I attributed to myself when in fact it was proved by someone else before me) by citing as unique example a result, namely the DeMorganization construction, which I discovered during my Ph.D. research, claiming that he had written a particular case of it in the *unpublished* draft of the third volume of his book in preparation! Quite astonishingly, when I discovered this result and the duality theorem as a Ph.D. student under his supervision he had confirmed me their originality (which actually corresponds to the truth, and to the declaration that I made in the context of the submission of my thesis) and the enthusiasm for my discovery of the DeMorganization was such that he immediately proposed to investigate it in the context of the theory of fields (this is how our joint paper “De Morgan’s law and the theory of fields” was born). However, knowing that I was the first person to publicly state and prove these results did not prevent him from diffusing the opinion amongst his colleagues that I was proving well-known results. For instance, Martin Hyland, the other professor of the category theory group in Cambridge, explicitly wrote in his official report on my Ph.D. thesis that "The title of the thesis refers to what is an essentially folklore (so-called) Duality Theorem (identifying subtoposes of a topos with quotients of a corresponding geometric theory)"; it is hard to imagine that he might have done that without the consent of Peter Johnstone, with whom he has always been on very good terms. Moreover, when I had asked him to confirm, as my Ph.D. supervisor, to the four editors of the Proceedings of the CT2010 who had brutally refused, after just two days of its receipt, my paper “Lattices of theories”, whose contents formed a part of my thesis, on the (completely unjustified) grounds that “much of its content was folklore”, he not only did not do this but, to my great astonishment, he sent a letter containing defamatory elements against me. I repeatedly asked him in the context of the debate organized by Joyal (see this section) to show this letter in order to clarify his responsibilities, but he never did. Nonetheless, he openly admitted having explicitly suggested in reference letters for me that I was full of myself to the point that nobody could appreciate me more than I did myself (cf. his response to my question n. 1). Needless to say, all of this greatly surprises and saddens me. I was a very respectful Ph.D. student who greatly admired her Ph.D. supervisor and always longed for advice and criticisms, as witnessed for instance by the very strong acknowledgements in my Ph.D. thesis and by all those in the papers written during my years in Cambridge. My invitation to Peter Johnstone to spend a part of his sabbatical in Pisa where I was working in 2010, which he had happily accepted, further testifies to my respectful and friendly attitude towards him. To be honest, I still cannot understand why he has chosen to betray the trust that I had in him, when a much more reasonable thing to do would have been to collaborate or at least to support me, given the fact that, as he has admitted in the context of the debate organized by Joyal, he considers me “an extremely talented mathematician”.

Sad to say, the other professor of the research group in Cambridge, Martin Hyland, shares a similar responsibility. In spite of the persistent conflictual attitudes which he had displayed during my Ph.D. studies (he even arrived to write in a recommendation letter for me, which I was not supposed to see, insulting expressions against my person which I prefer not to report here), I continued to believe in him and to look for the possibility of a fruitful scientific dialogue. I proposed to collaborate on various occasions, always receiving a refusal. The same thing I did with Johnstone after our joint paper “De Morgan’s law and the theory of fields”, obtaining the same result. In 2012, in spite of all these negative episodes, I still had enough trust in Hyland to ask him to write a reference letter supporting my application for a 6-month fellowship in Paris intended for “young researchers in Mathematics”. He accepted but, as I subsequently discovered, applied himself for the same position, in full competition with me; although being aware of the conflict of interest, he did neither inform me of his intentions nor declare the conflict of interest to the fellowship committee, and sent both his application (for the same position) and the reference letter for me, a letter which, according to a few professors who read it and who informed me about what appeared to them as a serious abuse of power at my disadvantage, contained several negative and ungrounded statements on my work. Following this event, not having received any excuses from him, I took the painful decision not to have any contacts with him anymore and I reported about this ethical misconduct to the University of Cambridge.

Reasons behind the denigratory campaign

One might wonder about the reasons behind such a persistent hostility towards my work of a number of prominent members of the category theory community. A possible explanation is that my work convincingly illustrates that, as advocated in the preprint “The unification of Mathematics via Topos Theory”, the subject of Morita-equivalences between geometric theories has an immense applicative potential. The problem that many category theorists of the old generation have with this is that, unlike me, they had not at all understood, or at least exploited, this potential for a very long time. Indeed, the theory of classifying toposes has been left essentially abandoned by category theorists for almost forty years; very few papers on the subject appeared in the years following the publication in 1977 of the book “First-order categorical logic” by Makkai and Reyes and, as a result, most mathematicians remained unaware of the existence and potential usefulness of this fundamental notion. The results obtained in my work thus implicitly show that a big mistake has been committed by the category theory community as a whole, and in particular by its leading experts, in not pursuing this rich line of research. Even though I have done my best to integrate myself with this community for instance by proposing collaborations to several main experts in the field and I have recognized my debt to the old generation for their important work on multiple occasions (in particular in my address at the CT 2010), the hostility has not decreased since 2010. All my collaborative proposals have been turned down and I had to leave Cambridge and find other mathematicians outside this community with whom I could share my ideas. A posteriori I can say that this has been very positive for me, since I have found in Paris and at IHÉS an ideal scientific environment with many actual and potential collaborators. However, this ‘expulsion’ from my community of origin has naturally had a negative impact on the development of my career. After all, any member of a recruitment committee who wants to seriously do his job in evaluating a candidate, or any young person that wishes to enter a given subject, is naturally led to ask the advice of the experts in the field and to rely on it (provided that, as it is often the case, he is not able, or does not have the time, to make an informed opinion by himself). In order to prevent my ideas from being understood and followed, some experts, such as Peter Johnstone (see his response here), not being able to criticize my work on a scientific ground, have resorted to personal attacks of the kind ‘she thinks of herself more highly than anyone’, ‘she thinks having invented the wheel as far as it concerns topos theory, ‘she oversells her work’ etc. Needless to say, the power which has been used by these experts at my disadvantage has caused numerous rejections of papers or job applications which were neither justified nor explicable on objective grounds.

Another reason for this hostility is the fact that my ideas are ‘heretical’ with respect to the Lawverian tradition in categorical logic which prescribes to consider theories only in invariant form (famous is Lawvere's statement that "a theory IS a category") discarding their presentations. Indeed, presentations play a crucial role in the technique ‘toposes as bridges’, which consists precisely in exploiting the duality between the invariant presentation of a theory and its different syntactic axiomatizations; see the following section for a detailed explanation of this point.

A bit of history

The idea of regarding Grothendieck toposes from the point of view of the structures that they classify dates back to A. Grothendieck and his student M. Hakim, who characterized in her book “Topos annelés et schémas relatifs” four toposes arising in algebraic geometry, notably including the Zariski topos, as the classifiers of certain special kinds of rings. Later, Lawvere's work on the Functorial Semantics of Algebraic Theories implicitly showed that all finite algebraic theories are classified by presheaf toposes. The introduction of geometric logic, that is of the logic which underlies Grothendieck toposes in the sense that every geometric theory admits a classifying topos and that, conversely, every Grothendieck topos is the classifying topos of some geometric theory is due to the Montreal school of categorical logic and topos theory active in the seventies, more specifically to A. Joyal, M. Makkai and G. Reyes.

After the publication, in 1977, of the monograph "First-order categorical logic" by Makkai and Reyes, the logical study of classifying toposes, in spite of its promising beginnings, stood essentially undeveloped. Very few papers on the subject appeared in the following years and, as a result, most mathematicians remained unaware of the existence and potential usefulness of this fundamental notion. Instead of pursuing this line of research, the category theory community oriented itself, as far as it concerns the logical study of toposes, mainly on the development of the more abstract theory of elementary toposes. The notion of elementary topos, introduced by F. W. Lawvere and M. Tierney, is an interesting one, but its level of generality is too high to shed light on problems arising in 'classical' mathematics. Indeed, besides the property of cocompleteness, the crucial feature that distinguishes Grothendieck toposes from their elementary generalization is the fact that the former admit sites of definition, i.e. they are categories of sheaves on a site. Sites allow to build toposes from a great variety of 'concrete' mathematical contexts (categories and Grothendieck topologies on them can be essentially be found everywhere in Mathematics), so Grothendieck toposes are susceptible of bringing insights into problems arising in such contexts. On the other hand, elementary toposes are essentially concepts of logical nature, which can be useful in investigating higher-order intuitionistic type theories (they are the classifiers of such theories) and shedding light on logical realizability. Whilst a certain amount of abstract sheaf theory and internal logic can be developed at the elementary topos level, this notion does not naturally yield applications in different mathematical areas, due to the lack of sites.

Two levels versus one

When I started my Ph.D. thesis at Cambridge in 2006, I decided to embark in a systematic study of Grothendieck toposes in order to bring the theory of classifying toposes back to life. In doing so (and as a result of a great number of concrete calculations that I had performed on sites), I have gradually developed a view of Grothendieck toposes as objects which can serve as unifying 'bridges' for transferring notions, properties and results across different mathematical theories. The notion of site, or more generally of any object which can be used for representing toposes from 'below', thus occupies a central role in this context. On the contrary, the classic tradition of categorical logic initiated by Lawvere has never attributed a central role to this concept, arriving to formulate a principle according to which theories should be only regarded in an invariant way (famous is Lawvere's statement that "a theory IS a category") and not in the classical Hilbert-style sense of presentation (i.e., axiomatization). There is in fact an important point in common between choosing elementary toposes over Grothendieck toposes and syntactic categories of theories over their presentations; indeed, sites correspond precisely to presentations of geometric theories in the theory of classifying toposes. Making a choice of one level *over* another, rather that deciding to work with *both* at the same time, certainly results in a more elementary theory, but the price to pay is an inferior depth and sophistication of the obtained results as well as a limited degree of applicability. Now, the theory of topos-theoretic 'bridges' consists precisely in exploiting this duality between the level of sites and that of toposes (or between the level of theories-as-presentations and that of theories-as-structured-categories), and as such it represents a technical implementation of a two-level view. On the contrary, the people following the Lawverian tradition in categorical logic have pursued a one-level approach essentially aimed at *replacing* the classical notions with the new, invariant, ones rather than *integrating* them with each other in a comprehensive way. It is therefore not surprising that the theory of topos-theoretic ‘bridges’ has not been well-received by the people of the old generation following these ideas.

A few selected episodes

I report on this page a few selected episodes with the purpose of giving an idea of the hostile behavior from some influential category theorists that I have suffered throughout the past years, and of their consequences.

André Joyal’s debate and concluding letter

In response to my letter of request for a clarification, André Joyal organized a private e-mail debate involving 9 persons to discuss the matter. The persons that took an active role in the debate were Joyal, myself, Peter Johnstone and Laurent Lafforgue. Several messages were exchanged. Lafforgue deserves a special thanks for his well-informed and extensive contribution to the discussion.

The debate concentrated on my “duality theorem”, which was cited by Joyal at the beginning of the discussion as an example of a folklore theorem that I published (in his original response to my question n.3). In the context of this debate, Johnstone insisted that my duality theorem (providing a natural canonical bijection between the quotients of a geometric theory and the subtoposes of its classifying topos) is “folklore”, whilst admitting at the same time that no statement let alone any proof of it appears in the literature preceding my preprint “Lattices of theories”, on the sole grounds that “all the elements in the proof were known long before you wrote them down” (which is certainly correct, but not at all a reason for labelling a result as “well-known” since every mathematical theorem arises from a combination of previously known results!) and on the pretention that any expert in the subject *could* have put them together to prove the theorem.

As pointed out by Lafforgue and myself, this is an inacceptable anti-scientific behavior, not to recognize credit for a result which is not a particular case of a published theorem nor a straightforward consequence of any result in the literature to the person that first publicly stated and proved it. This is all the more crazy if this is done by a professor with respect to a Ph.D. student who discovered the result under his supervision and from whom she had the assurance that the result was original.

Following the debate, Joyal decided, with admirable intellectual honesty (at least this is what I thought before the last events), to issue a public letter (available here) recognizing the originality of my “duality theorem” and of the methodology ‘toposes as bridges’, to replace his initial answer. The letter, which is written in French, concludes with the striking remark that the methodology ‘toposes as bridges’ represents a vast generalization of Klein’s Erlangen program. However, in the last months Joyal has taken an attitude towards me and my work which is substantially contradictory with that expressed in his letter (see this section for more details).

Responses to the initiative of clarification

I report on this page all the responses that I have received so far to my initiative of clarification.

Some updates

I am sorry to write the following apparenty polemical lines about a significant change in Joyal's attitude; I feel obliged to do so for the sake of transparency, and in order to alert people who may think, on the basis of his letter, that he has read and understood my work (while the opposite is unfortunately true).

In 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.







