**Reasons why I have undertaken
this initiative of clarification **

**
Reasons behind the denigratory
campaign**

**
André Joyal’s debate and concluding letter
**

**
Responses to the initiative of
clarification**

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 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.

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.

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.

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.

In spite of the 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.