Introduction
The research program of toposes as unifying 'bridges', which I first presented in an invited lecture at the International Category Theory Conference 2010, has generated controversy inside the category theory community, and an hostile attitude from some of its most influential members. Whilst there have never been criticisms as to the scientific solidity of the 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). These accusations have led to a number of difficulties in getting my papers published throughout the past years, as well as to a widespread attitude of suspect and denigration surrounding my work. Whilst there are many mathematicians, including several category theorists, that appreciate my research and are supportive of the above-mentioned program, some of the most influential exponents of the category theory community still indulge in rather ambiguous, if not downright hostile, positions in relation to it, and many of them systematically refuse any occasion of scientific dialogue or clarification, as if I were as a sort of dangerous "heretic" that does not accept the dogma that they represent and therefore needs to be pushed out of the community. I believe that this situation of conflict, division and lack of understanding which has persisted for almost five years by now is highly detrimental for the future progress of the subject, so I have decided to undertake an initiative of clarification with some of the most prominent members of this community, by sending them a list of questions to which they are kindly invited to answer individually. I will publish their responses, without any moderation, on this website. In case I do not receive an answer within a few weeks, I will also indicate it.
Some questions to category theorists
I realize that, when I first presented the methodology of
'toposes as
bridges'
at the CT 2010, the general vision outlined on that occasion
was not yet underpinned by a great number of examples and applications.
So I perfectly understand the feeling of exaggeration or excess of
enthusiasm in my presentation that many category theorists might have
resented. Still, as provocative and ambitious some claims in the paper
"The
unification of Mathematics via Topos Theory"
might have seemed, they
were rigorously justified on a purely technical ground. I would have
thus naturally expected a lively debate to arise, about in particular
the technical contents of the above-mentioned paper and the degree of
applicability of the general methodology. In fact, it has always
appeared to me as 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, but some of the leading category
theorists have pretended to completely ignore it, labelling it,
depending on the person, as
"absurd",
"uninteresting",
"irrelevant",
or "well-known".
What is even more unfortunate, is that, as the development of the theory
progressed and more applications have been obtained, this aprioristic
attitude of hostility did not decrease, and even amplified in some
cases. Now that many non-trivial applications of the
'bridge' technique
have been obtained in different fields of mathematics (such as topology,
order theory, model theory, proof theory, algebra and functional
analysis), to the point that it remains no reasonable doubt as to the
fruitfulness of this general methodology, a clarification is in order.
This is why I am posing the following questions to some of the main
experts of the subject.
Another possible
reason behind such negative reactions is the tendency to dismiss a
general theory in case it has not yet been applied for solving certain
specific problems which interest specialists of a given field.
Personally, I think that a
general theory should
rather be evaluated from a methodological point of view, in terms of its
technical soundness and potential fruitfulness. In the case of the
'bridge'
technique, the mere existence of non-trivial applications in a variety
of different mathematical contexts obtained in a very limited amount of
time appears to me a clear indication of its fruitfulness and potential.
Concerning the originality of results in a given research paper, it seems appropriate to recall that the standard accepted rule of the scientific community - to which I fully adhere - is to judge it on a purely objective basis; that is, if there exist anterior pieces of published work (or of recorded talks/abstracts) which contain precise statements or proofs of the given theorem, or of very strictly related results, the relevant reference should be cited in the article with a specific attribution to the cited author. Of course, as nobody is omniscient, it can happen that, in complete good faith, one might miss a relevant reference; in such cases, to remedy the problem, it suffices to point out the mistake and ask the author to correct it. Only if the author persists in attributing to himself a result that is demonstrably due to another author, one has the right to accuse him of "proving well-known things" and in particular of incorrect ethical conduct.
The questions that I
will pose are as follows.
(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)
?
In order to explain why I
am perceived by some of the
category theorists of the
old generation
as
"heretical", I will embark, in the following section, in a brief
historical digression.
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 certainly 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, due to the lack of sites, applications in different mathematical
areas.
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 very 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. In light of
this, it is not surprising that the method of topos-theoretic
'bridges'
has not been well-received by these people.