I report below
the answers by Marta Bunge to my six questions, together with some
commentaries by me. The answers are reported as they are and in their
entirety, that is without any moderation. I report the text of the
question in green, the parts of Bunge’s response in
italic and my
commentaries in normal text.

(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 the preprint available
here), such as to justify the resulting persistent hostility from
influential members of the category theory community? If yes, please
explain.

*No, of course not.*

(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 his/her own
results as "well-known by the experts" in spite of the lack of any
written reference.

*Certainly not. *

(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?

*By "well known" one may mean "folklore". I have several papers on
Morita equivalence, classifying toposes, stacks, and theories, and I
have used myself classifying toposes for getting results, particularly
when arguing that two fibered categories are equivalent since they are
both stack completions of a certain (small) category. For instance look
at my paper in the Math. Proc. Cambridge Phil. Soc. (on something like
"classifying toposes and descent"), or at a more recent one "Tightly
Bounded Completions" in TAC. I apologize for not giving you precise
references, but I am currently in Greece with difficult access to my
papers and files. If I had thought that you knew these results, I would
have written to you. *

(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 between theories naturally attached to such
representations? If not, please explain why.

*Well, sure, I do think that.*

(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 the preprint
available
here
? If not, please explain why.

*No, I do not as a first approximation. However, to emphasize that
many results can be obtained so simply is in my view an overstatement. A
lot goes into it before one can actually use as a last step the
conclusion in question. I am all for unification. For instance, my very
last paper "Pitts Monads and Lax Descent" (to appear shortly in Tbilisi
Mathematical Journal) uses a new notion ("Pitts KZ-monad") to unify
several known results about descent in topos theory and locales theory
and even new ones. However, this is possible only after verifying many
facts and even constructing new monads to apply my general theorem. It
is mostly the emphasis in the easiness that I believe most people object
to, not the results themselves. It is precisely that aspect that you
emphasize that has made you so popular in certain quarters, but in my
view it distorts a bit what working in category theory in general and
topos theory in particular, is like. You are young and naturally excited
about it all, but we, older people, have (so to speak) seen it all
before, even if we cannot pinpoint to a specific source. However, you
should not get animosity but advice, particularly from your advisor at
Cambridge. *

I do not think that the experts from the old generation
were familiar at all with building topos-theoretic ‘bridges’ (in my
sense), otherwise the literature of the past forty years would contain
at least a few notable examples. One thing is to *imagine* that toposes
could serve as unifying bridges in some situations (this, I agree, could
apply to some experts of the old generation), another is to develop a
whole set of methodologies which make such transfers of information
technically feasible and even ‘automatic’ in a number of cases. It is
hard to believe that the old generation of category theorists was fully
aware of the importance of investigating Morita-equivalences for
building topos-theoretic bridges across different theories when there
are no written sources witnessing it nor significant applications of
these techniques obtained by them in the past forty years.

Also, I do not think that saying that many non-trivial
(but not necessarily ‘interesting’) results can be obtained in an
‘automatic’ way through the ‘bridge’ technique is an exaggeration, for
the simple reason that it is true, and even easily verifiable. For
instance, there are topos-theoretic invariants I which admit site
characterizations of the kind ‘
satisfies I if and only if the site
satisfies a property
written in the language of
’ holding for all sites
;
in fact, in my work I have uncovered whole classes of such invariants,
see for instance the papers “Topologies for intermediate logics” and
“Site characterizations for geometric invariants of toposes”. For any
such invariant and any non-trivial Morita-equivalence connecting two
sites
and
,
one obtains the equivalence between the property
and
the property
.
Even though the property
and the property
are abstractly
the *same* property, namely I, they can look completely different from a
concrete point of view. The result asserting such an equivalence is
therefore in general a non-trivial one, lying in the field(s) of
mathematics to which the sites
and
belong.

(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 have no opinion on this except the following. Each of us should
develop what we believe in, even if others tell us that there are
better, alternative methods or programs. Why should there not be? I
would encourage you to continue your work, limit a bit the importance
you give to it, and ignore the criticism. Moreover, unless you have done
that already, I invite you to submit a paper to Cahiers de Top. et Geom.
Diff. Cat.. You can send it to me as editor and I promise that you will
get a very fair review.*

Thanks for your invitation to submit a paper to you as
Editor of the Cahiers de Top. et Geom. Diff. Cat. I look forward to
doing so.