Olivia Caramello's website

 

Unifying theory

Controversy with category theorists

Marta Bunge's responses to the six questions

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.