Dear Vaughan, On Mon, Jan 22, 2024, at 7:31 AM, Vaughan Pratt wrote:

Dear Francis,

I greatly appreciate your input to the categories list regarding the contributions of Jean Bénabou from half a century ago. I especially appreciated your "I downloaded a copy of these old texts via “WeTransfer” and they will remain available during one week via the link https://protect-au.mimecast.com/s/t0L3CD1vRkCV66pGuW4yUv?domain=we.tl ."

I did so and now have them in four folders. However they are all in French, whereas your three excellent volumes are in English.

Is anyone volunteering to translate Bénabou's texts into English?

At one time I was working on a translation of Bénabou's lectures on topoi, but I ran out of time. I might resume at some point, but perhaps someone with a better command of the subtleties French language would do a better job.

And on the matter of decidability, speaking as a complexiy theorist, is there a short summary of the essential points?

I wonder if Francis meant to type 'definability' rather than decidability. Definability is indeed one of Bénabou's most important and underrated ideas, which generalises certain notions of definability from logic and set theory. I have written some brief exposition on the topic here: https://protect-au.mimecast.com/s/MBluCE8wlRCDAAyXcwjUV9?domain=jonmsterling... but I do recommend Volume 2 of Francis' handbook for a more thorough account. One thing I learned from Thomas Streicher's paper on universes in toposes is that definability is related to descent — for instance, if you restrict the codomain fibration to a stable class of maps, you get a full subfibration, and definability in the sense of Bénabou is the gap between this subfibration and its stack completion. There is a lot of potential for this idea contributing to future works in category theory; for example, Mike Shulman has extended Bénabou's definability from "classes" of things in a fibration (i.e. properties) to a notion of definability that makes sense for structures; Andrew Swan has given a very interesting and thorough investigation of this generalised definability and its practical implications here: https://protect-au.mimecast.com/s/yeENCGv0Z6fMyynXFptjUl?domain=arxiv.org. All the best, Jon

Vaughan Pratt

On Sun, Jan 21, 2024 at 1:34 PM Francis Borceux <francis.borceux@uclouvain.be> wrote:

...

__ __ I had the privilege to have regular and often close contacts with Jean Bénabou during more than 50 years ; I learned much from him.____

__ __

It is a matter of fact that Jean did like very much lecturing, and he was doing that marvelously well … but he was quite reluctant to « freeze » a topic by writing it down.____

__ __

Early in the seventies, Jean gave several series of lectures at the University of Louvain: on monads (called “triples” in those days), multiplicative categories, distributors and toposes. Notes on these lectures were written down by some of the auditors and Jean accepted to have them published in the preprints of the mathematics department. I downloaded a copy of these old texts via “WeTransfer” and they will remain available during one week via the link https://protect-au.mimecast.com/s/t0L3CD1vRkCV66pGuW4yUv?domain=we.tl .____

__ __

Some years later, Jean also gave a series of lectures on fibred categories in Louvain-la-Neuve. As already mentioned by others, Jean-Roger Roisin wrote from these lectures a beautiful set of notes. These notes were never published because Jean, through the years, kept considering that he could still improve and complete his results before their publication.____

__ __

When I wrote the three volumes of my “Handbook of categorical algebra”, I wanted of course to include a chapter on fibred or indexed categories: I chose fibred categories (Chapter 8 of Volume 2). Before sending the chapter to the editor, I sent a copy of it to Jean, asking for his comments, but making clear that I was not at all asking permission to publish this chapter, and that I would be the only one (with the referees) to decide of the final form of the text and take the responsibility of it. I definitely wanted to avoid a new endless story, in the vein of what was happening to the Jean-Roger notes.____

__ __

I remember the upset answer that Jean gave me; he was focusing on three points.____

1. I do not appreciate that you include a chapter containing many results of mine before I myself publish them.____ 2. At least, it is a relief to notice that your text reflects faithfully my own views on this topic.____ 3. Thank you for putting emphasize on the notion of decidability, whose importance does not seem to have been recognized by the categorical community.____ I am thus confident that this chapter reflects quite faithfully several ideas of Jean, at the period where my book was written.____ __ __ Francis____ __ __ Francis Borceux____ 6 rue François____ 1490 Court-Saint-Etienne____ Belgique____ +32478390328 (portable)____ +3210614205 (fixe)____ francis.borceux@uclouvain.be____ __ __

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In the preprint that Jon mentioned I had group structures (or more specifically the forgetful functor from families of objects with group structures to families of objects) in mind as instances of algebras for a monad, which are definable whenever the fibration is locally small and the base has finite limits. That argument requires having enough internal logic to define the free group monad. However, I think the same ideas applied directly should show that group structures are always definable for locally small fibrations where the base has finite limits and the fibration has finite products, which I think would cover the examples you gave. Of course, that would only show existence without the nice concrete descriptions. Best, Andrew On Tue, 23 Jan 2024 at 01:03, Richard Garner <richard.garner@mq.edu.au> wrote:

...

One thing I learned from Thomas Streicher's paper on universes in toposes is that definability is related to descent — for instance, if you restrict the codomain fibration to a stable class of maps, you get a full subfibration, and definability in the sense of Bénabou is the gap between this subfibration and its stack completion. There is a lot of potential for this idea contributing to future works in category theory; for example, Mike Shulman has extended Bénabou's definability from "classes" of things in a fibration (i.e. properties) to a notion of definability that makes sense for structures; Andrew Swan has given a very interesting and thorough investigation of this generalised definability and its practical implications here: https://protect-au.mimecast.com/s/vucfC91W8rCBovRjhoxKeJ?domain=arxiv.org.

Definability is quite a fascinating thing. Peter Freyd's work on the core of a topos and the subsequent work on isotropy groups of toposes I find very pretty. One related thing that I am reminded of is the following cute fact. Possibly it is well known; I do not know the SGAs and EGAs very well.

Suppose I have a fibration E ---> B and some X in E(b). I can consider the existence of an "object G(X) of group structures on X". What this means is a map f: G(X) ---> b in B, together with a group structure on f^*(X) in E(G(X)), which is universal among such in the expected way. Clearly this is an instance of (the more general?) definability.

Let us consider this for the following fibration. The base B is the category of affine schemes, CRng^op. The fibre over a ring k is the category of formal affine k-schemes, i.e., the Ind-completion of the k-Alg^op. This is a full subfibration of the codomain fibration of Ind(k-Alg^op).

In the terminal fibre of this fibration we have the affine line L = Spec(Z[x]). This is an internal ring object in the fibre and so we can form its subobject N <= L of nilpotent elements ("Spec of Z[[x]]").

Now for any commutative ring k, to give a group structure on k^*(N) ("Spec of k[[x]]") in the category of formal affine k-schemes is to give a formal group law with coefficients in k. It follows that the "object of group structures on N" is Lazard's universal coefficient ring for a formal group law.

What is also quite fun is to compute the object of group structures on the generic group O in the (codomain fibration of) the group classifier [Grp_fp, Set]; this turns out to be O+O. This is because, given any group G, there are G+G ways of making it into a group. Indeed, each g in G yields two group structures on G, one with x * y = x.g^-1.y, and another with x * y = y.g^-1.x.

Richard

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