This page is dedicated to the loving memory of Aurelio Carboni and Bob Walters.

In the video below, Renato Betti, George Janelidze, Peter Johnstone and Ross Street share their memories on our beloved friends and colleagues.

George Janelidze slides here Ross Street slides here

Memories

Renato Betti

Dear friends,

I have been asked to remember two of our colleagues and friends who have passed away in recent years: Aurelio Carboni and Bob Walters. They were well known by all the researchers of our subject, they used to participate actively in the annual conferences of category theory and, in some occasions, have organized them: namely in 1990 and 2000 in Como and in 2010 at Genoa.

Both Aurelio and Bob are deeply connected to the development of categorical studies in Italy, and this is one of the reasons why they are remembered here today. Personally, I had the opportunity to hang around with them for many years not only for collaboration. On some occasions, as part of our friendship, we have been on vacation together with our families. So it is difficult for me to speak of them in a formal way, to find the right words in order to give you an idea of their personality, both as persons and mathematicians. Many impressions are mixed up in my memories.

Aurelio was, since the very beginning, an active member of the primordial group who initiated the studies and research on categories in Italy, as well as their dissemination in the initial stage when nobody in this country was still working on categories or even talking about them. Bob joined us in a second phase, by contributing in an essential way to enrich the body of ideas we were working on and helping to support new groups of people doing research about categories. This was first achieved with frequent exchanges between us and the Australian school, and then officially when he became a full professor in Italy.

Others will speak in more detail about their production, their research topics and their work, about their international collaborations and the results they achieved. From my side, I only intend to mention their work on motivations, the sense and the interest they developed not only with their personal activity but also with their friendship to people, individuals or groups of researchers in Italy. They made a big and warm contribution to the increase of new collaborations. A new consciousness about categories is largely due to them.

In this sense, paying homage to Aurelio and Bob means in large part telling the story of the birth of the categorical studies in Italy - or rather “a” story, the one I know because I have participated in it in some way. For this reason I will have to say something about myself, and I apologize for this.

So, let's consider mathematics in Italy in the early 70's, exactly 50 years ago. Perhaps, the first group that took an interest in categories was born at the University of Milan. I say “perhaps” because others in other places could justifiably say the same thing: I am thinking of Parma, Trieste, Rome and even Genoa, where we are now. I am thinking of all those Italian universities in which small groups or even individuals were looking for their own way to research on the then-little-established notion of category. They were looking for it in logic, in algebra, in geometry, in topology, as everywhere. Everyone with its own motivation, everyone on their own, often isolated. A sort of primordial chaos in the Italian category world.

Was it possible to summarize in such a notion all the previous experience and arriving at something new? This question was not so clearly formulated at that time, but it was essentially of this type. Not easy to answer if you are left alone to decide.

So, my story begins in Milan in the early ’70s of the last century. It would take a few years for us, being confined in Milan, to know people in other places and meet them, to become friends, understand that we had common research interests, exchange experiences with each other and collaborate. You may ask why it took so much time, even years. One may think that this was due to a lack of interest, but, as a matter of fact, this long period was due to a sort of intellectual inertia that was reigning in many Italian research centers in those years: a form of aging, both of people and of topics that affected – and in some sense is still affecting – many Italian universities.

My impression is that, especially in those years, mathematics in Italy was still connected to an old conception of research: a few academic “barons” – as they were called – imposed the research arguments. They were convinced that it was useless to develop contacts with other mathematicians and not necessary to travel and get to know new people. The implicit idea was that a mathematician works just by studying the classical papers written by old masters of our masters – the famous barons – and that a pencil is enough to write down slight improvements to old papers, namely those improvements suggested by our masters. New ideas were reserved to them.

So, the story I am telling you is also the story of how this intellectual inertia was broken, at least in the field of categories, thanks first to Aurelio and later with the involvement of Bob and a few others. At the beginning, in particular, the role of Aurelio was fundamental. He played a great role because he was, in a natural way, everyone’s friend. Nice, ready to joke, with a sparkling sense of humor that was not at all in contrast with the depth of his thinking in mathematics. On the contrary: when depth of thought comes together with the ability to connect emotionally with other people and with the pleasure of staying together, the result has always a great effect.

I had known Aurelio since the late ‘60s. I was preceding him by one year in my studies of mathematics at Milan university. So, when I got my degree, for one academic year I became a teaching assistant in an exceptional class of which Aurelio was a member. Many others of that class would become known mathematicians in different Italian universities and abroad. It is thank to them if the situation of Italian research slightly changed and is still changing.

Another young and brilliant member of that class who would play an important role in category theory was Giancarlo Meloni. He was especially interested in mathematical logic. At the end of that crucial year, when also Aurelio and Giancarlo graduated, they were offered a scholarship, while I continued to be an aspiring researcher, with enough old geometry behind me but few future perspectives.

Each of us was continuing on the path marked by his reference “baron”. Aurelio dealt with ordered structures, relations, equivalences and so on. Giancarlo was addressing the various types of logical calculus, while I, with the privilege of collaborating with the friend and colleague Massimo Galuzzi – who also dealt with categories for a certain period of time – was studying problems concerning the geometry of curves that the ancient and famous Italian geometers – Enriques, Chisini and others – had left to their pupils and descendants.

None of us was satisfied with their own work nor was seeing serious research perspectives. We discussed about that, and Aurelio was the voice of our dissatisfaction, using tough tones, so critical and sometimes so paradoxical to become humoristic. The humoristic side of any situation was always at his hands. That was his attitude: to take different arguments and show their intrinsic, unexpected connection, often highlighting their ridiculous aspects.

That's why, quite naturally, we decided to try to unite our forces, even though each of us remained involved in his own personal work: Aurelio, Giancarlo, Massimo and I. It was a good idea, it was a winning idea. After a few years, they would call us “the gang of four”, referring to the Chinese political situation at that time and the will to overcome our actual condition only by ourselves.

How did we arrive at categories? At a certain point, in the spring of 1972, we received a piece of news that seemed trivial at first sight: a young American professor had become visiting professor at the university of Perugia, invited by another young professor, an Italian whose name was Francesco Succi. Evidently Francesco had a wider vision than the other “barons”. After Perugia, Francesco would continue his career at the university of Rome. He passed away not long ago, and I take this opportunity to remember him with affection.

The young American professor at Perugia, a certain F. W. Lawvere, had been invited for two academic years and had announced a weekly seminar, every Wednesday, on an absolutely new subject, with a title that was fascinating for its synthesis but absolutely incomprehensible: “Category Theory over a Base Topos”. Who knew what it was all about? The only thing we were able to understand in his program was the unifying aspect of the subject. Enough for us? It seems that we decided it was worth trying to understand it better.

It is worth emphasizing that Bill Lawvere had been invited for two academic years. This is a remarkable fact that demonstrates once again the precarious situation of the mathematical research policy in Italy at that time, and how it has changed since then. Nowadays invitations, even of great mathematicians, are for one month or two, while at that time there were so few invitations that longer periods of time could be offered. A real stroke of luck for us and for categorical studies in Italy.

Regarding the notion of category, we had rough ideas about it, certainly vague. Some books described the notion as a special type of abstraction and shortly before, in 1970, a popular article in a well-known magazine – written by a well-known author – spoke about it as a “third level of abstraction”. In other words, putting together all the objects of a given kind you get the integer numbers (first abstraction), then putting together all numbers or similar things you get definite structures (second abstraction), now some mathematicians decided – who know why – to put together all the structures of the same type and to give rise to categories (third abstraction).

The paper was written in a really fascinating way, but essentially the content was such. A generality that was appearing unsatisfactory for our research topics if not ridiculous for all of mathematics: why not consider a forth abstraction then a fifth and so on. Is this mathematics?

But luckily there was also a book that had just come out (in 1971) with a different point of view: “MacLane’s Categories for the working mathematician”. We succeeded in putting our hands on it – another not-at-all-easy operation – and …wonderful, we discovered a new way of regarding mathematics, the correct generality one has to take in order to work in mathematics. The title of the book was correct and inspiring. From that book we discovered that categories could also be seen as a universal language for dealing with the different problems in which we were interested. Later, with the help of Bill and his teaching, we would understand that categories are much more than a good language because they enter deeply into explaining mathematical structures: they are not a “level of abstraction”, they are a new level of concrete understanding and possibility to develop new mathematics.

MacLane's book has remained a point of reference for us. It was providing the necessary algebra for Aurelio, the logical structures for Giancarlo and it gave a strong flavor of geometry for Massimo and myself. It was all cooked in a universal sauce that we felt useful and pleasant: categories. After a few years (in 1977), we managed to translate it for an Italian publisher, all four of us working on it. The original title, that we appreciated, was not difficult to render in Italian: “Categorie nella pratica matematica”, referring to the practical mathematical work.

The weekly seminar program written by Bill himself was hard to understand. Who, among us, decided to go and meet him in order to hear what the hell was a base topos? Certainly, Aurelio and Giancarlo were the most motivated. As a result, we decided that each of us would take turns in Perugia every week in order to follow Bill’s seminars on Wednesday. And so we did. We made acquaintance with Bill, Fatima and their children, Silvana and Danilo. The one of us in Perugia used to listen carefully to the seminar and returned to Milan with Bill's notes – the first ones handwritten, then typed by a student in Perugia at that time, Luciano Stramaccia, now a dear colleague. We had just a week to study the notes – or better to fight with them – so that the next one of us who would go to Perugia could easily – so to speak – follow the next week seminar.

I should note that Perugia is not well connected with Milan. The journey took approximately five hours by train (perhaps today a little less). Therefore, the one of us in Perugia had to sleep one night there on his own expenses, as a further confirmation of the situation of the university research policy in those years. But they were two exciting days, to enjoy later with the “gang”, while studying the notes, trying to clarify the doubts, writing on a notebook all the questions to ask Bill the following week.

“Category Theory Over a Base Topos”. Bill's weekly notes is an unfinished book. It was never published but is well known, read and appreciated by all scholars of the subject. To us, it opened the door. Through these notes, we learned the fundamental aspects of categories, the fact that they allow one to override traditional, often artificial boundaries between subjects, but more importantly, we sought to learn a new way to regard the world of mathematics and insert it into the real world, both a world of things and of rational thoughts.

Aurelio’s insight was able to clarify most algebraic passages and Giancarlo’s ability allowed us to explain any logical doubt. It remained the sense of the work and the pleasure we got each time when one of us was able to get a new idea about it. Bill's style taught us the way to consider and connect different things, it explained that there are no specific, unique and significant facts only for some special cases, that everything can be traced back to a conception, to a vision, to a policy. Everything should be regarded as a whole.

Hopefully we have learned enough of this style, of this way of thinking. You all understand what I am inadequately trying to say. There is no point for me to go on in this subject.

After that, our story continues in a more relaxed atmosphere and with much more satisfaction. We invited Bill to Milan and his frequent visits continued even when he finished his period in Perugia and returned to the United States, taking up a position in Buffalo. Perhaps not everyone knows that, during those years, Bill had the opportunity to officially become a full professor in Italy, in Rome, as the winner of a national competition: a chair in algebra. But communications were not easy at that time and – I imagine that's why – Bill learned of this opportunity when it was too late. We are left with the regret of not knowing what would have happened to the categorical research in Italy if he had known in time and accepted the position.

There were other events, other invitations, among which I would like to remember those with Anders Kock, who explained to us the most delicate passages with patience and precision, and with André Joyal’s creativeness. We developed other contacts in Italy, such as with the logical group of Mario Servi in Parma and with Fabio Rossi and Cristina Pedicchio in Trieste. Aurelio was often the authentic glue for these collaborations, always bringing to them a true flavor of friendship.

So, during the ‘70s, the world of categorical research in Italy was growing and unifying. The initial “gang of four” was structuring differently, because of different conceptions on how to carry out research, as is natural in all human affairs. It dissolved: Massimo Galuzzi dedicated himself to history of mathematics, Giancarlo Meloni tended to isolate himself with his students. Aurelio and I continued to collaborate together and with the help of people in different places. In Milan we had the opportunity to count on the collaboration of new colleagues, in particular Stefano Kasangian and others. It is not possible, as well as boring, to cite every place and everybody.

Now we arrive at a turning point. 1979, Bill invites Aurelio to Buffalo as a teaching assistant for one academic year and, during the same time, Kasangian gets a six-month scholarship from Max Kelly in Sydney. The result is, on one hand, that Aurelio receives the decisive push towards many international collaborations and, on the other hand, Stefano Kasangian motivated the whole group of Australian categorists – Max Kelly, Ross Street, Bob Walters and Murray Adelman – to follow the work of our groups in Italy, in particular in Milan where we were developing the new notion of enrichment in a bicategory.

As a result we got some good papers on this subject: it is worth citing just one of them, that we called “the four name paper”: “Variation through enrichment” by Aurelio, Ross, Bob and myself, a general synthesis of the topic, dense and complete, truly symbolic of our collaboration, written after a four month period of Ross visiting Milan and published in 1983. Someone, especially Max and Bob, would join our Italian groups in an almost regular way, practically every year since 1981.

In a short time, in the course of one year, we had achieved two big successes: the international authority of Aurelio and the concrete collaboration with the Australian friends. Aurelio was not only good and nice, but an original and deep-thinking mathematician. Bob Walters, besides being an original and deep-thinking mathematician, was a friend of Italy, the country where some years later he would come to live.

My story of the beginnings of the categorical studies in Italy is now at the end. It disappears and melts into the story of general categorical studies, like those performed by you, dear friends, with the research you are spreading in this conference. A conference waited for one year and now materialized thank to the perseverance of the organizers, in particular our friend Pino Rosolini. By the way, in the ’90s, Aurelio moved initially to Genoa and continued his intense collaboration with Pino and Marco Grandis and later, as the winner of a national competition, he moved to the university of Insubria at Como where Bob was already staying. The following years are rich of categorical results for us and for the new entries in the Italian groups, in particular let me mention Sandra Mantovani who has given rise to a fantastic and productive group, still greatly active, in Milan, where everything began.

My homage to Aurelio and Bob stops here. Any story ends when new stories are coming. Thank you so much.