F. William Lawvere
Farewell to Aurelio
Il Protagora,  July- December 2013, # 20, pp. 489-494.



My dear friend Aurelio Carboni was a force of nature. His great impulse to see and learn, to study, to experience, to work, to go out into the unknown for a scientific adventure got him far, and his inspection and fierce criticism that could be unforgiving had to be feared. I once incurred his wrath when I wrote a paper

for his 60th birthday. He tried to understand it, but after many tries he realized that I had made a mistake!  I regret that I wasted his time but the result of his effort was a corrective footnote to the published paper.

He had the gift to make friends with people of all kinds and ages. He had a connection with the young student, as he had with the leaders in the field, and the mighty in administration.  He was able to incite colleagues to pursue their projects or his projects with passion and seriousness. In his intellectual journeying, it was invigorating for me to try to illuminate for him that which was still obscure.

The 60's and 70's and 80's were a good time for mathematics, for category theory and for research. We had two respected teachers in Eilenberg and Mac Lane and many large groups of researchers working each on facets of knowledge that interested all category theorists. The meetings were amicable, completely concentrated on exchange of knowledge, freely given and taken, without mundane financial perturbations. We all had grants from our universities that paid for the travels for our encounters. There were no invited speakers or plenary talks. Whoever had something to say got a spot to talk. The attention to funding and priority, the shallowness and individualism, were foreign to our intellectual pursuit. The collaborations that resulted were far-reaching and exciting, spurring us on to satisfy the demands, and answer the questions that would be discussed at the next meeting. Aurelio was able to excite the colleagues to pursue projects and organize conferences, to collaborate on a serious basis with advanced ideas. We can continue working in his spirit as a fitting tribute to him, who 

!look at this room!

has still brought us together now, even after he has gone.

It all began for me when Aurelio Carboni with his friends Renato Betti, Massimo Galuzzi, Gian-Carlo Meloni arrived in Perugia to take my course in November 1972. They were four of the most passionate students I ever had. They had already worked together on a translation into Italian of Mac Lane's Categories for the Working Mathematician. Zealously pursuing both the foundations and the further developments in the application of category theory to mathematics, they made the long drive weekly from Milano to Perugia for these lectures, never tired of asking penetrating questions and demanding more explanation. By the moral pressure of their participation I was forced to start implementing my meager training in Italian in order to better communicate with them. This welcome imperative also brought about the weekly production on stencils of the Perugia Notes, the first chapter translated into Italian while I was still speaking in English, and then in English as I began lecturing in my 'bad Italian', for which they, unfortunately, seemed to have a certain affection. The course was also fortunate in having Eraldo Giuli, Stefano Guazzone, Rosanna Succi, Luciano Stramaccia and Anna Tozzi as active participants, who all are still my friends.

Aurelio had been most persistent in directing my conversation toward Italian and so I had managed (in the beginning) to explain and discuss mathematics in my bad, grammarless Italian with words that I learned from my 3-year old daughter Silvana and with invented expressions that they understood.

Aurelio and his friends remained faithful auditors and critics of my further lectures in Milan in the many following years.  Our friend Stefano Kasangian had become part of the group; (I only saw last year some notes that they had created of the 1977 lectures).

When in 1978, the University at Buffalo had a sudden opening for a calculus teacher, quickly before the semester started I verified that Aurelio was available and persuaded the mathematics department to invite him. His acceptance showed his considerable courage because the job required three sections of

calculus lectures each morning for three days a week, and his spoken English was still in its early stages. But he came through these many difficulties like a warrior; the students liked him very much, partly because he spoke more clearly than the other professors, but mostly because he was willing to sit on the steps and explain to shy and scared students whatever they did not understand in calculus. He came to appreciate very deeply the importance of calculus education. In fact, he took me to the Milan Cimitero Monumentale to see the tomb of Maria Agnesi who had actually advocated in 1750 that all Italian youth could and should learn Calculus. Aurelio worked for years on the draft of his proposed Italian Calculus textbook: Calcolo I, which I believe should be prepared for publication and made available to all Italian students. In this book he explains the intuitive aspect of calculus and pays great attention to logically coherent foundations.

During the second semester of his stay in Buffalo, in February 1979, Aurelio and I travelled to Puerto Rico, where for several years Jon Beck used to organize his conferences in February. A large part of the category theory community of Canada and the US gathered to exchange ideas. When Aurelio's ideas met with some skepticism from the old guard, he defended himself admirably. He later developed these ideas into influential publications. Later he remarked to me that the two happiest decisions he had made in his life were to marry Patrizia, and to go to Buffalo and Puerto Rico.

I believe this Puerto Rico Meeting represented the entry of Aurelio's steady presence into the world of active research in Category Theory.  A major snowstorm blocked our attempted trip back from Puerto Rico to Buffalo and the flight was diverted to a small town in North Carolina where we spent two days in a motel; the intense discussion that we had been having on the plane continued of course during that stay almost without interruption.

In 1982 Fatima organized a joint meeting of Synthetic Differential Geometry and Foundations of Continuum Physics, the summit of which took place at the old French Fort Niagara. A few days before the start of the meeting, we were immensely overjoyed to see Aurelio entering our back yard; he had decided to make the trip from Milan to attend that meeting, and we much valued his presence and introduced him to Noll, Owen, Coleman, Williams, and also to Froelicher, Kock, Reyes, and K.T.Chen. He had already met Clifford Truesdell during his Buffalo stay.

Our discussions continued in Australia in 1988, where Aurelio's preferred location was Bondi Beach. He had already been collaborating via e-mail with Bob Walters. We all participated together in the Walters, Kelly, & Street Sydney-Macquarie seminar.

Aurelio and Bob were shocked to find out that I still did not have e-mail; later they made a joint trip to Buffalo to persuade me of its importance and even took me to a store to purchase the correct modem. I continue to be touched by the generosity of my friends.

In 1989 Aurelio proposed to accompany me with his Alfa Romeo to visit Alexander Grothendieck in his stone hut in the middle of a lavender field near Mormoiron in the South of France. I had already visited Grothendieck in 1981 at that stone hut, but this time there was a specific item that needed to be negotiated. Grothendieck had stated that I might be the appropriate person to edit and publish his great work 'Pursuing Stacks'. But I had very serious questions about how that editing should be done in particular. Aurelio was attentively studying the situation. The discussion was complicated by the fact that Grothendieck was bound by a religious vow of silence, so that he could only communicate by writing (though he had given me a very friendly one-word greeting: Bill!)  

He wrote immediately that he was under a vow not to discuss mathematics. But his mathematical soul soon triumphed and he was writing mathematical statements and questions. (Aurelio observed all this with valiant equanimity.) The 1983 work of Grothendieck under discussion, concerned among many other questions the homotopy theory of presheaf toposes. It is a long manuscript, produced by the basic method of typing all night for several weeks. Naturally, mistakes occurred due to fatigue, but the fever to press on had meant that each morning when Alexander noted the previous evening's errors, he did not delete them, but simply kept them, and added the corrected version before proceeding. I said that as a conscientious editor I would have to delete the erroneous passages and give substantial explanations of my own. Grothendieck insisted that the errors be kept, so that students would have the opportunity to learn that even famous mathematicians make mistakes. I objected to this, pointing out that students have ample opportunity to see THAT and that learning the actual scientific material is difficult enough without the extra punishment that he was in effect advocating. But our parting was very amicable with the agreement to further consider the matter.  Aurelio agreed with my point. In the end the publication did not take place. Only several years later did Maltsiniotis and Cisinski work out the mathematics and bring this magnificent program to an initial fruition.

On the return drive from Mormoiron to Milano, we were curious to visit St. Tropez. After a long drive through the town we arrived at the waterfront to see a vast expanse of water choked with dozens of luxury yachts and cruisers to the extent that passage between them must have been difficult.

 Aurelio looked for a moment, then said: "I got the idea",

and we turned around and went back to the highway.

From then till now, we have found Aurelio's phrase 'I got the idea' to be immensely useful, like his Italian summing up: So tutto! It is not that he was opposed to listening to our long explanations; rather those phrases served as a kindness to us, a signal that our labors had already been successful.

As all those here know, Aurelio's sharp, useable formulations, in papers often written up together with colleagues, made exact completion and Schanuel's extensivity into household words.  The precise descriptions he helped develop of various classes of locally presentable categories are available for use. His contributions were broad, for example, he further perfected the topos-theoretic basis for non-standard analysis as proposed by Kock and Mikkelsen. His work with Marta concerning distributions on toposes will be discussed later in this meeting.

Though we must say farewell to him now, I should mention that perhaps one of the many accomplishments of which he was most pleased, and which is still far-reaching in its influence, is his organization, in collaboration with Maria-Cristina Pedicchio and Giuseppe Rosolini, of COMO 90, which turned out to be one of the most successful of the many excellent CTs.

                        When I walk in Milano near via Saldini, or around Viale Romagna, I sometimes imagine that I see Aurelio disappearing around a corner. And in the evening I imagine him playing soccer in the streets with Sammy Eilenberg, as he did when Sammy was in


                        If the young Aurelio were still with us, he would see our worries about the present and say: So tutto!  Let's get creative and solve the problems.