F. William Lawvere

*Farewell to
Aurelio
*

**FAREWELL TO AURELIO**

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 60^{th} 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

Milan.

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.