The n-Category Café

Aurelio Carboni, Robert Walters, and Richard Wood wrote:

We would like to add to Bill and Eduardo’s letters also our feelings of deep sadness at Max’s death.

Max Kelly’s Last Work

In due course Max’s last work will appear in a four-author paper. While it is not usual for coauthors to divulge who contributed what to a paper the present circumstances seem to warrant such, as an appreciation of Max’s extraordinary talents and tenacity.

Carboni, Kelly, Walters, and Wood, [CKWW] have for some time been extending the Carboni and Walters notion of ‘cartesian bicategory’ to the general case of bicategories that are not necessarily locally ordered. A cartesian bicategory B ultimately has a tensor product, a pseudofunctor $*: B \times B \to B$ that is naively associative and unitary. It is natural to ask whether such $(B,*$ is a monoidal bicategory, in other words a one-object tricategory in the sense of [Coherence for Tricategories; Gordon, Power, and Street]=[GPS].

Early in 2005 [CKWW] had shown that if a bicategory $A$ with finite ‘products’ $- \times -$ and 1, in the bilimit sense, has $(A,\times)$ a monoidal bicategory then a cartesian bicategory $B$ has $(B,\star)$ monoidal. In the course of polishing the paper it came to Max’s attention that nobody had proved the

Theorem: A bicategory with finite products is monoidal.

Nobody doubted the truth of this. In fact, experts in higher dimensional category theory said that if it were not true then the definition of tricategory is wrong! But when you consider the rather large amount of data that must be assembled and the many equations (some merely implicit in words such as pseudonatural and modification) that must be verified from the apparently rather weak universal property of finite products in the bilimit sense, it seemed like a rather thankless task to write out the details. This was to Max a completely unacceptable state of affairs. If nobody doubts the statement then it must be possible to find a good proof!

Now Max had no intention of redrawing any of the diagrams in [GPS]. For the last few years Max, with little central vision left as a result of macular degeneration, has been doing Mathematics using an 8-fold magnification monitor. This allowed him to see only a few square centimetres of a page at a time. Many [GPS] diagrams consume an entire page. His proof, that we were privileged to receive in the last few weeks, has no diagrams (though doubtless we will incorporate a few in a publishable version of the paper).

Max attributed the key idea in his proof to Ross Street. Briefly, this is how it goes: For $X$ a finite family of objects in the bicategory $A$, write $A(X)$ for the bicategory of product cones over $X$. Thus an object of $A(X)$ consists of an object $b$ of $A$, together with a family of arrows $p_i:b \to X_i$, such that for all $a$, the induced functor $A(a,b) \to \Pi A(a,X_i)$ is an equivalence of categories.

Lemma: $!:A(X) \to 1$ is a biequivalence.

(Recall that to say $B \to 1$ is a biequivalence is to say that:

i) there is an object $b$ in $B$

ii) for any objects $c$ and $d$ in $B$, there is an arrow $f:c \to d$

iii) for any arrows $g,h:c \to d$ in $B$, there is a unique 2-cell $g \Rightarrow h$.

It follows that in a bicategory biequivalent to 1, every arrow is an equivalence and every 2-cell is an isomorphism.)

Next, Max observes that if $A$ has finite products then, for any $B$, the bicategory $[B,A]$ of pseudofunctors, pseudonatural transformations, and modifications also has finite products, given ‘pointwise’ by the products of $A$. $- \times -$ is an object of $[A^2,A]$. We can use $(a \times b) \times c$ and $a \times (b \times c)$ as names for objects in $[A^3,A]$ and applying the Lemma to $[A^3,A](a,b,c)$ deduce the existence of the associator equivalence, pseudonatural in $a,b$, and $c$. The associator gives rise to two arrows (abbreviating somewhat) $$((ab)c)d \Rightarrow a(b(cd))$$ in $[A^4,A](a,b,c,d)$ and between these we have a unique invertible modification, the $\pi$ of [GPS]. The coherence of $\pi$ is chiefly the Stasheff non-abelian 4-cocycle condition (again see [GPS]) and for this we need only apply the Lemma to $[A^5,A](a,b,c,d,e)$ to see that the two modifications in question are equal. Of course the other data and equations are handled with similar appeals to the Lemma.

Max was not content to stop here. In his last few days he had been learning the rather subtle definition of symmetric monoidal bicategory and constructed the requisite braiding equivalence and syllepsis isomorphism for a bicategory with finite products. Everything follows from the universal property but Max has shown us how so that we can calculate with these things. His insights show us the way to deal with coherence issues arising from birepresentability generally and weak n-representability when the need arises. Max’s personal copy of [GPS] was autographed by Ross with the words “To Max Kelly, a master of coherence”. Yes, he was.

Aurelio Carboni, Robert Walters, and Richard Wood

André Joyal wrote:

Dear Max,

I feel deeply sad that you have left.
Now that you are gone, I realise how much you mean to me.
I regret not telling you that.
I wish to repair that by writing you this letter.
If I send it to Imogen and to your friends,
it will reach you in some way.

Your work has been a constant source of inspiration for me.
It combines beauty, rigor and depth.
It is fundamental, I use it every day.
It will last forever.
You were a great mathematician.

I also want to thank you for inviting me to Australia.
I did some of my best work there.
You were a great host.
I made many friends.

I wish we could meet again.
I will talk with you in my dreams.

Yours, André

Bill Lawvere wrote:

I am deeply saddened by the loss of Max. In our field he was a rock of reliability and a fountain of imagination. I will miss my lively, warm, kind, and sometimes mischievous friend.

Bill Lawvere

Eduardo Dubuc wrote:

I am deeply saddened by the death of Max Kelly. When I saw the subject in Ross posting, and before opening the message, my heart already felt anguish. I am more saddened with his loss that what I have been by the loss of any other member of our category theory community. In fact, I loved Max. I admired his courage, his independence of thought, his lack of hypocrisy, and I loved him simply by the way he was. I am proud that he considered me his friend. For me, our community is not the same without Max.

Eduardo J. Dubuc

Peter May wrote:

Max visited Saunders Mac Lane in Chicago in 1970-71, and conversations with him then were both great fun and greatly influenced my work. To quote from the preface to “The geometry of iterated loop spaces”, in which I introduced operads, “The notion of `operad’ defined in Section 1 arose simultaneously in Max Kelly’s categorical work on coherence, and conversations with him led to the present definition”. It is a pity that, due to ill-advised suggestions by a referee a little later, his January, 1972, preprint “On the operads of J.P. May” was not published until 2006! It contains many often rediscovered insights. See:

http://www.tac.mta.ca/tac/reprints/articles/13/tr13abs.html.

We also had many conversations about his upcoming role as chair in Sydney. In those days, before e-mail and even xerox, the problem of relative isolation down under was much on Max’s mind, and he thought that this was one good reason for following his heart and working to make Sydney a home for the development of the then underappreciated area of category theory that he so much loved. We are all in his debt for the marvelous way that he succeeded.

Peter May

Steve Vickers wrote:

I only met Max a couple of times, but I vividly remember a particular phrase of his. He would ask, “What’s the deal?”, and that was a prelude to cutting right through to the mathematical essence of an argument. The phrase has stayed with me ever since.

Steve Vickers

See also this webpage set up by Max’s son:

• Max Kelly’s Perpetual Web Page