Max Kelly, a master of coherence

Max Kelly, a master of coherence

[RJ Wood]

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 *:BxB--->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'
-x- and 1, in the bilimit sense, has (A,x) a monoidal bicategory then a
cartesian bicategory B has (B,*) 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--->X_i, such that
for all a, the induced functor A(a,b)--->\Pi A(a,X_i) is an equivalence of
categories.

Lemma: !:A(X)--->1 is a biequivalence

(Recall that to say B--->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--->d
iii) for any arrows g,h:c--->d in B, there is a unique 2-cell g--->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. -x- is an object of [A^2,A]. We can use (a x b) x c  and  a x (b x 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 ===> 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