Subject: categories: Appreciation of Jon Beck

From: Michael Barr <[email-masked]>
Date: Wed, 29 Mar 2006 08:54:48 -0500 (EST)

It would seem appropriate for me to write something on Jon's
mathematics.  His career can be divided into three parts.  The first was
on (co)homology theories, (co)triples, and resolutions and is really the
only one I understand.  The second was on things related to H-spaces and
there were only two published papers [LNM #88, 139--153 and LNM #196,
54--62].  One thing I know he claimed (and never got credit for) was
that an A_\infty space is homotopic (or maybe weakly homotopic) to a
topological monoid.  Since I never really understood this work, I will
leave it to others to comment on.  The third was his attempt to apply
homotopy theory to understand the operations of a finite calculator.
This was evidently an outgrowth of the second, since the operations of a
finite calculator are not associative, but perhaps more like an A_\infty
space.  Since that work was not perfected, I will say no more about it.

Jon's thesis has many things in it, but here are the parts I remember
best.  One of the most ingenious was the idea of a Beck module.  He
observed that if *C* is a category and C an object, then the category
Ab(*C*/C), the abelian group objects in the category of objects over C
is defined to be the category of C-modules.  Here is what that amounts
to in specific cases:  Groups, left C-modules; rings, C-bimodules;
commutative rings, left C-modules; Lie algebras, left C-modules.  It is
apparent that in each case (including that of commutative algebras whose
cohomology theory appeared only in 1962 and Jon did not know of until
later), these Beck modules are exactly the modules used as coefficients
in the cohomology theories.  It also turns out that if C' --> C is an
object over C, and M is a C-module, then Hom(C',M) in the category *C*/C
can be identified in all those cases as the group Der(C',M) of
derivations.  I think that this brilliant observation alone would have
deserved a PhD, but it was only one chapter (of five) in Jon's thesis.

Given a cotriples \G on the category *C*, define a resolution of an
object C as the simplicial object
      --->      --->
   ... ..  G^3C  ..  G^2C ===> GC
      --->      --->
(This is an awful medium to do this in.  You also have to imagine
degeneracies.)  and the cohomology H^.(C,M) as the cohomology groups of
the chain complex
  0 ---> Der(GC,M) ---> Der(G^2C,M) ---> Der(G^3C,M) ---> ...
Jon proved that H^0(C,M) = Der(C,M) and that H^1(C,M) could be
interpretated as the group of "singular extensions" (he had to find the
appropriate definitions of that too) of C with kernel M. These
definitions are thus, in low dimensions, closely related to the
classical cohomology groups modulo the usual shift in dimensions.  He
also defined what you might call the inhomogenous chain complex that
used the triple in the underlying category to define the same groups.
Along the way, he also proved the PTT, the theorem that characterizes
the category of Eilenberg-Moore algebras for a triple.  Although all
this was available in a draft dated 1964, the final version was
completed only in 1967.  Fortunately, the thesis is now available as a
TAC reprint.  Until that reprint, the only source were various
n-generation photocopies that were being passed hand to hand.

I spent the two years 1962-64 at Columbia, but I don't recall that I had
much contact with Jon until the spring term of 1964.  And even then,
Sammy ordered me to keep away from Jon because he wanted him to write up
his thesis.  But we talked about showing that the higher dimensional
groups he had defined were the same as the higher dimensional groups
that had been defined originally.  We had no handle on the question.
This discussion continued through the fall of 1964 when we had both
moved to Urbana.  The two resolutions looked so different that we just
could not see any way to procede.  In mid-December, we both went east
for the vacation, Jon to NY and me to Philadelphia and eventually also
in NY.  One day, I ran into Jon absolutely accidentally at the Times
Square subway station, if you can imagine, and Jon asked me if I had ever
heard of acyclic models.  I hadn't.  Jon had spoken to Harry Appelgate,
another Eilenberg student, who was writing a thesis on a categorical
version of acyclic models.  When we got back to Urbana, we looked at
acyclic models a la Appelgate.  As I recall it, Appelgate's version used
a Yoneda extension to induce a triple on a functor category.  Since we
were starting with a triple, it turned out to be much simpler for us and
we quickly (well, fairly quickly) carried out the required
verifications.  We both spoke on this at the first midwest category
meeting in Chicago in April of 1965 and Jon spoke on it at the La Jolla
meeting and it appeared in the La Jolla prodeedings [Springer, 336-343].

The final collaboration that Jon and I had was that of using a
simplicial resolution, not derived from a cotriple, to define cohomology
and derived functors.  This was a long paper in LNM #80 (informally
called the "Zurich Triples Book") [345--435].  I think we did this after
(but I cannot recall) seeing Michel Andre's "step-by-step" resolution
for the particular case of commutative algebras.

One more notable thing he did in that period was discover distributive
laws between triples, since they do not naturally compose [LNM #80,
119-140].  This concept was crucial to my analysis of Shukla cohomology,
which required the original distributive law between multiplication and
addition in rings, moved up to the level of triples.

After that our interests diverged and I leave it to others to comment on
his later work.

Let me just make a comment on the word "triple".  Although Jon never
thought it was a good term he thought that "monad" was at least as bad
and didn't approve of the idea of replacing one poor term by another.  I
went along as a gesture of solidarity, although I don't have any deep
feeling on the subject (although my title TTT is really nice).  In the
end, of course, Mac Lane prevailed.  I once asked Sammy why he gave the
idea such a poor name.  Especially given the care that he and Steenrod
had taken to naming "exact".  His answer was that the concept seemed to
have no importance, so he and John Moore didn't spend any time on it!

Michael