Subject: categories: Demystifying the categorial approach

From: Vaughan Pratt <[email-masked]>
Date: Mon, 03 Apr 2006 08:09:23 -0700

Reinhard Boerger wrote:
  Of course, category
> theoty usrather abstract and can hardly be explained to non-mathematians.

Would this not be both true and false of any mathematical subject worth
its salt?  Such a subject will include deep results and/or abstract
concepts that are inaccessible even to many mathematicians, let alone
nonmathematicians.  At the same time it should be possible to trace the
chains of reasoning motivating those results and concepts back to
origins that should be easy to motivate for the nonspecialist and/or
nonmathematician.

As a case in point, category theory can be motivated by presenting it as
an approach to axiomatizing sets and functions (and more generally
algebraic structures and homomorphisms as their structure-respecting
functions, but one need not start there).  In that approach, instead of
defining a function to be a binary relation of a certain form, one
postulates functions as primitives in their own right, axiomatized by
the laws of composition and the existence of identities.

One should not feel obliged to axiomatize in full the morphisms of Set,
certainly not at the outset, and moreover not if one plans to apply the
same ideas to the morphisms of other categories at some point.  This
central tenet of category theory should be one of the factors guiding
the order of development, with associative composition foremost.

Along similar lines one can point up the parallels with how the very
words with which we speak combine associatively to form phrases and
sentences, and how addition and multiplication of both integers and
reals also obey those laws but differ in furthermore being commutative.
  It should be possible to make such an account of the
category-theoretic axiomatization of functions very accessible both to
non-mathematicians and to eighth-graders with some interest in mathematics.

One should also avoid the common jargon of the category theory
literature.  Everyday language about everyday concepts is to be
preferred to new notions that need to be defined before they can be
understood.

One could also dwell on the politics of the status quo, were it as easy
to explain ZF.  Indeed, when one compares the compositional approach
with how functions are introduced into mathematics founded on ZF, one
has to wonder how the latter came to be preferred over a framework that
axiomatizes functions as primitives.  To any save those that have long
since come to live and breathe the ZF-based definition, it is unnatural,
unmotivated, and hard to explain to nonmathematicians by comparison with
associatively composing functions.

One hypothesis for why set theory is the preferred basis today for the
concept of function is that the primitive-function approach is just too
simple to take seriously.  But that can't be true, as we saw at the
Universal Algebra and Category Theory conference at MSRI 17 years ago,
where every algebraist there understood the motivation for defining
formal languages and relation algebras abstractly as monoids, a really
simple abstraction with a rich literature of implications.  Few of them
however seemed to fully appreciate the power of applying the same
abstraction, subject to essentially the same laws, to the definition of
function, being content with the ZF picture of functions.

Or am I grossly misinterpreting what we all witnessed at that meeting?
Walt Taylor, of McKenzie, McNulty, and Taylor, "Algebras, Lattices,
Varieties: Volume I", spoke on his highly developed theory of varieties
back to back with Fred Linton's talk on monads.  It was like ships
passing in the night.  I pointed this out to George McNulty at lunch
after Fred's talk, but though we both struggled mightily and without
rancour to communicate, we just could not get to square one, in the time
available before the first afternoon talk, with the concepts of monad,
adjunction, or their relevance to what Walt had just spoken about.  And
it's not as though George and I can't communicate at all, as can be
inferred from my profuse acknowledgements of his considerable help at
both the start and end of http://boole.stanford.edu/pub/iowatr.pdf,
followed up by http://boole.stanford.edu/pub/jelia.pdf (work done
entirely in the universal algebra tradition with no mention of
categories, since that was the audience for those papers).  Except for
his being the master and I the student, George and I are very much on
the same wavelength in algebraic matters, it was only category theory
that was a closed channel between us then.  Ralph McKenzie seemed to be
more keen than Walt or George for algebraists to embrace category
theory, but even for the prime mover of tame congruence theory it seemed
to be something of an uphill battle.

It's not just monads and adjunctions that noncategorist mathematicians
have a hard time with.  In his welcoming remarks at the start of the
meeting, MSRI director and formidable geometer Bill Thurston expressed
his discomfort with Set^op, and the incredulity of half the audience was
palpable for a second.

Even today category theory labors under the dual impressions that it is
too abstract to explain to the non-specialist, yet too trivial to take
seriously.  This screwy situation is not unlike the days when oxygen was
perceived only as the absence of phlogiston, oxygen being harder to
"see" than phlogiston despite our constant immersion in it, just as we
are constantly immersed in associatively composing functions.  The
chemists of the pre-oxygen era, who believed that burning wood gave up
phlogiston to the air, would have found quite mystical the idea that the
absence of phlogiston constitutes 20% of air.  Category innocents are in
much the same boat.

Complicating matters is that this works both ways.  It is hard to
understand someone who finds it hard to understand the "obvious,"
whether true or not.   How should a categorist understand a talented
mathematician unable to process the concept of the opposite of a
concrete category?  (Useful trick: define a function to be a binary
relation and consider its converse.  Why didn't Thurston think of that?
  Why didn't someone suggest it to him?)

Marta should look forward to a time when functions are taught as
associatively composing entities as routinely as nitrogen and oxygen are
taught today as the principal constituents of air.  High school math
teachers will look back at the 21st century and marvel at how confused
people were about the nature of functions in those days.

It might of course never happen, with the concept of graph of a function
forever taken as prior to the function concept itself.  One big obstacle
is that the concept of binary relation is sufficiently well motivated in
its own right as to justify being introduced first (but in that case try
to at least get *that* defined via relation algebra, Tarski's program,
which so far has not taken hold).  Another is that there does not seem
to be any knockdown argument making composition prior to application,
and the mathematical world is currently strongly wedded to the opposite.

But the biggest obstacle is that such a sea change can't happen via an
independently written book focused on the need for that change.  Rather
the viewpoint needs to be integrated into the existing literature,
either by adapting existing texts or providing superior alternatives
whose primary purpose is to meet the extant curriculum requirements and
which accepts that getting the definitions right is only a matter of
good hygiene and not a subject in its own right to be added to an
already crowded curriculum.  It has to be an inside job.

Vaughan Pratt