Subject: categories: Johnstone review of Clark/Davey

From: Peter Freyd <[email-masked]>
Date: Tue, 29 Feb 2000 13:58:30 -0500 (EST)

I am chagrined to find myself as one of two category people
  in the bibliography of the work reviewed below and -- worse --
  Peter says that they ascribed schizophrenic objects "not 
  altogether appropriately" to me. He's right.

CMP 1 663 208 (99:06) 18-02 (03G25 06D05 08C15 18A40)
Clark, David M.(1-SUNYP); Davey, Brian A.(5-LTRB) 
Natural dualities for the working algebraist. (English. English
summary) Cambridge Studies in Advanced Mathematics, 57. Cambridge
University Press, Cambridge, 1998. xii+356 pp. $64.95. ISBN
0-521-45415-8

I began reading this book (conventionally enough) with the first two
paragraphs of the Preface:

  "In 1936 Marshall Stone published a truly novel theorem ... What
  Stone discovered was a representation for all Boolean algebras which
  gave algebraists a usable understanding of their structure, using
  topological spaces to construct the representations ... In fact,
  Stone proved much more than a representation theorem ... In modern
  language, he proved that the category of Boolean algebras is dually
  equivalent to the category of Boolean spaces ...".

Finding in these words what I presumed to be a conscious echo of the
words:

  "This book is about a particular theorem -- the Stone representation
  theorem for Boolean algebras -- and some of the mathematical
  consequences which have developed from it ... Stone's key idea was
  the introduction of topology ... this was a really bold idea ...
  Moreover, [Stone's representation formed] one of the earliest
  nontrivial examples of an equivalence of categories ..."

with which I had begun the Introduction to my own book Stone spaces,
published in the same C.U.P. series in 1982 [Cambridge Univ. Press,
Cambridge, 1982; MR 85f:54002], I naturally turned next to the
Bibliography to check that my book was listed there. After all, though
my book dealt with other subjects besides duality, it did have a whole
chapter devoted to dualities of the type studied by Clark and Davey,
and so it ought to be one of their standard references.

My name does not appear in the Bibliography. Indeed, its only
occurrence in the entire book is in the list of previous volumes in
the series which appears opposite the title page.

I mention this at the outset of this review, in order to give the
reader a fair chance to make allowance for what he may construe as
"sour grapes" on my part. For a related reason, I have departed from
Mathematical Reviews tradition by writing the review in the first
person; I cannot pretend to have achieved the standards of objectivity
and dispassionateness that would be implied by a third-person review.

However, it is not only the works of Johnstone which have been
overlooked by Clark and Davey. I looked next in the Bibliography for
John Isbell's ground-breaking 1972 paper on general functorial
semantics [Amer. J. Math. 94 (1972), 535--596; MR 53 #580], and his
later work characterizing concrete dualities in terms of commuting
subtheories of a ruled theory: neither is present, and Isbell is
represented only by a much less important 1980 paper on median
algebra. Again, the 1982 paper in which Harold Simmons coined the term
"schizophrenic object" [Topology Appl. 13 (1982), no. 2, 201--223; MR
83f:18006], for a set with two commuting algebraic structures, is not
there, although Clark and Davey freely use this term in their text.
Having discovered this, I began studying the Bibliography more
systematically, and soon realized its salient feature: all works
written by category-theorists, or making serious use of categorical
ideas, are excluded from it -- with the twin exceptions of Saunders
Mac Lane's classic Categories for the working mathematician [Springer,
New York, 1971; MR 50 #7275] (which, after all, the authors could
hardly have left out, given their indebtedness to his title), and of
Peter Freyd's 1966 paper on algebra-valued functors [Colloq. Math. 14
(1966), 89--106; MR 33 #4116] (to which I shall return below).

This, then, is a book on a categorical subject -- duality -- by
authors who are not category theorists, and presumably intended for
such a readership.  However, the editors of Mathematical Reviews have
asked a category-theorist to review it, so the review is written from
that point of view.

The subject of the book is the construction of dual equivalences, or
more generally contravariant adjunctions, between algebraic categories
$\scr A$ and categories $\scr X$ whose objects are compact topological
spaces equipped with compatible algebraic and/or relational structure.
Here one encounters the first restrictive feature of the authors'
outlook: for them "algebraic" means "finitary algebraic" (an attitude
already debunked by Marshall Stone in 1947), and so the idea of a
compact topology as a kind of algebraic structure (indeed, the
universal example of a structure which commutes with all finitary
structures) cannot be expressed. Nor can the concomitant "unity of
opposites" idea that $\scr A$ and $\scr X$ are two categories of the
same kind -- they are doomed to remain forever separate, like the
lovers on Keats's Grecian urn.

Clark and Davey follow the common tradition in universal algebra that
the underlying set of an algebra is not allowed to be empty. Having
thus amputated the initial objects from many of their categories 
$\scr A$, they are forced to amputate the terminal objects from the
dual categories $\scr X$, leading to unnecessary complications in the
descriptions of these categories as quasivarieties. Since they have
not amputated the terminal object from $\scr A$, they are forced (in
appropriate cases) to allow the empty space as a member of $\scr X$.
Thus, for them, a topological algebra does not necessarily have an
underlying discrete algebra!

As already mentioned, Clark and Davey make heavy use of schizophrenic
objects. However, as far as most of the book is concerned,
schizophrenic objects are simply a convenient ad hoc way of
constructing a contravariant adjunction; the fact that every
contravariant adjunction between categories of the type they consider
is induced by mapping into a schizophrenic object is not mentioned
until page 162, where it is ascribed (not altogether appropriately) to
the paper of Freyd mentioned earlier, and stated in such a convoluted
way as to be almost unrecognizable. Again, all the schizophrenic
objects the authors consider are finite (and topologically discrete);
thus, although they refer to Pontryagin duality in their Preface (and
Pontryagin appears, along with Birkhoff and Stone, as one of the three
mathematicians to whose memory the work is dedicated), they are able
to describe only the special case of Pontryagin duality for abelian
groups of some fixed finite exponent.

All the above criticisms are, in a sense, trivial ones; but their
cumulative effect is nontrivial. Category theory is often criticized
for consisting entirely of trivialities; but, as Freyd long ago
observed, its real function is to demonstrate that the trivial parts
of mathematics are trivial for trivial reasons, and that is a valuable
service which it performs for the mathematical community. Reading
Clark and Davey's book, anyone new to the field would find it
difficult or impossible to distinguish between the trivialities and
the results with genuine content, since the former are so often
presented in ad hoc ways that obscure the underlying pattern. This is
not a book that I would recommend a graduate student to read.

One might argue that the book's shortcomings are not of major
importance in relation to its declared purpose. After all, it contains
a wealth of detailed information on particular techniques for
establishing duality theorems, and "working algebraists" (read:
working universal-algebraists) will undoubtedly find it immensely
useful to have all these techniques collected together in one place. I
admit the force of that argument; but I also find it seriously
worrying, for it carries the implication that universal-algebraists
have given up the attempt to engage in dialogue with the rest of the
mathematical community. As the authors remark at the end of their
Preface, there is much that remains to be done in studying and
classifying concrete dualities; but this book is not likely to inspire
anyone outside the closed circle of universal-algebraists to take up
the task.

                        Reviewed by Peter Johnstone

             Copyright American Mathematical Society 2000