RANDOM THOUGHTS ON THE RECENT UA-CAT WORKSHOP AT MSRI
Saunders Mac Lane Dept. of Math, Univ. of Chicago, Aug, 1993
In addition to drafting a final report on this workshop, I'll venture here to
ask some provocative questions about the interrelation between Universal
Algebra (UA), Category Theory (CAT) and the connections of both subjects to
all
(!) the rest of mathematics. I hope thereby to provoke both strong dissent
and
some progress and innovation.
First those minor remarks about terminology.
CLONE vs. ALGEBRAIC THEORY. The notion of a clone - for each n, all n-ary
operations of the intended type of algebra - is due to Philip Hall (apparently
unpublished by him). Lawvere's notion of an "algebraic theory" (his thesis,
1963) is a category, objects the natural numbers, maps n-->m given by all
m-tuples of term operations. These two notions are essentially equivalent; the
word "clone" is short and convenient, while the Lawvere notion includes more
composites and so is more flexible and easier to formulate and to generalize.
It follows one of the tacit slogans of CAT: With every map, present at once
both the domain and the codomain, by this effective presentation making the
whole situation more visible and showing what composites are possible. This
is
the useful adage "Draw Diagrams! This art should be easy to learn; but as for
the choice of words, I suppose that both "clone" and "Algebraic theory" will
continue in use, hopefully with a more general recognition of their
equivalence.
"GROUPOID" as the name of a category with every arrow an isomorphism,
apparently appeared first in 1926 in a study of the ideals of algebraic
integers in a non-commutative algebra (Brandt). This notion has vital uses in
many parts of mathematics, as for example for the fundamental groupoid of a
space, and so combining the fundamental groups at each base point. It appears
also in differential geometry and in mechanics, as a means of transporting
structures along one of the paths of the space at hand. In UA, on the other
hand, a groupoid has meant a set with a single binary operation--and no
required identities; it may be that this usage first appeared in a book by R.
H. Bruck. Such objects do need a name, but clearly the categorical use of
"groupoid" is the widest and most important. I hope that the other use can be
replaced by some other suitable term; possibly "binary algebra" or perhaps
even
"binoid".
The "empty algebra" has been proscribed in UA. Since there is now occasion to
consider models of algebras not just in sets (possibly in topoi or in regular
categories) this early proscription seems too narrow.
WHAT IS AN ALGEBRA? CAN WE LOOK BEYOND THE PRESENT? The Clone-algebraic theory
notion is now evidently too narrow; no infinitary operations. The algebras
for
a Monoid (=triple, as in Linton's lecture) are somewhat more general and much
neater (adjoints again!, cf. my "Categories Work:"). The algebras described by
sketches are considerably more general. Recall the lecture by Wells, where a
sketch together with its models is described by a graph, together with a set
of
diagrams there (to commute in the models), a set of cones (to be limits) and a
set of cocones (colimits). This notion is flexible, especially because it
comprises colimits as well as limits. The lecture by John Gray indicated how
this works for computer science; see also recent books e.g., by Barr-Wells.
The
Workshop did not get to consider how standard concepts of UA would work out
for
sketches. Much more might develop here!
WHERE DO THE MODELS OF ALGEBRAS LIE? Classical model theory says: "in sets,"
with a spectrum. There have been a few starts at taking models in a general
topos. Does this provide for greater "Variety"?
MALTSEV ALGEBRAS are conventionally described by those familiar term
identities--but their bearing on the composition of relations is relevant,
while a recent paper of Carboni, Lambek and Pedicchio (JPAA, 69, 1991)
indicates how a category of such algebras can be characterized as a kind of
regular category. This offers a striking connection; see also a preprint,
Kelly et al, on Goursat categories. Many prospects might open up here.
TAME CONGRUENCE THEORY provides remarkable and surprising results for finite
algebras (those astonishing five types of possible quotients for a covering).
But finite algebras have a somewhat limited scope; perhaps something equally
"Tame" could be managed for some classes of infinite algebras, possibly by the
well-tried method of introducing a topology?
FINITE CLONES on two elements (E. L. Post) bear on the problems of "truth
values"; but there appear to be too many clones on bigger finite sets; it is
not yet clear to me how they relate to other mathematics.
COMMUTATOR THEORY is a fascinating development, especially in view of its
relation to commutators in group theory--and what about Lie algebras and the
Lie bracket here? At the end of the workshop M. C. Pedicchio mentioned to me a
somewhat more categorical description of such commutator theory.
SUBOBJECT LATTICES and CONGRUENCE LATTICES seem to arise almost separately in
UA. One merit of categories is that they bring together the inclusions
(subobjects) and the epimorphisms (quotients), both appearing as arrows. Does
this view have consequences? Is it simpler?
INTERACTIONS. Some of the most striking developments of mathematics come from
unexpected connections (collisions) between different subjects. I know of
several such collisions in CAT: The use of categories in the drastic
reformulation of algebraic geometry by Grothendieck, the connection to Quillen
Model structures in topology (Tierney's lecture) and the recent use of
coherence theorems for commuting diagrams as they appear in quantum field
theory and quantum groups (which are really not groups but deformed Hopf
algebras). Then there is computer science: sketches, categories expressing
data types and the newer uses of topoi there. I do not know of similar
outside
connections for UA. How should one search and in which other fields? In Logic?
SIMPLICIAL SETS were designed for calculations for Eilenberg-Mac Lane spaces,
defined as functors, and now are used as surrogate spaces!
CONCEPTS: Emmy Noether's "modern algebra" achieved much, and in particular
emphasized the astute use of general concepts--the homomorphism theorems, for
example, to illuminate and make understandable known mathematical results.
This was in particular the case with Galois theory, which before had been
expressed in terms of obscure "substitutions", but was then organized in terms
of Automorphisms to become more perspicuous, as later in Emil Artin's
presentation. The same clarification came to ideal theory, done jointly for
algebraic numbers and for algebraic varieties. I personally hold that a major
role of "abstract" mathematics is this sort of clarification and
understanding--both for its own sake and for the new results it can and does
encourage. The recent reformulation of algebraic geometry is a striking
example of this.
The notion of an ADJOINT FUNCTOR (for posets, Galois correspondence) is one of
the most pervasive and effective notions of CAT (recall Joyal's lecture on the
Witt vectors as an adjoint).
TWO-DIMENSIONAL categories (with 2-cells to represent things such as
homotopies)
and the related higher dimensional categories, originally seemingly
complicated, have now found much use (Street's lecture), as in the solutions
of
Yang-Baxter equations, and in coherence theorems.
PROBLEMS vs. CONCEPTS. Sometimes mathematics is viewed as the construction and
astute solution of hard problems. When they are indeed of central interest
and
make use of powerful methods, as in the recent case of Fermat, this is
splendid. On the other hand, not every hard problem is necessarily relevant
to
the progress of knowledge. How to select?
The PARTICULAR vs. the GENERAL. Some categorical studies are troubled by a
sort of "dual" objective--the study of special problems vs. the formulation of
general notions directed at wide aspects of mathematics - what is a theory,
for
example. Lawvere's lecture at the Workshop emphasized the general, in the
guise of a Doctrine (a category with structure) or a hyperdoctrine. These
questions may have some bearing on education: in teaching, should we present
just the particulars and just the corresponding skills, or somehow try to
convey the general in its particular guise. Such questions are provocative
and
contentious, but nevertheless useful. Learn it all young! This is now
possible;
recently, Lawvere and Schanuel have prepared an elementary text on "Conceptual
mathematics."
A TOPOS is a category with good structure (finite limits, exponentials.
subobject classifier). It encompasses sheaf theory, yields an internal logic
with a Heyting algebra of truth values, while sheafification gives a
formulation of Cohen forcing, Topoi have recently been used in modelling
computer science; a topos thus exemplifies many connections.
Thus arises my hope that we can search here and elsewhere for decisive new
interactions. The growth of ideas can be splendid!
Saunders Mac Lane
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