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of a clone - for each n, all n-ary operations of the intended type of algebra - is due to Philip Hall (apparently unpublished by him).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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++