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