Besides Carl Futia's question as to the expressiveness of categories (if I understood its drift) there is also the less judgmental question of what it feels like to use category theoretic methods at any given level of sophistication, from the viewpoint of a classically trained mathematician, scientist, engineer, or technologist. Bill Lawvere's early pre-topos (in the nontechnical sense) work on the categorical axiomatization of sets creates something of a straw to grasp at for non-category theorists looking for insights into this question. But while it is natural to view topos theory as a maturation of this program, there is a bifurcation in the maturation process that should be kept in mind. Toys are the natural playthings of the institution of childhood. Maturation can shift the focus of that energy either to more sophisticated toys (and I lump preoccupation with fast cars together with love of good engineering or architectural design here) or to the support of institutions -- teaching, management, etc. This is not purely bimodal but is something of a spectrum, with entrepreneurs, technology visionaries, research university professors, and project team leaders occupying intermediate positions along it. Taking ordinary sets as the toys of late 19th century foundations, it seems to me that set theory and algebra stand in an analogous relationship. Set theory continues the early preoccupation with sets at a far more mature and technically demanding level. I'm not suggesting that set theory's motto should be "he who dies with the largest cardinal wins," but rather that its success has created a certain technically focused inertia that has sustained set theory for over a century in close to its original form as conceived by its founders. Algebra is more institutional in its outlook. Whereas set theory sits patiently inside the same house it was born in, algebra can view that house from outside if it wishes. In this regard I lump category theory together with algebra, with category theory strongly supporting the inside-out perspective via the Duality Principle as generalized from posets to categories (reverse the arrows -- the Duality Principle is like glass, so transparent as to make its substance invisible). This is actually easier for algebra as it was not born in that house, predating set theory by half a century. The symmetric group of finite order n is more easily seen to have an underlying set than say a Lie algebra, and one might even take the view that Lie algebras have been unjustly dragged into the house and taken hostage by set theory, with pointset topology intervening to prevent their torture. Isbell's example of the intersection of the rationals and the irrationals as a nonvacuous locale makes the point that a faithful functor to Set^op, as a house on the other side of the street, might offer a more hospitable home than the Set house for some denizens of mathematics. The fact that many algebraists still have a strongly set-theoretic view of their world is more inertial than intrinsic, as evidenced by an increasing willingness of the mainstream algebra community to accept categorical perspectives as a legitimate point of view. This is not to say a set theoretic outlook on algebra is bad, rather that it puts traditional algebraists in a middle position not unlike that of technology visionaries or team leaders. But without recognition of the Duality Principle, a geometer such as Bill Thurston, who as then-director of MSRI kicked off the 1989 UACT meeting with the insightful confession that he felt ill contemplating Set^op, is missing a major part of the big picture, just as is the visionary entrepreneur who fails to see the need for marketing expertise. It should be a rite of passage to category theory that the initiate be told that a topos is not intended as a universe of sets so much as of institutions (again in a non-technical sense) of a particular character. Even those who insist that a graph is a set (on the ground that everything is a set) will surely allow that it is a set with structure. And in fact they would not object to calling a graph a pair of sets V and E of vertices and edges as long as you humored them by allowing that it was therefore the set {{V,E},{V}}, as the von Neumann encoding of the pair (V,E) as a single set. But if you could put this encoding in perspective for them as being less central (in whatever sense impresses them: pedagogical, natural, etc.) to the graph concept than the basic sets V and E themselves, you are off to a good start explaining what toposes are about. If this post had been directed to non-category theorists it would be necessary to make it somewhat longer. The present audience should need nothing beyond the distinction between sets per se such as V and E vs. presheaves of sets to continue it to its intended conclusion (though it's a nice question how many conclusions might result). Those with experience of hostile appointment and promotion committees may already have had some practice with that. Simplicial complex theory as an extension of reflexive graph theory (both as petits toposes) is a nice next step avoiding the implication that graphs constitute the core paradigm here. The same needs to be pointed out concerning the role of reflexive graphs as the algebraic stepping stone from sets to categories, which is only the beginning of that story when one looks ahead to n-categories. While I'm still unclear as to the optimal order of things for training the working mathematician in category theory, a short introduction to category theory for the benefit of the other kind of mathematician might profitably organize itself around a manageable part of the background material presumed by the above point of view, with the goal being insight into why presheaves are not sets and how toposes abstract them. Unless of course you're one of those rugged left-field individualists who believe that linearly distributive categories are more important than toposes. ;) Vaughan Pratt 14-Mar-2005 11:01:08 -0400,3248;000000000000-00000000