Greetings. These days I'm going through a little categories-refresher course. Among my reading matter are the "Categories, Functors..." by Arbib and Manes, and Manes' "Algebraic Theories". My three questions are related to these. Q1: In "Algebraic Theories", there are several propositions (say 4.22, 4.23, 4.24) which include the following assumption: "Let (K,E,M) be a regular category _such that every equalizer is in in M_." This mystifies me, because an equalizer is _always_ an M-monic, for any image-factorization system (E,M). In any category, if an equalizer f factors as f = ge (composition to the left...) with e epi, then e is an isomorphism. This is quite trivial; since I cannot imagine that this could have been overlooked for as long as image-factorization systems were around up to the Manes' book, I am starting to doubt my ability to spot a mistake in my reasoning... Q2: How can I reach Manes or Arbib? They're not on the structures mailing list. I have collected quite an errata on their book which they still might find useful. Q3: This last question is about Universal Algebra vs. Monads and friends. My knowledge of the relevant literature is rather narrow, so this question has been probably adressed somewhere; I'd like the relevant pointers. As a kind of self-imposed exercise I have tried to fill in the gap I felt there exists between the old good techniquies of UA and that of high-flying categorial treatment involving monads and such. I developed a little "elementary theory of UA in categories" which uses arrows as axioms, in the following sense. Let C be any category, and A a set of arrows in C. Say that an object c of C _satisfies_ A if for every arrow p: a ---> a' in A, any arrow f: a ---> c factors through p as f = f'p. Denote by C:A the full subcategory on all the objects that satisfy "axioms" A. Then, playing around with conditions imposed on the category C and on the axioms A, I can, step by step, reconstruct most of the "standard" results of UA, up to and beyond the Birkhoff's theorem. All along the reasoning stays close to that in AU, just translated into the language of arrows. At certain point a monad emerges, and then I can make the jump upwards to more rarefied regions. For me this treatment by arrows as axioms provided the "missing link". On the road up to the final ascent to monads there are plenty of points where one can go off in some other direction and still get some benefit by transplanting part of UA techniques "in abstract" to an un-algebraic matter. One can also generalize: there are other conditions on an object expressible by a single arrow, one can allow restricted boolean combinations of one-arrow conditions as basic statements (say Horn implications), and so on. If this was already done, where can I look it up? If not, is it worth publishing, and where? ------------------------------------------------------------------------ France Dacar Email: france.dacar@ijs.si Computer Science Department Phone: +386 61 1-259-199 / 768 Jozef Stefan Institute Fax: +386 61 1-258-058 Jamova 39, 61000 Ljubljana, Slovenia ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++