In reply to André : What seems reasonable to do is analysis, namely what is behind the success of category theory and how is this success related to the progress of mathematics. Which implies asking questions of mathematics, some of which have been aired in this discussion list. In this way, it should be possible to avoid seeming partisan, but to ask serious questions, which should help to steer directions, or suggest new ones. Of course lots of great maths does not arise in this way, but by following one's nose, but that does not mean that such analysis of direction is unhelpful. I know some argue that this excursion into what might be called the theory of knowledge, or into methodology, seems unnecessary to some. In reply I sometimes point to remarks of Einstein on my web site www.bangor.ac.uk/r.brown/einst.html or more mundanely retort that normal activities normally require some meta discussion: if you want to go on a holiday, you do some planning, you don't just rush to the station and buy some tickets. I develop this theme in relation to the teaching of mathematics in an article What should be the output of mathematical education? on my popularisation and teaching page. I gave a talk to school children on `How mathematics gets into knots' in the 1980s, and a teacher came up to me afterwards and said: `That is the first time in my mathematical career that anyone has used the word `analogy' in relation to mathematics.' Yet abstraction is about analogy, and very powerful it is too. This was part of the motivation behind the article 146. (with T. Porter) `Category Theory: an abstract setting for analogy and comparison', In: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. pdf There is also interest in the question of how category theory comes to be successful, and more successful than, say, the theory of monoids. This seems connected with the underlying geometric structure being a directed graph, i.e. allowing a `geography of interaction'. A category is also a partial algebraic structure, with domain of definition of the operation defined by a geometric condition. Is this enough to explain the success? It is worth noting that the article Atiyah, Michael, Mathematics in the 20th century, Bull. London Math. Soc., {34}, {2002}, 1--15, suggests that important trends in the 20th century were: higher dimensions, commutative to non commutative, local-to-global, and the unification of mathematics, but does not include the words `category' or `groupoid', let alone `higher dimensional algebra'! This kind of analysis needs to be presented to other scientists, and to the public, not only to mathematicians. There is a hunger for knowing what mathematics is really up to, in common language as far as possible, what new concepts, ideas, etc., and not just `we have solved Fermat's last theorem'. If your analysis of what category theory should do suggests some gaps, then that is an opportunity for work! Good luck Ronnie Brown Joyal wrote: Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research. It is probably the most important mathematical developpement of the 20th century. But we cant say that publically. André Joyal [For admin and other information see: http://www.mta.ca/~cat-dist/ ]