In response to Jean Benabou's question about the terminology for what some call "cofinal" functors, may I refer him to Section 4.5 of my book "Basic Concepts of Enriched Category Theory", where such notions are considered in considerable generality? In so far as we deal with functors - meaning "V-functors" in the context of V-enriched category theory - the terms I used, which are those common here at Sydney, are "final functor" and "initial functor". These notions, however, make sense only when V is cartesian closed; for a more general symmetric monoidal closed V, what is said to be initial is a pair (K,x) where K is a V-functor A --> C and x is a V-natural transformation H --> FK, where H: A --> V and F: C --> V are V-functors with codomain V, and thus are "weights" for weighted limits. The 2-cell x expresses F as the left Kan extension of H along K if and only if, for every V-functor T: C --> B of domain C, the canonical comparison functor (induced by K and x) between the weighted limits, of the form (K,x)* : {F,T} ----> {H,TK}, is invertible (either side existing if the other does); the book contains a third equivalent form making sense whether the limits exist or not. When these equivalent properties hold, the pair (K,x) is said to be INITIAL. The point is that, in this case, the F-weighted limit of any T can be calculated as the H-weighted limit of TK. When V is cartesian closed, we have for each V-category C the V-functor C ---> V constant at the object 1, limits weighted by which are the CONICAL limits, which when V = Set are the classical limits. For such a V we can consider the special case of the situation considered above, where each of H and F is the functor constant at the object 1, and where x is the unique 2-cell between H and FK; we call the functor K "initial" when this pair (K,x) is so; equivalently when the canonical lim T ---> lim TK is invertible for every T (for which one side exists -- or better put in terms of cones), or equivalently again when colim C(K-,c) == 1 for each object c of C. When V = Set, this is just to say that each comma-category K/c is connected. When the category C is filtered, a fully-faithful K: A --> C is final (dual to initial) precisely when each c/K is non-empty. The book goes on to discuss the Street-Walters factorization of any (ordinary) functor into an initial one followedby a discrete op-fibration. The above being so, it seems that Jean's good taste has led him to suggest the very same nomenclature that recommended itself to us at Sydney. I should have been happier, though, if he had recalled the treatment I gave lovingly those many years ago. There are many other expositions in the book that I am equally happy with, and which I am sure Jean would enjoy. By the way, someone spoke recently on this bulletin board of the book's being out of print and hard to get; I've been meaning to find the time to reply to that, and discuss what might be done. The copyright has reverted to me; but the text does not exist in electronic form - it was written before TEX existed, and prepared on an IBM typewriter by an excellent secretary with nine balls. I suppose I could have some copies - one or more hundreds - printed from the old master, after correcting the observed typos. But the photocopying and binding and the postage would cost a bit. I'ld be happy to receive suggestions, especially from such colleagues as would like to get hold of a copy. By the way, I sent out preprint copies to about 100 colleagues back in 1980 or 1981; if any of those are still around, I point out that they contain the full text. So too do those copies which appeared in the Hagen Seminarberichte series. Once again, I look forward to any comments, either in favour of or against making further copies. Max Kelly.

Why not get it retyped in TeX .... This would only be worth it if you can get a "subscription" system to offset that fee of course. ... R.A.G. Seely

I was thinking of suggesting the same myself - I'd be very interested in subscribing. All the better if, assuming Max is willing, you can find subscribers public spirited enough to allow free electronic availability once it's TeXed. (Perhaps subscribers get a bound copy signed by the author?) What would it cost? Can OCR scanning help the initial input for such texts? Steve Vickers.