On Friday, July 5, 2002, at 10:54 AM, Bruno Marchal wrote: > But, perhaps more importantly at this stage I must recall the book > "Mathematics of Modality" by Robert Goldblatt. It contains fundamental > papers on which my "quantum" derivation relies. I mentionned it a lot > some time ago. > And now that I speak about Goldblatt, because of Tim May who dares > to refer to algebra, category and topos! I want mention that Goldblatt > did wrote an excellent introduction to Toposes: "Topoi". (One of the big > problem in topos theory is which plural chose for the word "topos". > There > are two schools: topoi (like Goldblatt), and toposes (like Bar and > Wells). :) > > Goldblatt book on topoi has been heavily attacked by "pure categorically > minded algebraist like Johnstone for exemple, because there is a remnant > smell of set theory in topoi. That is true, but that really help for an > introduction. So, if you want to be introduced to the topos theory, > Goldblatt Topoi, North Holland editor 19?(I will look at home) is > perhaps the one.

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Yes, this is an excellent book. It has more of an expositional style than many books on category and topos theory. It's out of print and Amazon has been looking for months for a used copy for me. (Amazon can search for books which become available. I also have them searching for a copy of Mac Lane and Moerdijk's book on sheaves, logic, and toposes, also out of print.) Fortunately, I live near UC Santa Cruz, which has an excellent science library. The category and topos theory books I actually _own_ (bought through Amazon) are: * Cameron, Peter, "Sets, Logic and Categories, 1998. An undergraduate level primer on these topics. One chapter on categories. (By the way, most modern algebra books, e.g., Lang's "Algebra," Fraleigh, Dummitt and Foote, etc. have introductory chapters on category theory, as this is the "language" of modern abstract algebra.) * Lawvere, F. William, Schanuel, Stephen H., "Conceptual Mathematics: A first introduction to categories," 1997. This is a fantastic introduction to categorical thinking. The authors are pioneers in topos theory, but the presentation is suitable for any bright person. There is not much on applications, and certainly no mention of quantum mechanics a la Isham, Markopoulou, etc. But the conceptual ideas are profound. (And this should be read before tackling the "formalistic" presentations in other books.) * Pierce, Benjamin, "Basic Category Theory for Computer Scientists," 1991. A thin (80 pages) book which outlines the basics. Includes material on compilers, the "Effective" topos of Hyland and others, cartesian closed categories, etc. * Mac Lane, Saunders, "Categories for the Working Mathematician, Second Edition," 1971, 1998. Wow. A dense book by the co-founder of category theory. As someone said, reading along at 10% comprehension is better than reading other books at full comprehension. I find the book sort of dry, on historical and conceptual motivations, but Mac Lane has written many longer expositions in MAA collections of reminiscences...I just wish mathematicians would do more of what John Baez in his papers: show the reader the motivations. (Much of mathematical writing came out of the tradition of "lecture notes." In fact, the leading publisher of mathematics, Springer-Verlag, calls their series "Lecture Notes," or, more recently, "Graduate Texts." Brilliant mathematicians like Emil Artin and Emmy Noether would have their lectures on algebra transcribed by grad students or post-docs, like Van der Waerden, who would then republish the notes as "Moderne Algebra," the first of the "groups-rings-fields" modern algebra books. Which is why one of E. Artin's students, Lang, writes so many dry books! These books are often very short on pictures or diagrams, very short on segues and motivations. It's as if all of what a good teacher would do in class, with drawings on blackboards, with historical asides, with mentions of how material ties in with material already covered, with mention of open research problems and unexplored territory...it's as if all this material is just left out of these texts. Too bad.) * Lambek, J., Scott, P.J., "Introduction to higher order categorical logic," 1986. Way too advanced for me at this point. So no comments on content. But it's useful to glance at topics so as to get some idea of where things are going (part of the issue of motivation I raised above). * Taylor, Paul, "Practical Foundations of Mathematics," 1999. Another advanced book, covering logic, recursive function theory, cartesian closed categories, and a lot of the second half I can't comment on. A wonderful browsing book, as he has lots of tidbits and asides. There are 3-4 other books I'd like to get, including the Goldblatt book (he is giving permission to Xerox his book, so I may do that), the Mac Lane and Moerdijk book, and a few others. Peter Johnstone wrote the defining book on toposes in 1977...long-since out of print and long-since overtaken by newer results. Ah, but he is about to have his massive 3-volume set of books on topos theory published: Here's John Baez's summary in his Week 180 column: "2) Peter Johnstone, Sketches of an Elephant: a Topos Theory Compendium, Cambridge U. Press. Volume 1, comprising Part A: Toposes as Categories, and Part B: 2-categorical Aspects of Topos Theory, 720 pages, to appear in June 2002. Volume 2, comprising Part C: Toposes as Spaces, and Part D: Toposes as Theories, 880 pages, to appear in June 2002. Volume 3, comprising Part E: Homotopy and Cohomology, and Part F: Toposes as Mathematical Universes, in preparation. "I can't wait to dig into this. A topos is a kind of generalization of the universe of set theory that we all know and love, but topos theory is really a wonderful way to unify and generalize vast swathes of mathematics - you could say it's the way that logic and topology merge when you take category theory seriously. I've really just begun to get a glimmering of what it's all about, so I'm curious to see Johnstone's overall view of the subject. " (end of John Baez's comments) The first two volumes are due this month or next, according to Oxford University Press (_not_ Cambridge!) and Amazon. Cost for the two is a whopping $295. But the books are 750 and 850 pages, respectively. I'm am steeling myself to buying them. A lot of money, but this stuff is more entertaining to me than spending the same amount for 1-2 nights in a hotel, or lots of other things people spend their money on. And obviously 1500 pages is a lot of reading! And to better understand these things, I've been brushing up on my math background. A lot of algebra texts (mentioned above), topology (Munkres, Hocking and Young, Alexandroff, various Dover editions), and algebraic topology (Massey, Bredon, Fulton, etc.). My background is mostly physics, but I fortunately had some good exposure to analysis and measure theory, the stuff that can provide the assumed "mathematical maturity" for further study. I wish I'd spent more time studying this stuff...but wishing about changes in the past is pointless. I'm here now, in my one and only present, and this category and topos theory is turning out to be enjoyable and stimulating as a goal unto itself, and as a tool for, I think, better understanding things I want to understand. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks