--On Thursday, July 05, 2001 1:32 PM +0100 Richard Dybowski <rdybowski@btinternet.com> wrote:

I would be grateful if members of this list could give me their recommendations for introductory books on category theory. In particular, what is your opinion of "Conceptual Mathematics: A First Introduction to Categories" by Lawvere and Schanuel?

Richard

Hello Richard, Lawvere and Schanuel is certainly an easy read and covers the very basics. It is definitely the text that I would use for, say, introducing some undergraduates to the basics of CT in a advanced general topics course. My personal favorites are the following, in the order of my favor. ArbibManes75 is, IMHO, by far the most clear exposition of basic category theory available. I constantly reference it and it is a joy to (re-)read. AspertiLongo91 is above Pierce91 due to application, rather than CT, reasons. The book is also a bit more complete than Pierce's work. Taylor99 is an overwhelming tome of amazingly good stuff that I'm in the middle (well, truth be told, the first quarter) of. I *highly* suggest it because it does not focus solely on CT but places it in the proper (IMO) context - as a fundamental tool used to describe other pervasive structures in mathematics. RydeheardBurstall88 is a great text if you are (or want to be) an ML programmer. "Programming" categories is a great interactive way to learn about CT. @Book{ArbibManes75, author = { Arbib, M. and Manes, E. }, title = { Arrows, Structures, and Functors: The Categorical Imperative }, publisher = pub-ap, year = 1975, read = { Feb, 2001 }, reviewed = { 2-13-01 }, review = { An excellent introduction to category theory. The book is written for the mathematically astute student, but doesn't expect the reader to be a pure mathematics graduate student like [MacLane71] or similar. The work is very example-oriented, covering the basics in the first two chapters then moving to examples for the next several chapters (e.g. monoids, groups, metric and topological spaces). Additive categories are then covered, then structured sets. The book then moved to part two which covers functors and adjoints and then more examples: monoidal and closed categories and monads and algebras. This is one of my favorite intro-to-CT books, and not just because the authors were UMass profs. } } @Book{AspertiLongo91, author = { Asperti, Andrea and Longo, Giuseppe }, title = { Categories, Types, And Structures: An Introduction to Category Theory for the Working Computer Scientist }, publisher = pub-mit, year = 1991, series = { Foundations of Computer Science }, callno = { QA76.7 .A766 1991 }, read = { Briefly reviewed early Feb, 2001 }, reviewed = { 2-13-01 }, review = { Provides a general introduction to category theory from a classical computer science perspective. Chapters 1 through 7 cover all the standard topics (categories, universal constructions, functors, natural transformations, categories derives from functors and natural transformations, universal arrows and adjunctions, cones and limits, and indexed and internal categories. The second half of the book focuses on types as objects - formulae, types, and objects, reflexive objects and the type-free lambda calculus, recursive domain equations, second-order lambda calculus, and some examples of internal models. The book is typeset somewhat poorly which can be a distraction to the reader, otherwise it is a comprehensive, fairly readable text on the topics. } } @Book{Taylor99, author = { Paul Taylor }, title = { Practical Foundations of Mathematics }, publisher = pub-cup, year = 1999, volume = 59, series = { Cambridge Studies in Advanced Mathematics }, read = { started the week of Jan 1, 2001 }, review = {}, notes = {} } @Book{RydeheardBurstall88, author = { Rydeheard, { D.E. } and Burstall, { R.M. } }, editor = { {C.A.R.} Hoare }, title = { Computational Category Theory }, publisher = pub-ph, year = 1988, series = { Prentice Hall International Series in Computer Science }, read = { Feb, 2001 }, review = { Excellent basic introduction to category theory from a computer science point of view. Basic notions are captured and described by programming them in ML. Brief summary of categorical semantics in/with OBJ3, GTTS (a type theory), and Hagino's Category Programming Language. Really an excellent text. } } @Book{Pierce91, author = { Pierce, Benjamin C. }, title = { Basic Category Theory for Computer Scientists }, publisher = pub-mit, year = 1991, series = { Foundations of Computer Science }, callno = { QA76.9.M35 P54 1991 } } And of course I love the following two just for the titles: @Book{AdamekHerrlichStrecker90, author = { Ji\v{r}\'{i} Ad\'{a}mek and Horst Herrlich and George E. Strecker }, title = { Abstract and Concrete Categories: The Joy of Cats }, publisher = pub-wil, year = 1990 } @Book{FreydScedrov89, author = { Peter J. Freyd and Andre Scedrov }, title = { Categories, Allegories }, publisher = { Elsevier Science Publishers }, year = 1989, volume = 39, series = { North-Holland Mathematical Library } } I cannot comment on their contents because I have only read introductory chapters. Note that I am a Computer Scientist first and a Mathematician second, thus the more "hard core" CT books are beyond my means at this point in time. Let me defend my thesis though and I'll be plowing through some of the more classic texts (e.g. BarrWells84|85|95, Borceaux94a|b|c, MacLane71) in no time. You can find my full CT bibliography at http://www.cs.caltech.edu/~kiniry/projects/papers/kiniry/bibliography/categ ory.bib and my full research bibliography at http://www.cs.caltech.edu/~kiniry/projects/papers/kiniry/bibliography/index .html for more information. Have fun, Joe (aka reads-too-much-for-his-own-good) Kiniry -- Joseph R. Kiniry http://www.cs.caltech.edu/~kiniry/ California Institute of Technology ID 78860581 ICQ 4344804