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 ------------------------------- Richard Dybowski, 143 Village Way, Pinner, Middlesex HA5 5AA, UK Tel (mobile): 079 76 25 00 92
--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
I am surprised neither Mike nor Charles has replied to this thread, but I would remind folks that their two books (which have been mentioned by others) are available (very cheaply in fact): TTT is in fact free on-line, and CTCS is now an inexpensive publication of the CRM. The details may be found on the Montreal Category Theory page <http://www.math.mcgill.ca/triples/>. (Click on the last two entries under "What is Category Theory?".) -= rags =- ================== R.A.G. Seely <rags@math.mcgill.ca> <http://www.math.mcgill.ca/rags>
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
I'm surprised no one has mentioned is Bob Walters' "Categories and Computer Science" (Cambridge University Press). It's a useful little book, unbeatable value, with a wealth of interesting and unusual examples from computer science. Steve Vickers.
I'd say that the best intro book on CT is a pair (\P,\B) where \P is a problem (in algebra, topology, logic, computer science, whatever) in resolving which you are really interested and suspect that CT would be useful to approach it, and \B is a set of chapters taken from a set of CT-books/papers that you suspect is reasonable to read to approach \P. Normally, it'd work as follows. Having \P in mind, you have some main interpretation of CT-constructions you've read in \B (actually, they have multiple non-isomorphic interpretations) and it'd guide you thru \B (a qualified advice here would be very useful). It allows you to reformulate \P more precisely in CT-terms as \P' and it gives rise to \B', and so on. Hopefully, at the end you'll have read a set \B! and interpret it not only in terms of your problem \P but much wider, with a few interpretations in mind, and in abstract CT-terms as well. Then you may well consider your have some working knowledge of some fragment of CT. Well, for someone some strong curiosity itself may provide enough motivation for CT-studies (probably, there are as many ways to and in CT as there are people willing to follow such a way), but I suspect that without \P CT-studies might be really hard with any \B. ZD
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?
Another source is the set of notes compiled by Mario Caccamo, based on lecture courses given by Martin Hyland and Glynn Winskel, at http://www.brics.dk/~mcaccamo/ Being lecture notes, they're quite "tight" and to the point, thus not particularly chatty. Perhaps they'd be a good complement to the Lawvere-Schanuel book (which personally I like a great deal). Tom 10-Jul-2001 14:34:24 -0300,2105;000000000000-00000015
On 5 Jul, Richard Dybowski 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
If you read french, you can try my diploma work at http://iiun.unine.ch/people/mamiguet/index.html (third downloadable ps file) It is intended to be a manageable introduction to CT for undergraduate french-speaking science students (yes, that's true, especially computer science students). I tried to keep the examples simple (vector spaces, graphs, etc.) It's highly inspired by Barr&Wells' book, but meant to be a little bit more simple (and less complete). If you try it, please let me know if you find it useful. Matthieu 12-Jul-2001 07:40:35 -0300,886;000000000000-0000001a
participants (7)
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Joseph R. Kiniry -
Matthieu Amiguet -
Richard Dybowski -
Robert A.G. Seely -
S.J.Vickers@open.ac.uk -
Tom Leinster -
Zinovy Diskin