What are the best books and lecture notes on category theory?
2026-04-01 06:30:27.1775025027
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Good books and lecture notes about category theory.
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There are 18 best solutions below
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Categories for the Working mathematician by Mac Lane
Categories and Sheaves by Kashiwara and Schapira
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Arbib, Arrows, Structures, and Functors: The Categorical Imperative
More elementary than MacLane.
I don't know very much about this, but some stripes of computer scientist have taken an interest in category theory recently, and there are lecture notes floating around with that orientation.
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Another book that is more elementary, not requiring any algebraic topology for motivation, and formulating the basics through a question and answer approach is:
An added benefit is that it is written by an expert!
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I've read a fair amount of Sets for Mathematics and found it to be a gentle introduction.
http://www.amazon.com/Sets-Mathematics-F-William-Lawvere/dp/0521010608/ref=pd_sim_b_5
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I'm also a fan of Tom Leinster's lecture notes, available on his webpage here. In difficulty level, I would say these are harder than Conceptual Mathematics but easier than Categories and Sheaves, and at a similar level as Categories for the Working Mathematician.
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Wikipedia has some nice free texts linked at the bottom. There's an online version of Abstract and Concrete Categories, for example.
Steve Awodey has some lecture notes available online too. (Awodey's newish book is expensive, but probably rather good)
Patrick Schultz's answer, and BBischoff's comment on an earlier answer also have good links to freely available resources.
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Awodey's new book, while pricey, is a really pleasant read and a good tour of Category Theory from a logician's perspective all the way up to topos theory, with a more up to date view on categories than Mac Lane.
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Barr and Wells, in addition to Toposes, Triples and Theories, have written Category Theory for the Computing Sciences, a comprehensive tome which goes through most of the interesting aspects of category theory with a constant explicit drive to relate everything to computer science whenever possible.
Both books are available online as TAC Reprints.
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The first few chapters of Goldblatt's Topoi: the categorial analysis of logic provide another fairly elementary introduction to the basics of category theory.
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MATH 4135/5135: Introduction to Category Theory by Peter Selinger
(17pp). Concise course outline. Only wish it covered more topics. Available in PS or PDF format.
http://www.mscs.dal.ca/~selinger/4135/
Handbook of Categorical Algebra (Encyclopedia of Mathematics and its Applications) by Francis Borceux. Rigorous. Comprehensive. This is NOT free, but you can see the contents/index/excerpts at the publisher's web site, listed below. This is a three volume set:
(v. 1) Basic Category Theory, 364pp. (ISBN-13: 9780521441780)
http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521441780
(v. 2) Categories and Structures, 464pp. (ISBN-13: 9780521441797)
http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521441797
(v. 3) Sheaf Theory, 544pp. (ISBN-13: 9780521441803)
http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521441803
Reprints in Theory and Applications of Categories (TAC). This site has 18 books and articles on category theory in PDF, including several by F.W. Lawvere.
http://www.tac.mta.ca/tac/reprints/index.html
Abstract and Concrete Categories-The Joy of Cats by Jirı Adamek, Horst Herrlich, and George E. Strecker (524pp). Free PDF. Published under the GNU Free Documentation License. Mentioned already by Seamus in reference to Wikipedia's external links for Category Theory, but worth repeating, because it's very readable.
http://katmat.math.uni-bremen.de/acc
A Gentle Introduction to Category Theory (the calculational approach) by Maarten M. Fokkinga (80pp).
http://wwwhome.cs.utwente.nl/~fokkinga/mmf92b.html
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As a young student, I enjoyed Peter Freyd's fun little book on abelian categories (available online as a TAC Reprint). The nice thing about Freyd's book is it isn't boring, and it has little pieces of wisdom (opinion) such as the remark that categories are not really important, you just define them so you can define functors. And in fact you just define functors so you can define natural transformations, the really interesting things.
Of course you may disagree, but blunt debatable assertions (like this one) always make for more interesting reading. Another provocative remark by this author is the observation that he himself seldom learnt math by reading books, but rather by talking to people.
From the nice link above I learned that Goldblatt also quotes a remark (which may have inspired Freyd's) by Eilenberg and Maclane that categories are entirely secondary to functors and natural transformations, on page 194 where he introduces these latter concepts.
Leinster's notes linked by Patrick, look nice - a bit like an introduction to Maclane's Categories for the working mathematician, chatty and full of debatable assertions, (many of which I disagree with, but enjoy thinking about). He does not give much credit, but I believe the adjoint functor theorems he quotes without proof, (GAFT,...) may be due to Freyd. Leinster's notes are easy reading and informative.
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Lawvere, Rosebrugh. Sets for Mathematics.
Pierce B. C. Basic category theory for computer scientists.
José L. Fiadeiro. Categories for Software Engineering.
Martini. Elements of Basic Category Theory.
Burstall, Rydeheard. Computational category theory. Requires ML background.
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Paolo Aluffi, Algebra: Chapter 0 has category theory woven all through it, particularly in Chapter IX of course. I can tell that randomly sampled pieces of the text are well-written, although I have never systematically read longer parts of it.
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A relatively new source tha I believe hasn't been mentioned: Notes on Category Theory with examples from basic mathematics by Paolo Perrone.
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field.
Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. From the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Finally, monads and comonads are treated on an equal footing, differently to most literature in which comonads are often overlooked as "just the dual to monads". Theorems, interpretations and concrete examples are given for monads as well as for comonads.
Lang's Algebra contains a lot of introductory material on categories, which is really nice since it's done with constant motivation from algebra (e.g. coproducts are introduced right before the free product of groups is discussed).