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What are the best books and lecture notes on category theory?

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    $\begingroup$ Community wiki? $\endgroup$ Jul 21, 2010 at 20:23
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    $\begingroup$ I can give you an "anti-recommendation": don't get Cameron's "Sets, Logic and Categories". While it's a nice short introduction to some set theory and logic, the final chapter on category theory is too short and not at all well explained. It is however a neat little book for logic and sets... $\endgroup$
    – Seamus
    Aug 3, 2010 at 13:11
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    $\begingroup$ Best with respect to what metric...and for whom? This is a very fuzzy question. $\endgroup$ Feb 13, 2011 at 7:03
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    $\begingroup$ Programmating Reading Guide by Stanford Encyclopedia of Philosophy. It is a supplement to this article. $\endgroup$
    – M. Vinay
    Mar 9, 2016 at 8:39

26 Answers 26

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Categories for the Working mathematician by Mac Lane

Categories and Sheaves by Kashiwara and Schapira

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    $\begingroup$ pfft, only punks read Cat & Sheaves. These lecture notes by Schapira are better: people.math.jussieu.fr/~schapira/lectnotes/AlTo.pdf They use some nice specific examples and problems to develop category theory. With the added bonus of learning some nice alg top along with it. :D:D:D $\endgroup$
    – BBischof
    Jul 21, 2010 at 23:19
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    $\begingroup$ Uh-why do only "punks" read Cat & Sheaves? I happen to think it's a terrific and modern introduction to homological algebra and related fields. $\endgroup$ Sep 13, 2011 at 18:35
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    $\begingroup$ What do you mean by punks? What's wrong with the book. Your comment really made me not want to read the book (especially because it has 7 upvotes, although it doesn't explain anything. $\endgroup$
    – Asinomás
    Mar 10, 2013 at 17:26
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    $\begingroup$ I like Schapira/Kashiwara. But probably not for an introduction $\endgroup$ May 27, 2016 at 13:17
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    $\begingroup$ fwiw, "pfft, only punks ..." sounds like the beginning of a joking remark to me. $\endgroup$ Jan 21, 2018 at 16:34
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And when you get bored of reading, let the Catsters take over. (78 videos on Category theory!)

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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).

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    $\begingroup$ Lang's algebra also has significant typos that make it frustrating to read if you do not have enough mathematical maturity. $\endgroup$
    – user126
    Jul 24, 2010 at 11:07
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    $\begingroup$ re the typos: this is one instance where buying a used book that someone has marked up can be a really good idea. $\endgroup$
    – Isaac
    Jul 24, 2010 at 16:08
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    $\begingroup$ Bergman has a HUGE (ca 200 pages) companion to Lang's Algebra book math.berkeley.edu/~gbergman/.C.to.L . But even with this companion, I am not really sure if Lang is a good textbook to learn from. It does the right things, but it often does them in sloppy and/or subtly wrong way. I wouldn't say it does very much category theory either. $\endgroup$ Dec 31, 2012 at 21:51
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    $\begingroup$ I once heard from a rather well-known professor that he and another professor didn't quite get why a proof of a theorem was "obvious". It turned out that it wasn't so obvious, and they wrote an article about the proof. Quite fun. I think Lang has an absolutely fantastic idea of what topics to include, but the exposition is not ideal. $\endgroup$
    – Dedalus
    Jun 7, 2013 at 10:47
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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:

Conceptual Mathematics

An added benefit is that it is written by an expert!

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    $\begingroup$ Could someone edit this to give the link a name, so people immediately know what is being linked to? It's Lawvere and Schanuel's Conceptual Mathematics. Which is a really wonderful book for learning some category theory if you don't have the background to understand the heavy duty algebraic topology etc examples that enter into some discussions of CT at a very early stage. $\endgroup$
    – Seamus
    Aug 3, 2010 at 13:07
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    $\begingroup$ @Seamus Ok done! $\endgroup$
    – BBischof
    Aug 3, 2010 at 15:06
  • $\begingroup$ Are you implying that usually books are not written by experts? :-? $\endgroup$ Aug 22, 2010 at 13:24
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    $\begingroup$ @Andrea hehe no, but Lawvere is particularly great! $\endgroup$
    – BBischof
    Aug 22, 2010 at 18:37
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    $\begingroup$ One particular prolific author was rumoured to write (and publish) many of his books in order for him to learn their subjects :-) $\endgroup$ Aug 23, 2010 at 18:35
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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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    $\begingroup$ I discovered those notes recently and in my opinion they are great! $\endgroup$
    – Pandora
    Dec 6, 2011 at 21:32
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    $\begingroup$ He also has a book coming out: Basic Category Theory. $\endgroup$
    – J W
    Jul 21, 2014 at 19:45
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    $\begingroup$ @JW: It's published. See maths.ed.ac.uk/~tl/bct. And it will be available free (online) in January, 2016. $\endgroup$
    – eltonjohn
    Aug 9, 2014 at 1:46
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    $\begingroup$ @eltonjohn: Thank you; that's useful to know. I note that it's in arrangement with the publisher and that the book will be both freely downloadable and freely editable. $\endgroup$
    – J W
    Aug 9, 2014 at 1:58
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Emily Riehl's recently published book Category theory in context is a fantastic introductory text for those interested in seeing lots of examples of where category theory arises in various mathematical disciplines. Understand the examples from other branches of mathematics requires some mathematical maturity (e.g., a bit of exposure to algebra and topology), but these examples aren't strictly necessary to understand the category theory; even the less advanced reader should have no problem understanding the categorical content of the text. It stresses the importance of representability, an understanding of which is crucial if the reader wants to go on to learn about $ 2 $-categories in the future. It's elegantly written, well-motivated, uses very clear notation, and overall is refreshingly clear in its exposition.

The current version of the text is available at http://www.math.jhu.edu/~eriehl/context.pdf and errata in the published version are being updated. The text is new, so it's not as well-known as other texts, but it's so well-written that it seems very likely that it will soon become a mainstay in the world of category theory texts.

9 July 2017 Edit. Updated the link to the text.

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The nLab is a great resource for category theory.

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    $\begingroup$ I take the OP to be asking about introdutions to category theory. nLab is not a good introductory text... $\endgroup$
    – Seamus
    Aug 4, 2010 at 7:30
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    $\begingroup$ @Seamus You're right. But it's a good general reference, in the same way that wikipedia is a good general reference but shouldn't be used as a text. $\endgroup$ May 4, 2013 at 16:03
  • $\begingroup$ nlab ist one of the worst resource that any begginner can find!! $\endgroup$ Mar 23 at 22:14
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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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    $\begingroup$ Peter Smith has criticised Awodey's book for being pitched too high to be an introduction to categories logicmatters.net/2008/06/awodeys-category-theory-ch-1 $\endgroup$
    – Seamus
    Aug 5, 2010 at 10:37
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    $\begingroup$ I will admit to having mainly used Awodey as source material while putting together my own introduction to categories lecture course material. As such, it was highly pleasant - but I am not a good example of what a newcomer'll need... haskell.org/haskellwiki/User:Michiexile/MATH198 $\endgroup$ Aug 5, 2010 at 11:44
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    $\begingroup$ +1 for Awodey,which is the only book I would consider to teach category theory to undergraduates. $\endgroup$ Sep 13, 2011 at 18:39
  • $\begingroup$ I wanted to add that his book is now available in paperback at half the price of the hardcover edition: Amazon $\endgroup$
    – user4536
    May 30, 2012 at 18:06
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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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    $\begingroup$ I have gone through this book very carefully. It is indeed an excellent algebra book, but the last chapter is not very good. $\endgroup$
    – Hui Yu
    Apr 30, 2013 at 6:49
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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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    $\begingroup$ Goldblatt's book (which is very beautifully written, by the way) is available online in its entirety here. $\endgroup$ Aug 23, 2010 at 6:51
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    $\begingroup$ I struggled for years to understand category theory until I met Goldblatt's book; then the struggle was over. $\endgroup$
    – MJD
    Mar 11, 2014 at 0:23
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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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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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    $\begingroup$ Eilenberg and Mac Lane's original paper: General theory of natural equivalences says that they defined "category" to define "functor", and "functor" to define "natural transformation". But I get the impression that the category theorists of today don't take that remark all that seriously. $\endgroup$
    – Uday Reddy
    Dec 7, 2013 at 18:44
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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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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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Last year the book Basic Category Theory by Tom Leinster was published by Cambridge University Press. I think it can serve very well as an introduction to Category Theory. It covers much less than Mac Lane's Categories for a working mathematician, but motivates concepts better.

And it's also available on Arxiv.

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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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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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There's also this Category Theory for Programmers by Bartosz Milewski with the companion video lectures

https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ https://www.youtube.com/playlist?list=PLbgaMIhjbmElia1eCEZNvsVscFef9m0dm

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  • $\begingroup$ really engaging video lectures with a computer programming (Haskel) bend; I'm not a mathematician but I really enjoyed them enough to look for textbooks on category theory $\endgroup$ Sep 23, 2018 at 10:03
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First Chapter of Jacobson's Basic Algebra -II.

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I'm surprised that this hasn't been mentioned already.

"Category Theory: An Introduction" by Herrlich and Strecker. You can find this book in either the Allyn and Bacon Series in Advanced Mathematics or Sigma Series in Pure Mathematics.

Herrlich and Strecker co-authored another book called "Abstract and Concrete Categories: The Joy of Cats" which is not nearly as good as the former book.

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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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"Basic category theory"is a script by Jaap van Oosten from Utrecht university (u can find more scripts on topos theory and intuitionism there). Advanced is Introduction in Higher order categorical logic by Lambek & Scott. The 3 vols. from Borceux aswell as Johnstone: Sketches of an elephant, 1-2 are very readable reference for looking up proofs and technical details. Toposes and local set theories by Bell is availlable in Dover prints.

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"Algebra:Rings Modules and Categories" by Carl Faith has alot about category theory,which dos'nt need any topology to understand,but is mixed with all the stuff about algebra,which is also writen in a catigorcal way.

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There is also this Introduction to Applied Category Theory Course offered by MIT. The lectures conducted by David Spivak and Brendan Fong are recorded and posted here. I personally find this a much better introductory material than the other one I posted, despite being a software engineer.

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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.

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