Category theory: online lecture notes, etc.

The links below are to a selection of freely (and legitimately!) available online resources for those interested in category theory at an elementary/intermediate level.


Three excellent introductions

Where to start? That must depend on your mathematical background, and one size won’t fit all. Here are three highlights among freely available books, which would I think be widely agreed to be excellent in their different ways:

  • Over the years, many have found the very accessible early chapters — say the first three, 74pp. — of Robert Goldblatt’s Topoi (originally published 1979, now a Dover pbk) a particularly helpful entry-point.
  • Then, a step up, Tom Leinster’s short Basic Category Theory (CUP 2014) is indeed basic and is rightly very well-regarded.
  • Emily Riehl’s Category Theory in Context (Dover 2017) is rather more challenging, in part because it assumes rather more mathematical background, but is also outstanding.

Another gentle introduction ..?

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A dozen years ago, I wrote some introductory notes Beginning Category Theory. Versions of those notes were downloaded rather startlingly often — which was really a bit embarrassing, as I knew all along the notes were in a pretty rackety state! So I have been revising these notes, under a new title.

The result is a hopefully accessible introduction in three parts. Part I says something about what can be found inside individual categories (products, quotients, limits more generally, exponentials and the like). Part II says a very little about those categories which are elementary toposes. Part III, which can be read independently of Part II, introduces the distinctive categorial  ideas of functors, natural transformations, the Yoneda Lemma, and adjunctions.

I have gone for a fairly conventional mode of presentation but at a pretty gentle pace (which makes for quite a long book — but I make no apology for that: faster-track alternatives are available in you want them!). The result is aimed at those who want an entry-level warm-up before taking on an industrial-strength graduate-level course, or perhaps just want to get an idea of that the categorial fuss is about. You can freely download the current version, to see whether the level and expositional style is to your taste.

A print-on-demand version will be inexpensively available (Amazon only, to minimize cost) in due course. Comments and corrections are still hugely welcome.

A print-on-demand version of Part I (and the beginning of Part III) is already available under the title Category Theory I: A Gentle Prologue (again Amazon-only, ISBN 1916906389).


Selected lecture notes

I used to list here over 30 sets of available lecture notes, without much comment.  That was probably rather unhelpful. So let me now give a rather shorter selection of lecture notes that do seem to me likely to be particularly helpful for one reason or another. (But your mileage of course may vary, which is why the original longer list is still available here.)

Notes of P.T. Johnstone’s Lectures for his famed Cambridge Part III course:

  1. Notes by Bruce Fontaine (pp. 52: version of Nov. 2011).
  2. Notes by David Mehrle (pp. 80; lectures given 2015, notes revised 2016).
  3. Notes by Qiangru Kuang (pp. 68, 2018)

Other online notes An idiosyncratic list, in alphabetical order by lecturer:

  1. Michael Barr and Charles Wells, Category Theory Lecture Notes for ESSLLI (pp. 128, 1999: a cut down version of their Category Theory for Computing Science which is also available online: see below).
  2. Daniel Epelbaum and Ashwin Trisal, Introductory Category Theory Notes (pp. 56, 2020).
  3. Julia Goedecke, Category Theory (pp. 63, lecture notes for her Cambridge Part III Maths course, 2013: related materials on her website here).
  4. Valdis Laan, Introduction to Category Theory (pp. 52, 2003).
  5. Bartosz Milewski, Category Theory for Programmers (series of long blogposts, available in book format, linked below: also see also his videos, also linked below).
  6. Jaap van Oosten, Basic Category Theory and Topos Theory (pp. 123, Utrecht 2016).
  7. Paulo Perrone, Notes on Category Theory (pp. 181, 2021)
  8. Uday S. Reddy, Categories and Functors (pp. 47, Lecture Notes for Midlands Graduate School, 2012).
  9. Pavel Safronov, Category Theory (pp. 56 — Oxford lecture notes, 2015).
  10. William R. Schmitt, A Concrete Introduction to Categories (pp. 60).
  11. Thomas Streicher, Introduction to Category Theory and Categorial Logic (pp. 116, 2003/4).
  12. Daniele Turi, Category Theory Lecture Notes (pp. 58, Edinburgh, 2001).

Selected books and articles, etc.

Some books and other longer published works on category theory These are e-copies of paper publications, at introductory or intermediate level, which happen also to be officially available to download. I’ll keep this list respectable by passing over in silence those copyright-infringing pdf repositories that, of course, none of us use …  For a somewhat longer list, see here.

In addition, then, to the books already mentioned at the top of this page by Goldblatt, Leinster, and Riehl, you might find some of these particularly helpful.

  1. Jiri Adamek, Horst Herrlich and George Strecker, Abstract and Concrete Categories: The Joy of Cats (originally published John Wiley and Sons, 1990).
  2. Andrea Asperti and Giuseppe Longo. Categories, Types and Structures: Category Theory for the working computer scientist. MIT Press, 1991.
  3. Michael Barr and Charles Wells, Category Theory for Computing Science (originally published Prentice Hall, 1995: particularly clear and useful).
  4. Tai-Danae Bradly, Tyler Bryson and John Terilla, Topology: A Categorial Approach (online version of a book published by MIT Press, 2020: a short, elementary, book — the categorial approach is illuminating of both category theory and topology).
  5. Horst Herrlich and George Strecker, Category Theory (originally published Allyn and Bacon, 1973; third edition 2007: more introductory than their later book with Adamek listed above.)
  6. Bartosz Milewski, Category Theory for Programmers (book version of his blog posts, 2018)
  7. David I. Spivak, Category Theory for the Sciences (online version of book published by MIT Press, 2014)

Some handbook essays on categorial logic in particular

  1. Samson Abramsky and Nikos Tzevelekos, Introduction to Categories and Categorical Logic (as above). [Clear intro. to categories: but when it turns to logic rather rushed and oddly focused.]
  2. John L. Bell, The Development of Categorical Logic (more advanced: published in D.M. Gabbay & Franz Guenthner, eds, Handbook of Philosophical Logic, 2nd edition, Volume 12, Springer 2005).
  3. Andrew Pitts, Categorical Logic (in S. Abramsky, D. Gabbay, T. Maibaum, eds, Handbook of Logic in Computer Science Vol 5, OUP 2000).

Page of links to reprints, including some classic articles

Web resource

I can’t finish listing text resources without mentioning the massively useful wiki, the nLab. See in particular category theory in nLab.


Videos

  1. There is a fun and instructive series at an introductory level by The Catsters (Eugenia Cheng and Simon Willerton).
  2. Steve Awodey has an excellent series, aimed a little higher (with a compsci flavour), going a little further.
  3. B. Fong and D. Spivak: elementary lectures on applied category theory.
  4. Bartosz Milewski has a series of videos (again with a compsci flavour).

I have only listed here substantial enough material of roughly the right level that is, to repeat, officially available online. I don’t plan to be completist — but do please let me know of errors and omissions and newly available lecture notes, etc.

Links last updated 3 February 2024

21 thoughts on “Category theory: online lecture notes, etc.”

  1. How about Lawveres and Schanuels book – Sets for mathematicians? and if I’m not mistaken Maclanes book Categories for the working mathematician is not in your list!

  2. Hi, nice blog and nice set of notes. Would you be so kind as to share the latex template you’re using to write “Category Theory: A gentle introduction”?

  3. I just wanted to thank you Dr Smith for your notes on category theory, they get right the always difficult balance between depth and readibility. Without these it would have been almost impossible for me to give a talk at our undergraduate seminar on dual spaces and dual categories, being specially useful the discussion in the section on naturally isomorphic functors.

  4. Will the final version of your notes on Category Theory still be available on this page? I mean, do you plan to remove the link when (if at all) these notes are transformed into a book like your An Introduction to Formal Logic?

    1. Well, it’s a hopeful thought that there will be a final version! But if it does come to the point of official publication, I guess it would depend on arrangements with the publishers. (CUP is increasing allowing authors to leave late versions online, or to make their books available online after a certain interval.) But all that’s in the future … at the moment, things seem to be going a lot more slowly than I would like.

  5. I’ve spotted a potentially interesting book, An Invitation to General Algebra and Universal Constructions, by George M. Bergman, that turns out to be a category theory text. (That link points to a page provided by the author that provides links to PDF versions of the book.)

    From the back-cover blurb:

    Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader’s grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions.

      1. I’d looked at the longer list but somehow failed to notice it. (!)

        Do you remember why you relegated the book? It looked it took an approach you might like.

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