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 more challenging, in part because it assumes rather more mathematical background, but is also outstanding.

Another gentle introduction

My own notes also aim to provide a very accessible entry-level introduction.

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

I have gone for a fairly conventional mode of presentation but at a very gentle (painfully slow?) pace. This makes for a long book — but I make no apology for that: faster-track alternatives are available in you want them!

For encouragement to dip into the PDF to see whether the book might work for you,

And for those like me who prefer to work from a printed text, an inexpensive paperback version is now available, Amazon print-on-demand only (sorry!) to minimize cost: ISBN 1068346701. 

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 useful at an introductory level. (But your mileage of course may vary, which is why an updated version of the original longer list is available here.)

Notes of P.T. Johnstone’s Lectures for his famed Cambridge Part III course (the topics of later lectures differ from year to year):

  1. Notes by David Mehrle (pp. 80; lectures given 2015, notes revised 2016).
  2. Notes by Qiangru Kuang (pp. 68, 2018)
  3. Notes by Sky Wilshaw (pp. 69, 2024)

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. Pavel Safronov, Category Theory (pp. 56 — Oxford lecture notes, 2015).
  9. Thomas Streicher, Introduction to Category Theory and Categorical Logic (pp. 116, 2003/4).

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. NB: I’ll keep this list respectable by passing over in silence those copyright-infringing pdf repositories that, of course, none of us use …

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/accessible. (Again, for a longer list, see here).

  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)

Page of links to reprints, including some classic articles

Wiki

I can’t finish listing text resources without mentioning the massively useful wiki,

A more comprehensive list 

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).
  5. Not introductory, but there over eighty videos of talks at various levels of accessibility, and ranging widely, from the New York City Category Theory Seminar.

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 28 June 2025. I have now deleted some older comments, that are no longer so relevant: but thanks for all those past suggestions!

11 comments:

  2. Rowsety Moid

    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.

    Reply
      1. Rowsety Moid

        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.

        Reply
  3. Richard

    Hello — how “locked” are the first “n” chapters of your “Introducing Category Theory” text, which looks like it will be published this month or next?

    Reply
    1. Peter Smith

      (1) Well, nothing is irrevocably “locked”, though I do hope that the first forty or so chapters are in a reasonably good state and I have no particular plans to change them (though sometimes I add stuff or rephrase stuff to fit better with the last few chapters which I’m still juggling with). So all comments are still very gratefully received.

      (2) As to publication … hey, ho, the time frame seems to be slipping again (various distractions here, good and not so good!). But the plan and fervent hope is STILL to have a complete late draft online by the end of this month. And then it is a question how much time and effort I put into final read-throughs for sense and for typos before making a printed paperback available.

      Reply
  4. Sara H

    Hello! I am still reading this, but I just wanted to say, this book is a fantastic introduction! I recently finished my bachelor’s in computer science, and I wanted to dip my toes into some branches of mathematics that I didn’t get to explore as an undergrad. I tried a few different books and video series, but none of them really went into the depth that I was looking for. This, however, is incredibly thorough! You do a fantastic job of explaining not just the core concepts of category theory, but how to arrive at those core concepts from common mathematical structures.
    The asides on mathematical foundations are also very welcome, for a lover of formal logic.
    As a show of my gratitude, (with the energy of a cat bringing you a dead bird) here’s a typo I found: “Theorem 48 – There are categories with initial objects where the products 0 × X and X × 0 exists but are not generally isomorphic to 0.” pg. 108. Should say “exist”, not exists?

    Reply
  5. Sayantan Roy

    I think among the introductory books on Category Theory, Eugenia Cheng’s recent book “The Joy of Abstraction” would be an excellent recommendation.

    Reply
    1. Peter Smith

      Perhaps. But not it’s really appropriate to mention Eugenia Cheng’s book in a page of links “to a selection of freely (and legitimately!) available online resources”.

      Some may like the book: but others may find the non-category-theoretic remarks irritating and distracting, and I’m not sure the book has enough serious content to be recommended to those who want an actual introductory book on category theory.

      Reply
      1. Sayantan Roy

        I agree that sometimes the non-category-theoretic remarks could be too distracting for the reader. In any case, let me say this, even though you have probably heard this several times: your book is excellent! The level of detail with which you explain everything is truly amazing.

        Reply
  6. Benson Bear

    Dude, I like your blog, and all your great expository work, which is provided free of charge to boot, but this is the INTERNET you know, and when I see a link on your home page (previously unnoticed) that says “CATS”, I click on it expecting to find a whole lot of cute videos or at least jpegs, and instead I see article on category theory?

    C’mon be better!

    Reply

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