Category theory: online lecture notes, etc.

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

Gentle introductions?

As you will see, there are nearly fifty books and sets of notes listed below. Where to start? That must depend on your mathematical background and one size won’t fit all. Here are three highlights:

• Over the years, many have found the very accessible early chapters — say the first three — of Goldblatt’s Topoi a particularly useful first introduction.
• Then, a step up, Leinster’s short Basic Category Theory is basic, excellent, and much recommended.
• Riehl’s Category Theory in Context is more challenging, in part because it assumes rather more mathematical background, but is also outstanding.

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

All of Part I has now been revised, and there is even a paperback version for those like me who much prefer to work from a paper copy (this is a low cost but very acceptable quality Amazon print-on-demand book ISBN 1916906370). Though please take this to be a beta version of work in progress (comments/corrections still particularly welcome).  Part II is being slowly revised: as of Sept 17, the first eight chapters (73pp.) have been updated.

• Category Theory I: Notes towards a gentle introduction.
• Category Theory II: More notes towards a gentle introduction.

It being a law of nature that books have typos and thinkos, we need of course

• Category Theory I: corrections to the printed versions.

For those wanting even more accessible routes into category theory and/or links to philosophical discussions, here is a page of

• Introductory readings for philosophers

Lecture notes on Category Theory

Notes of P.T. Johnstone’s Lectures for the 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 of notes/expositions of various styles that I happen to have come across and that might in varying degrees be useful (I’ve only listed the more substantial lecture notes available, which are sufficiently discursive to stand alone without the lectures they were intended to accompany, and which don’t tangle too much/too quickly with more advanced topics).  In alphabetical order by author:

1. Steve Awodey and Andrej Bauer, Category Theory (pp. 44, 2022, primer on category theory in draft textbook-in-progress on categorical logic).
2. John Baez, Category Theory Course (pp. 59, 2019: past course page here).
3. 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).
4. Mario Cáccamo and Glynn Winskel, Lecture Notes on Category Theory (postscript file, pp. 74, 2005; pdf version: notes for a course inspired by Martin Hyland’s Part III Mathematics course).
5. Robin Cockett, Category Theory for Computer Science (pp. 107, 2022). And by the same author, a significantly different set of notes Categories and Computability (pp. 100, 2014).
6. Daniel Epelbaum and Ashwin Trisa, Introductory Category Theory Notes (pp. 56, 2020).
7. Rafael Villarroel Flores, Notes on Categories (pp. 77, 2004).
8. Julia Goedecke, Category Theory (pp. 63, lecture notes for her Cambridge Part III Maths course, 2013: related materials on her website here).
9. Chris Hillman, A Categorical Primer (pp. 62, 1997).
10. Randal Holmes, Category Theory (pp. 99, 2019).
11. Robert Knighten, Notes on Category Theory (about pp. 160 of unfinished notes, followed by appendices including useful information about many books: 2011).
12. Ryszard Kostecki, An Introduction to Topos Theory (pp. 93, with first half on categories more generally, 2011).
13. Valdis Laan, Introduction to Category Theory (pp. 52, 2003).
14. Günter Landsmann, Basic Theory of Categories (pp. 65, 2012).
15. Bartosz Milewski, Category Theory for Programmers (series of long blogposts, available in book format, linked below: also see also his videos, also linked below).
16. Ed Morehouse,  Basic Category Theory (pp. 77, 2016).
17. Jaap van Oosten, Basic Category Theory and Topos Theory (pp. 123, Utrecht 2016).
18. Prakash Panangaden, Brief notes on category theory (pp. 36, 2012).
19. Paulo Perrone, Notes on Category Theory (pp. 181, 2021)
20. Benjamin Pierce, A Taste of Category Theory for Computer Scientists (pp. 75, 1988: earlier version of this book).
21. Uday S. Reddy, Categories and Functors (pp. 47, Lecture Notes for Midlands Graduate School, 2012).
22. Pavel Safronov, Category Theory (pp. 56 — Oxford lecture notes, 2015).
23. Pierre Schapira, Algebra and Topology (pp. 157, 2008 — largely category theory).
24. Peter Smith, Category Theory I: Notes towards a gentle introduction (pp. 226, 2023) and Category Theory II: More notes towards a gentle introduction (pp. 153, mostly still 2015).
25. William R. Schmitt, A Concrete Introduction to Categories (pp. 60).
26. Thomas Streicher, Introduction to Category Theory and Categorial Logic (pp. 116, 2003/4).
27. Daniele Turi, Category Theory Lecture Notes (pp. 58, Edinburgh, 2001).
28. Ravi Vakil, Some category theory (pp. 57: from Ch. 1 of The Rising Sea: Foundations Of Algebraic Geometry Notes. Latest version available here, 2022).
29.  Mariusz Wodzicki, Notes on Category Theorem (pp. 205, 2016). 

Books and Articles on Category Theory

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 …

1. Samson Abramsky and Nikos Tzevelekos, Introduction to Categories and Categorical Logic (pp. 101: 2011 arXiv version of their chapter in Bob Coecke, ed. New Structures for Physics, Springer 2010).
2. Jiri Adamek, Horst Herrlich and George Strecker, Abstract and Concrete Categories: The Joy of Cats (originally published John Wiley and Sons, 1990: recommended).
3. Andrea Asperti and Giuseppe Longo. Categories, Types and Structures: Category Theory for the working computer scientist. MIT Press, 1991.
4. Michael Barr and Charles Wells, Toposes, Triples and Theories (originally published Springer, 1985).
5. Michael Barr and Charles Wells, Category Theory for Computing Science (originally published Prentice Hall, 1995: particularly clear and useful).
6. George M. Bergman, An Invitation to General Algebra and Universal Constructions (online version of book published by Springer, 2nd end 2015: this is about recurrent ideas in algebra and the way category theory unifies them).
7. 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).
8. Brendan Fong and David Spivak, Seven Sketches in Compositionality:An Invitation to Applied Category Theory (pp. 341, 2018: published  as CUP book in 2019).
9. Peter Freyd, Abelian Categories (originally published Harper and Row, 1964: not exactly elementary — but a classic).
10. Robert Goldblatt, Topoi (originally published North-Holland, 1979/1984: an expository classic – also available as cheap Dover book).
11. 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.)
12. Tom Leinster, Basic Category Theory (originally published CUP, 2014: warmly recommended).).
13. Bartosz Milewski, Category Theory for Programmers (book version of his blog posts, 2018)
14. Bartosz Milewski, The Dao of Functional Programming (draft book, largely about categories, 2022)
15. Birgit Richter, From Categories to Homotopy Theory (pp. 384 in the PDF version, 2019: published by CUP in 2020 —  this gets advanced, but Part I ‘Category Theory’ is pretty accessible).
16. Emily Riehl, Category Theory in Context (pp. 240: 2016  Dover book: warmly recommended).
17. Andrei Rodin, Axiomatic Method and Category Theory (2012 arXiv version of book published by Springer 2104: not an exposition of category theory but discusses something of the history and philosophy behind its development).
18. 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. Jean-Pierre Marquis & Gonzalo E. Reyes, The History Of Categorical Logic 1963-1977 (in Dov Gabbay et al., eds, Handbook of the History of Logic Vol 6: Sets and extensions in the twentieth century, North-Holland 2012). [Over-detailed and consequently rather impenetrable: probably only useful if you already know a lot.]
4. 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

• Reprints in Theory and Applications of Categories

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).
5. Ed Morehouse: four basic level lectures to accompany his 2016 notes listed above.

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 25 January 2023