Category Theory: Lecture Notes and Online Books

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

There is supplementary page, introductory readings for philosophers, for reading suggestions for those looking for the most accessible routes into category theory and/or links to philosophical discussions.

A gentle introduction?

My Category Theory: A Gentle Introduction is intended to be relatively accessible; in particular, it presupposes rather less mathematical background than some texts on categories. The version of January 29, 2018 is x + 291 pp. long, and is very much work-in-slow-progress, at an uneven level. I then had to leave it on the back burner while finishing of IFL2 for the press: but I now hope to return to it.

The current version incorporates a raft of corrections of the previous version, but everything of course still comes with the warning caveat lector. However, although I started writing really as an exercise in getting myself a bit clearer about some basic category theory, I hope that others will find something of interest and use here. Obviously I’d very much welcome comments and corrections.

There are a lot of possible follow-up materials listed here. But if you want something just a step or two up from my notes but still tolerably gentle, let me highlight two books listed below. One is Steve Awodey’s Category Theory (chapters available on his website here). The other is Tom Leinster’s Basic Category Theory.

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 that might in varying degrees be useful (I’ve only listed the more substantial lecture notes available).  In alphabetical order:

  1. John Baez, Category Theory Course (pp. 59, 2019: past course page here).
  2. 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.)
  3. Mario Cáccamo and Glynn Winskel, Lecture Notes on Category Theory (postscript file, pp. 74, 2005: notes for a course inspired by Martin Hyland’s Part III Mathematics course ).
  4. Robin Cockett, Category Theory for Computer Science (pp. 107, 2016). And by the same author, a significantly different set of notes Categories and Computability (pp. 100, 2014).
  5. Rafael Villarroel Flores, Notes on Categories (pp. 77, 2004).
  6. Maarten M. Fokkinga, A Gentle Introduction to Category Theory: The Calculational Approach (pp. 78, 1994).
  7. Julia Goedecke, Category Theory (pp. 63, lecture notes for her Cambridge Part III Maths course, 2013: related materials on her website here).
  8. Randal Holmes, Category Theory (pp. 99, 2019).
  9. Chris Hillman, A Categorical Primer (pp. 62, 1997).
  10. Robert Knighten, Notes on Category Theory (about pp. 160 of unfinished notes, followed by appendices including useful information about many books: 2011).
  11. Valdis Laan, Introduction to Category Theory (pp. 52, 2003).
  12. Bartosz Milewski, Category Theory for Programmers (series of long blogposts, available in book format, linked below: also see also his videos, also linked below).
  13. Ed Morehouse,  Basic Category Theory (pp. 77, 2016).
  14. Jaap van Oosten, Basic Category Theory and Topos Theory (pp. 123, Utrecht 2016).
  15. Paulo Perrone, Notes on Category Theory (pp. 132, 2019)
  16. Benjamin Pierce, A Taste of Category Theory for Computer Scientists (pp. 75, 1988: earlier version of this book).
  17. Prakash Panangaden, Brief notes on category theory (pp. 36, 2012).
  18. Uday S. Reddy, Categories and Functors (pp. 47, Lecture Notes for Midlands Graduate School, 2012).
  19. Andrea Schalk and Harold Simmons, An Introduction to Category Theory, in four easy movements (pp. 126, plus solutions of exercises: 2005 notes for an MSc course in math. logic).
  20. Pavel Safronov, Category Theory (pp. 56 — Oxford lecture notes, 2015).
  21. Pierre Schapira, Algebra and Topology (pp. 157, 2008 — largely category theory).
  22. Pierre Schapira, Categories and Homological Algebra (pp. 120, 2002-15: presupposes some background in algebra etc., but fairly introductory on categories).
  23. William R. Schmitt, A Concrete Introduction to Categories (pp. 60).
  24. Greg Stevenson, Rudimentary Category Theory Notes (pp. 28).
  25. Thomas Streicher, Introduction to Category Theory and Categorial Logic (pp. 116, 2003/4).
  26. Daniele Turi, Category Theory Lecture Notes (pp. 58, Edinburgh, 2001).
  27. Ravi Vakil, Some category theory (pp. 57: from Ch. 1 of The Rising Sea: Foundations Of Algebraic Geometry Notes. Latest version available here, 2017).

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.

  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. Steve Awodey, Category Theory (versions of the chapters from the 2010 second edition of this useful book in the Oxford Logic Guides available here).
  5. Michael Barr and Charles Wells, Toposes, Triples and Theories (originally published Springer, 1985).
  6. Michael Barr and Charles Wells, Category Theory for Computing Science (originally published Prentice Hall, 1995: particularly clear and useful).
  7. 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).
  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).
  13. Bartosz Milewski, Category Theory for Programmers (book version of his blog posts, 2018)
  14. Bodo Pareigis, Categories and Functors (originally published Academic Press, 1970).
  15. Birgit Richter, From Categories to Homotopy Theory (pp. 327, 2019: to be published by CUP in 2020: stretching a point to include this as it gets advanced, but it starts off relatively accessibly).
  16. Emily Riehl, Category Theory in Context (pp. 240: 2016 version of  her lecture course at Johns Hopkins, now published as a Dover book).
  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. D.E. Rydeheard and R.M. Burstall, Computational Category Theory. (postscript file: originally published Prentice-Hall, 1988).
  19. Harold Simmons, An Introduction to Category Theory (late version of book published by CUP, 2011: includes answers to most of the exercises).
  20. 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

Web resource

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


  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 material of roughly the right level that is, to repeat, officially available online (I have omitted links to some short sets of notes, and we must here pass over in silence copyright-infringing repositories). 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 checked, deleted, revised, and added 26 November 2019

14 thoughts on “Category Theory: Lecture Notes and Online Books”

  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.

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