Category Theory


The links below are to various freely (and legitimately!) available online resources for those interested in category theory at an elementary/intermediate level. See this supplementary page, introductory readings for philosophers, for reading suggestions for those looking for the most accessible route into category theory.


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 February 8, 2016 is x + 273 pp. long: the last three chapters need considerable polishing, having just been pasted in from some earlier, and now mostly superseded, Notes. (Future plans involve some additional preliminary chapters, better chapters on adjunctions, and then chapters on monads, sets-done-categorically, and on categorical logic.)

Having found a few gruesome errors in the earlier notes as I work on the Gentle Introduction, everything of course still comes with the warning caveat lector. However, although I am 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 links mentioned below. If you want something just a step or two up from my notes but still tolerably gentle, let me highlight two books. I’d have selected these two anyway on their merits, but an additional plus-point is that they are now both freely available. These are Steve Awodey’s Category Theory and Tom Leinster’s Basic Category Theory.


Lecture notes, etc.

Past Cambridge notes of lectures for the Part III course:

  1. P. T. Johnstone’s notes as transcribed by Bruce Fontaine (pp. 52: version of Nov. 2011. Later pages do contain, however, a number of serious typos.)
  2. Eugenia Cheng’s notes (pp. 71: 2002)
  3. Julia Goedecke’s notes (pp. 63: 2013/14). There are other materials available on JG’s teaching page (including extra examples, and video solutions to some exercises).

Other online notes An idiosyncratic list of notes/expositions of various lengths and styles that I happen to have come across that I have found in varying degrees useful. In alphabetical order:

  1. 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 ).
  2. Robin Cockett, Category Theory for Computer Science (pp. 105, 2009). And by the same author, a significantly different set of notes Categories and Computability (pp. 100, 2014).
  3. Rafael Villarroel Flores, Notes on Categories (pp. 77, 2004).
  4. Chris Hillman, A Categorical Primer (pp. 62, 1997).
  5. Robert Knighten, Notes on Category Theory (about pp. 160 of unfinished notes, followed by appendices including useful information about many books: 2011).
  6. Valdis Laan, Introduction to Category Theory (pp. 52, 2003).
  7. Bartosz Milewski, Category Theory for Programmers (ongoing series of long blogposts, and see also his videos, linked below)
  8. Ed Morehouse, Introduction to Categorical Semantics for Proof Theory (pp. 91, 2015: notes for a course, mainly an introduction to category theory).
  9. Jaap van Oosten, Basic Category Theory (pp. 86, Utrecht 2002).
  10. Benjamin Pierce, A Taste of Category Theory for Computer Scientists (pp. 75, 1988: earlier version of this book).
  11. Uday S. Reddy, Categories and Functors (pp. 47, Lecture Notes for Midlands Graduate School, 2012).
  12. Emily Riehl, Category Theory in Context (pp. 240: 2016 version of notes for her lecture course at Johns Hopkins, now published as a Dover book).
  13. 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).
  14. Pierre Schapira, Categories and Homological Algebra (pp. 120, 2002-15: presupposes some background in algebra etc., but fairly introductory on categories).
  15. William R. Schmitt, A Concrete Introduction to Categories (pp. 60).
  16. Thomas Streicher, Introduction to Category Theory and Categorial Logic (pp. 116, 2003/4).
  17. Daniele Turi, Category Theory Lecture Notes (pp. 58, Edinburgh, 2001).
  18. Ravi Vakil, Some category theory’ (pp. 48: Ch. 2 of The Rising Sea: Foundations Of Algebraic Geometry Notes. Latest version available here, 2015).
  19. The Stacks Project at Columbia, Categories (pp. 82: chapter of open-source textbook on algebraic stacks and the algebraic geometry needed to define them , 2015).

Books and articles

Some books and other longer published works on category theory These are e-copies of paper publications, at introductory/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 series can be downloaded 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. Robert Goldblatt, Topoi (originally published North-Holland, 1979/1984: an expository classic – also available as cheap Dover book).
  9. Tom Leinster, Basic Category Theory (originally published CUP, 2014).
  10. Bodo Pareigis, Categories and Functors (originally published Academic Press, 1970).
  11. 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).
  12. D.E. Rydeheard and R.M. Burstall, Computational Category Theory. (postscript file: originally published Prentice-Hall, 1988).
  13. Harold Simmons, An Introduction to Category Theory (late version of book published by CUP, 2011: includes answers to most of the exercises).
  14. 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


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. Bartosz Milewski has a series of videos (again with a compsci flavour).

I don’t at all plan to be completist and try to list everything relevant of roughly the right level that there is officially available online (I have omitted links to some short sets of notes, and we must pass over in silence copyright-infringing repositories). But do let me know about omissions of anything good, substantial, but not too wildly advanced, that I haven’t mentioned. (Thanks to those who have already suggested additions.)


12 Responses to Category Theory

  1. Mozibur Ullah says:

    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. People might also be interested in other material available on my teaching page from when I lectured the course in 2013. Such as lots of extra examples, and some video solutions to some easy exercises. https://www.dpmms.cam.ac.uk/~jg352/teaching.html

  3. Would you please consider uploading versioned copies with permalinks? Maybe that’s overkill, but I just linked to theorem 68 of the current version of your notes — in this post:
    https://www.reddit.com/r/ocaml/comments/3ifwe9/what_are_ocamlers_critiques_of_haskell/czsri44 (but I won’t try to explain what divergence means, it makes no sense unless you care about practical programming languages, as I also sometimes do).

  4. Dion says:

    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”?

  5. Fybove says:

    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.

  6. Souvik Das says:

    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?

    • Peter Smith says:

      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.

  7. Dick swenson says:

    MAcLanes book is available on line. Search for it. I found it @
    http://Www.maths.ed.ac.uk/~aar/papers/MAcLanes.pdf
    Forget the capitals.

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