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:

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:

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.

Some handbook essays on categorial logic in particular

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.] 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). 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.] 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

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