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

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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 but also outstanding.

A dozen years ago, I wrote some introductory notes Beginning Category Theory. Versions of those notes have been downloaded rather startlingly often (7500 times last year) — which has been a bit embarrassing, as I knew all along the notes were in a pretty rackety state! So this year, I am slowly revising the notes, under a new title, and they are now in two parts. Currently (Apr 29) all of Part I has been revised, though it is still work in progress:  Part II is quite unrevised.

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

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

• Introductory readings for philosophers

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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 Gra