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* was a substantial set of notes originally from 2015 (revised in 2018). As the title suggests, those notes were intended to be very accessible; they went slowly, and presuppose rather less mathematical background than some texts on categories. I started writing them as an exercise in getting myself a bit clearer about some basic category theory, and I put them online because I thought that others might perhaps find them useful. Which seems to have happened! Those notes have been downloaded rather startlingly often — e.g. just under a thousand times during January 2022. Which was really getting a bit embarrassing, as the notes were/are in a pretty rackety, half-baked, unfinished form. So, with some other projects out of the way, it is time to start work revising/improving them. So here then is:

As of 27 April 2022, the first thirteen chapters are in a revised form, and then these are followed by the rest of the old chapters (with a warning about their unrevised status headlining each page!). Even the early chapters are still *very* much work in progress; but I’m incorporating newly revised chapters once I am sure that they are at least *better* than the corresponding chapters in the earlier notes (and not waiting for perfection!).

In a bit more detail, the plan is, first, to go through the old *Gentle Intro*, correcting all the typos and thinkos I’ve been told about, improving the presentation wherever I can, to make a better version of the existing material. Second, I’ll get back to doing quite a lot more reading and rereading in category theory, so I can round out the existing material e.g. with more examples, more applications, and additional related ideas at the same level. Then third, I’ll have to decide which further topics (if any) I should add at the end to make a more satisfying book (though given the current length, already 300 pages, I guess I wouldn’t want the book to extend its reach *very* far into new territory).

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:

- Notes by Bruce Fontaine (pp. 52: version of Nov. 2011).
- Notes by David Mehrle (pp. 80; lectures given 2015, notes revised 2016).
- 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). In alphabetical order by author:

- John Baez, Category Theory Course (pp. 59, 2019: past course page here).
- 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). - 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 ).
- 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).
- Daniel Epelbaum and Ashwin Trisa, Introductory Category Theory Notes (pp. 56, 2020).
- Rafael Villarroel Flores, Notes on Categories (pp. 77, 2004).
- Maarten M. Fokkinga, A Gentle Introduction to Category Theory: The Calculational Approach (pp. 78, 1994).
- Julia Goedecke, Category Theory (pp. 63, lecture notes for her Cambridge Part III Maths course, 2013: related materials on her website here).
- Chris Hillman, A Categorical Primer (pp. 62, 1997).
- Randal Holmes, Category Theory (pp. 99, 2019).
- Robert Knighten, Notes on Category Theory (about pp. 160 of unfinished notes, followed by appendices including useful information about many books: 2011).
- Valdis Laan, Introduction to Category Theory (pp. 52, 2003).
- Günter Landsmann, Basic Theory of Categories (pp. 65, 2012).
- Bartosz Milewski, Category Theory for Programmers (series of long blogposts, available in book format, linked below: also see also his videos, also linked below).
- Ed Morehouse, Basic Category Theory (pp. 77, 2016).
- Jaap van Oosten, Basic Category Theory and Topos Theory (pp. 123, Utrecht 2016).
- Prakash Panangaden, Brief notes on category theory (pp. 36, 2012).
- Paulo Perrone, Notes on Category Theory (pp. 181, 2021)
- Benjamin Pierce, A Taste of Category Theory for Computer Scientists (pp. 75, 1988: earlier version of this book).
- Uday S. Reddy, Categories and Functors (pp. 47, Lecture Notes for Midlands Graduate School, 2012).
- Pavel Safronov, Category Theory (pp. 56 — Oxford lecture notes, 2015).
- Pierre Schapira, Algebra and Topology (pp. 157, 2008 — largely category theory).
- Pierre Schapira, Categories and Homological Algebra (pp. 115, revised 2022: presupposes some background in algebra etc., but fairly introductory on categories).
- William R. Schmitt, A Concrete Introduction to Categories (pp. 60).
- Greg Stevenson, Rudimentary Category Theory Notes (pp. 28).
- Thomas Streicher, Introduction to Category Theory and Categorial Logic (pp. 116, 2003/4).
- Daniele Turi, Category Theory Lecture Notes (pp. 58, Edinburgh, 2001).
- 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. I’ll keep this list respectable by passing over in silence those copyright-infringing pdf repositories that, of course, none of us use …

- 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). - Jiri Adamek, Horst Herrlich and George Strecker,
*Abstract and Concrete Categories: The Joy of Cats*(originally published John Wiley and Sons, 1990: recommended). - Andrea Asperti and Giuseppe Longo.
*Categories, Types and Structures: Category Theory for the working computer scientist*. MIT Press, 1991. - Steve Awodey,
*Category Theory*(versions of the chapters from the 2010 second edition of this useful book in the Oxford Logic Guides available here). - Michael Barr and Charles Wells,
*Toposes, Triples and Theories*(originally published Springer, 1985). - Michael Barr and Charles Wells,
*Category Theory for Computing Science*(originally published Prentice Hall, 1995: particularly clear and useful). - 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). - Brendan Fong and David Spivak, Seven Sketches in Compositionality:An Invitation to Applied Category Theory (pp. 341, 2018: published as CUP book in 2019).
- Peter Freyd,
*Abelian Categories*(originally published Harper and Row, 1964: not exactly elementary — but a classic). - Robert Goldblatt,
*Topoi*(originally published North-Holland, 1979/1984: an expository classic – also available as cheap Dover book). - 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.) - Tom Leinster,
*Basic Category Theory*(originally published CUP, 2014). - Bartosz Milewski,
*Category Theory for Programmers*(book version of his blog posts, 2018) - 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). - Emily Riehl,
*Category Theory in Context*(pp. 240: 2016 version of her lecture course at Johns Hopkins, now published as a Dover book). - 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). - David I. Spivak,
*Category Theory for the Science**s*(online version of book published by MIT Press, 2014)

**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 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 checked, deleted, revised, and **added 24 March 2022*

BorisGreat notes, books and videos, thank you:-) See this yt series from Richard Southwell:

https://youtube.com/playlist?list=PLCTMeyjMKRkoS699U0OJ3ymr3r01sI08l

Peter SmithThanks for this info. Apparently there is a related book too … but self-published at an absurd price (unless Amazon have got the decimal point in the wrong place).

Rowsety MoidJust noticed this:

Topology: A Categorical Approachby Tai-Danae Bradley, Tyler Bryson and John Terillahttps://www.math3ma.com/blog/topology-book-launch

Note link to a free open access version. (You can download individual chapters as pdf.)

FrankyRandall R. Holmes has a free Category Theory textbook.

https://web.auburn.edu/holmerr/8970/Textbook/CategoryTheory.pdf

Peter SmithIt’s linked … a bit arbitrarily under lecture notes rather than books.

Steven ShawBartosz Milewski now has a series of videos on youtube:

https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_

Souvik DasWill 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 SmithWell, it’s a hopeful thought that there

willbe 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.FyboveI 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.

DionHi, 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”?

Peter SmithIt’s just using the memoir class, with the default \pagestyle{ruled} with minor tweaks.

Paolo G. GiarrussoWould 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).

Julia GoedeckePeople 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

Peter SmithYes, thanks, I indeed should have linked this before!

Mozibur UllahHow 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!

Peter SmithThese are links to books which are freely and legally available to download. Neither Lawvere and Schanuel, nor Mac Lane, are thus available. Both books however are mentioned in the linked reading list.