## Introduction

Philosophers will have come across claims that category theory – or in particular, that division of it which is topos theory – provides a new foundation, or a different sort of foundation, to mathematics, in some sense rivalling set theory in its sweep and generality. So philosophers with foundational interests may well want to know what the fuss is about. (OK, the cool kids may now think that that ‘homotopy type theory’ is where the foundational action really is: but you will still need to have some background in category theory to understand all that.)

But perhaps ‘foundations’ is the wrong metaphor for category theory. Thus Tom Leinster has written as a two-sentence characterization,

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.

So this would have category theory *organizing* mathematics (with sweep and generality, indeed) by finding unifying patterns, but not necessarily by seeing mathematics as reducible to, or embeddable into, or *underpinned* by, one grand foundational story.

It is too soon to worry about ‘foundations vs unifying overview’, however, before we even see category theory at work! But Leinster’s remark does point up why many philosophers, in particular, might find category theory hard work: for it is indeed often presented against an assumed wide-ranging knowledge of modern mathematics, and by drawing out patterns in a background which philosophers may well lack. But perhaps we can edge our way in.

*The Stanford Encyclopedia of Philosophy* has a brisk introductory encyclopaedia article notionally addressed to philosophers by Jean-Pierre Marquis, ‘Category Theory’. But while piece this is a useful source of pointers forward to different areas of interest, I suspect most beginners won’t be enlightened at all, as it is already pretty abstract (and indeed will probably only reinforce the impression that you need to know a *lot* of mathematics before you can get to grips with categories — not so). Better to return to this article once you are already moderately acquainted with category theory. So where to begin?

## Entry-level expositions of category theory

I’ll start with three introductory books with different virtues, in something like order of difficulty:

- F. William Lawvere and Stephen H. Schanuel,
*Conceptual Mathematics: A First Introduction to Categories*(CUP 2nd edn. 2009). A genuinely gentle introduction that slowly introduces you to new categorial ways of thinking about some familiar things. In fact, you might well find the book rather too slow and are than a bit condescending at the outset (does the likely reader really need to have it explained, at length, over a hundred pages into the book, that*if P then Q*and*if not-Q then not-P*are equivalent?). However, if you can ignore such irritations, you do end up after a pretty painless journey having learnt rather more than you realize. (The second edition is notably more useful than the first because it adds chapters that ease the transitions to more advanced topics.)

Still, many/most might prefer to start a step up and work through the first hundred or so pages of an older, more conventional book:

- Robert Goldblatt,
*Topoi: The Categorial Analysis of Logic*(North-Holland 1979, very inexpensively reprinted by Dover 2009, and also available online ). Overall, this book (as the title suggests) is angled towards aspects of category theory of particular interest to logicians. It eventually covers a good deal of ground (with the difficulty level then going up accordingly, of course), but for the moment I’m suggesting you start off by just reading the*first three chapters*which cover much of what you’d pick up from Lawvere and Schanuel in far fewer pages. As mathematical exposition, this still strikes me as first rate. (I’m inclined to say that if you don’t find these opening chapters accessible and intriguing, then category theory is probably not for you. If you*do*like them, then at some later point you’ll no doubt want to read more of the book.) - Harold Simmons,
*An Introduction to Category Theory*(CUP, 2011) is also highly accessible, and introduces some central notions of category theory you won’t find in Lawvere and Schanuel or in the opening three chapters of Goldblatt. In particular, it will give you a first handle on the key notion of adjoint functors. This is written for self-study with a lot of exercises (there are online solutions here). It still has as something of the conversational tone of the lecture room, and you could very well find it an engaging and helpful introduction. Try the*first five chapters**.*

Also starting from scratch, and at something like the same level of accessibility of Goldblatt’s early chapters and/or Simmons’s book, there is:

- My unfinished, sporadic, work-in-progress
*Category Theory: Notes towards a gentle introduction*aims to be rather more discursive/explanatory at various points. The coverage will eventually go further than Simmons and in a somewhat different direction to Goldblatt. Since I also try to keep the required mathematical background down to a bare minimum, this hopefully will be particularly useful to philosophers and others.

A book which is often been highly recommended as an entry-point for beginners is

- Steve Awodey,
*Category Theory*, Oxford Logic Guides 49 (OUP 2nd ed, 2010).

However, I should say that a reading group with a few very smart maths students who had already done a category theory course confirmed my impressions this is a very considerably bumpier ride than the author intends (and indeed that was one spur to me writing my own notes). It is too often a bit less than ideally clear, and some of the expositional choices seem to me debatable. However, it should work well if you have e.g. early read those early chapters from Goldblatt.

What next? Here is another book which is again intended for relative beginners — but this time as in beginning graduate students who haven’t previously met category theory but can be assumed to be mathematically pretty competent. So this is somewhat tougher, being a bit brisker and more compressed. But I do think that this text is a model of brilliant exposition at its level.

- Tom Leinster,
*Basic Category Theory*(CUP, 2014).

Finally, I should certainly mention a book which is not strictly speaking a general introduction to category theory but it introduces enough category theory for its purposes, and you’ll probably want to read it quite early in your explorations of category theory.

- F. William Lawvere and Robert Rosebrugh,
*Sets for Mathematics*(CUP 2003) presents a different way of thinking about sets, and shows that an approach via category theory can throw new light on familiar territory. However I do find this quite badly written (philosophers will find too much sloppy talk — you have to read past that).

## Some philosophical discussions

The Stanford Encyclopaedia article by Marquis mentioned at the start also has lots of references to philosophical discussions. There are also two substantial books which promise some extended philosophical discussions:

- Ralf Krömer,
*Tool and Object: A History and Philosophy of Category Theory*(Birkaüser, 2007) - Jean-Pierre Marquis,
*From a Geometrical Point of View: Study of the History and Philosophy of Category Theory*(Springer, 2009).

Maybe at some point I’ll write book notes on both, but for the moment I’ll list them just with the comment that neither is easy! Some of the essays in

- Giandomenico Sica (ed),
*What is Category Theory?*(Polimetrica, 2006)

will also be of interest to philosophers. And the recent

- Elaine Landry (ed),
*Categories for the Working Philosophy*(OUP, 2017)

has articles (of various degrees of accessibility) on aspects of category, pure and applied, of philosophical interest.

Back to the main category theory page

*Page updated 9 July 2023*

joseph tanegaMany thanks for the list! I agree with your choices. I’m not trained in maths but have read L&S, Simmons and Goldblatt’s 4 chapters and about half of L&R. Awodey’s lectures on youtube are excellent. Marquis book made me wonder a lot, eg, “presentation invariance” (somewhere past page 200) might be used a lot outside of maths per se…i’m trying to work my way through Robert Harper’s “Computational Trinitarianism” where CT I suppose plays the part of the Holy Ghost. I think once we get use to transposing type theory verbal statements into category theory diagrams and vice versa, we get a scientific foundation for any object field of relations. My lazy work is to find something some way of laying a TypeCat foundation for law and finance. Anyway, many, many thanks for your notes!

samuel sayagMany thanks for this so useful and readable site !

A little correction, you wrote: “F. William Lawvere and Robert Rosebrugh, Sets for Mathematicians (CUP 2003)”

… and the title is just:

“SETS FOR MATHEMATICS”

Have a good day !

Joseph TanegaMany thanks for your recommendations above which I wholly agree. I’d add Bartosz Milewski’s site on category theory. His notes and videos are wonderfully approachable. I’d say a compendium of maxims and diagrams common to Lawvere & Schanuel (identities and systems), Goldblatt (through chapter 3), Simmons (especially adjunctions) arranged in a kind of “parrallel translation” with diagrams on the left side and narrative on the right side, would be helpful as a digest (a kind of Rosetta Stone). I found Marquis’ book much more exciting and integrative than Kromer’s. It’s too bad Marquis book is so expensive! Marquis sees the big picture, Kromer would like us to connect certain scholarly controversies and the questions he asks at the end are not so big. I’m working on trying connect category theory to law and finance. kind regards.

David P EllermanHere is another recent paper of mine on category theory and philosophy. https://www.ellerman.org/how-category-theory-works/

AbelThanks a lot! I have in particular similar feeling as you on Goldblatt’s.

I am not sure if it is best to post here, but I recently find “Sets, Logic and Categories” by Peter J. Cameron. And it seems a good intro. to both math. logic and categories.

It would be greatly helpful to know your ideas on this book. Thanks again.

MatthewThanks for sharing :) I would add E. Cheng’s ‘The Joy of Abstraction’ as an accessible introduction for the motivated layperson.

Peter SmithI do have my reservations about Eugenia Cheng’s book — https://www.logicmatters.net/2022/11/03/back-to-business/. But I guess it may work for some readers!

ChuenHi thank you for sharing. What kind of math background do you think is necessary before studying category theory? Do I need to study undergraduate math level linear algebra before studying category theory? Thank you for your attention

Peter SmithAa I say in the Preface to

Category Theory I,The same will apply to other introductions to category theory. You’ll need to know at least a little undergraduate mathematics to understand the point of it all, with different books presupposing different amounts of background knowledge. However, you won’t need in-depth knowledge of any one area like linear algebra (though some general knowledge of abstract algebra like a little bit of group theory could be particularly useful).

ChuenHi Peter thank you for the explanation!