Categories for the Working Mathematician

Graduate Texts in Mathematics #5

by Saunders Mac Lane

Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions.

This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

Genres: Mathematics, Algebra, Textbooks, Logic, Reference, Theory, Nonfiction

330 pages, Hardcover

First published January 1, 1971

Reviews

[lolineka · Rating 2 out of 5]

Being one of the fathers of category theory, Mac Lane did much more to averse people from CT, than many category haters. Poorly structured text, esoteric stuff, many proofs left as exercises to the reader and uneven writing make this book bad choice for beginners. This book contrasts with 'Algebra' written in collaboration with Birkhoff. If you want to start with Category Theory, you'd want to begin with Awodey or even better choice IMO - Tom Leinster. While being pretty complete and widely acknowledged handbook on CT, it's thorns and pure evil for beginners.

[John Kameas · Rating 2 out of 5]

One of three books I continuously referred to throughout the year as I worked on my thesis. It's pretty hard to oversell just how important Category Theory has become in mathematics. For something that was essentially formalised in the 1950s — a small-little baby when compared to the geriatric status of most other theories in mathematics — it's incredible how quickly it has become ubiquitous. Seriously. There's not a single modern mathematics textbook that doesn't give you a quick primer on functors, natural transformations, adjoints, duals, etc. It is just essential to get anywhere nowadays in maths. The problem with learning about Category Theory in this ad-hoc fashion, is that the definitions feel utterly austere. If you're looking for any more information, intuition, or justification, then bad luck — go look elsewhere.

So, what the hell do-you-do if you actually need to learn the nuts-and-bolts of the damn thing? You probably go to one of the two guys that literally created it: Mac Lane. Who else could do it better than the person who came up with the all the definitions, axioms and view points? This is no doubt one of the reasons why everyone who is even slightly interested in maths, knows about this book: it's a treat to have, as your tour-guide, the person who laid down all the bricks in the palace you now stand in. The problem, however, is that Mac Lane created fucking Category Theory. He naturally sees so much further than everyone-else that he has forgotten the problems of the myopic-riddled masses. He'll be describing to you the beautiful mountain range 3000kms away that perfectly and succinctly explains his point, whilst you're still trying to poke your head out from under the dense foliage.

If you're in dire need of a reference to all things Category Theory, then look no further. If you need absolutely anything else — barring some trifle that you wish resolved by the most authoritative voice on the matter — then this really isn't worth your time.

[K Sadov · Rating 4 out of 5]

I've tried to learn category theory multiple times, and Chapters I-VII are the best introduction to the subject that I've encountered. CftWM doesn't rely overmuch on examples from other branches of math, so it's accessible to readers without a background in abstract algebra. The exercises are useful and should be doable for any undergrad, but some of the proofs are concise to a fault; most egregiously, Mac Lane's proof of the Yoneda Lemma. I recommend referencing Bartoz Milewski's Category Theory for Programmers and Math3ma's category theory blog posts for more intuitive explanations.

[Alex Lee · Rating 3 out of 5]

This was a pretty incredible book. Towards the last half my understanding was exceeded but up until that point I understood most of what I was reading and was able to follow. The tightness of what was said is pretty impressive, but it will be some time before I return to this to retackle all the mysterious relationships in this book.

Still, Mac Lane could have written it in a way that is less tense, and offered more in the way of teaching than anything else, than just a book of proofs.

[Cameron · Rating 4 out of 5]

Learn from the master himself

[Steve Miller · Rating 5 out of 5]

Holy Crap! Everything you ever wanted to know about category theory in one astoundingly clear and concise book. This is truly one stop shopping for category theorists and it has been a good friend during my thesis writing. I was just now thumbing through for the specifics on enriched categories when I felt a serious appreciation for this book. I highly recommend "Categories for the Working Mathematician" not only to my fellow working mathematicians but also to anyone with an interest in higher mathematics as there are almost no prerequisites for this self-contained masterpiece.

[Pedro Carvalho · Rating 3 out of 5]

While very complete and concise, it's not very intuitive (the words "stiff," "boring" and "formal" come to mind) and makes for a very hard time actually digesting and understanding the concepts presented.

[Tyler · Rating 5 out of 5]

Part of the RedditU course on Category Theory.

[Sandor Molnar · Rating 4 out of 5]

Well, it's a tough one indeed... but clearly a definitive in terms of laying the foundations.... super sturdy foundations... ones, that you can build a complete abstract cathedral on. Which the author does so.