Basic Category Theory (Cambridge Studies in Advanced Mathematics Book 143) 1st Edition by Tom Leinster (PDF)

Description

At the heart of this short introduction to category theory is the idea of a universal property, important throughout mathematics. After an introductory chapter giving the basic definitions, separate chapters explain three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties all three together. The book is suitable for use in courses or for independent study. Assuming relatively little mathematical background, it is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations. Copious exercises are included.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Category theory has an unfounded reputation of being very difficult. I think that may be because people start with MacLane and get scared away by how terse it is.This book is sort of the dual to MacLane– It has amazing clarity and presentation, while not covering quite as much material. Reading it feels more like a private lecture than a textbook. If you are just reaching the point in your mathematical education where you are learning material on your own, this is a fantastic place to start.One point of caution, however– people coming from a functional programming background should be aware that this book does NOT cover monads. However, once you are finished with it, learning about them will not take you very long.

⭐Exactly what Robert Sievers said in his review.It’s focused and well-written.With Awodey, it’s often hard to tell what’s really important and what’s not.With Leinster, everything is important.

⭐This is a slightly edited version of the authour’s lecture notes in category theory still available from his home page. (*** Added, the 23rd February, 2016 *** The original notes are gone, and the authour says a free online version will be released soon. Thus one will be able to download the entire book for free, but perhaps the printing cost will dictate one to purchase instead.)I wrote “for ordinary students” because the intended audience are general students of mathematics who are not particularly interested in category for the sake of category.As everybody is aware, there are textbooks on category theory with established reputations: namely ones by Mac Lane and Awodey respectively.Compared to those two, Leinster’s book covers somewhat less both in width and depth: for example, he has put the proof for Adjoint Functor Theorem in the appendix and merely touches upon Special Adjoint Functor Theorem without giving a precise statement and its proof.But it is much more accessible than Mac Lane and Awodey: Mac Lane’s book should be titled “Category for the category theorist”, while Awodey has put an emphasis on the relation between category and logic which is beyond the scope of most students of mathematics.The bottom line is, unless you specialise in a heavily category-oriented area, this is probably the only textbook on category theory you will ever need.

⭐Somehow the book is much more focused than all others I have read. And I think ,despite the fact that it lacks some breadth, it is really an advantage. It is ideal for students who have spent some time in category theory and really want to comprehend the relation of limits, adjunctions and representables – which is core category theory. I would give it 5 stars if the intro was a bit broader in examples.For a CS student who has not much intuition on the structures in the initial examples, the book can be a bit off-putting.If you already know some category theory though, you understand that the constructions presented have a “polymorphic”/ parametric nature and the structures themselves are irrelevant. I am not sure though if someone with no prior experience in categorical concepts and not from a Math background will appreciate them.In a nutshell, great book but with some changes it could become a teaching standard in the field.

⭐I don’t disagree with any of the very positive reviews here. But ifyou’re truly a beginner, let me warn you about what mathematiciansmean by words like “basic”, “simple”, “easy introduction to”,etc. Being a real beginner means: 1. You don’t know anything about thesubject to begin with; 2. You are trying to learn the terminology ofthe field by reading about it.The very first (substantive) sentences of this book say:”Our first example of a universal property is very simple.Let 1 denote a set with one element.”Already, WTF!? To the beginner, 1 is a number that I learned aboutin elementary school. It is not a set (despite ZFC’s definition, whichas a beginner, I don’t know about).So you glean that they’re using the name “1” in some special way that youdon’t quite understand. You’re immediately disoriented. You don’t knowwhat things you already think you know about 1 to hold onto and whichto redefine for this special context.Yes, it is “mathematically legal” to introduce a new definition for anold name. You can call it anything you damn well please. And yes,calling it 1 evokes to the “essential singleness” of a singletonset. And yes, that kind of thing doesn’t upset the “mathematicallymature” (BTW, I’ve got a BS in Math from MIT).But is it wise to do so in a book for absolute beginners? Hell, no.It’s not “very simple” at all for the beginner. Why cause needless confusion?And that’s what we’re dealing with. Many math books mean “simple”,”basic”, etc. only in the sense of having few prerequisites inprior knowledge of specific mathematical fields. It doesn’t mean thatthey are easy to understand for beginners.To beginners: Don’t let it stop you from reading this book.Don’t say to yourself, “They tell me it’s simple, so if I don’tunderstand it, I’m too stupid or I’m not cut out for math”, etc.Read carefully. Get a good math teacher to explain to youhow to think about mathematical definitions. Ask “why” questions.To mathematicians who write introductory books: Think hardabout what a real beginner does and does not understand. Don’tsweep confusion under the rug by saying you require “mathematicalmaturity”. Most importantly: User test it. If your teenager who likesmath and took one calculus course doesn’t think it’s simple, it’s yourfault. Fix it.

⭐Tom Leinster is a very modern category theorist and this book is a great introduction to the subject.

⭐Provides by far the most digestible introduction to category theory and covers the fundamental topics, in a very clear and concise way.

⭐La pasta es una mala impresión de la pasta dura forrada en tela, clásica de los libros de Oxford. Las páginas no están cosidas sino pegadas y temo que se desprendan con el uso. Pareciera una edición en pasta blanda a la que pusieron pastas duras de mala calidad.El contenido es hermoso.Great to grasp the basics of Category Theory for anyone with a bit of undergrad level math experience.As I said on the title there’s a more recent and slightly corrected version of this book openly shared by the author himself online, with the .tex source file too. Still but this if you want to support the author for his work, or even just to have a nice hardcover copy.

⭐Well-written and clear. I enjoyed reading this book and found it useful for review and ideas in general on the applications of category theory.

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