Basic Category Theory (Cambridge Studies in Advanced Mathematics) (英語) ハードカバー – 2014/7/24
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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.
Tom Leinster has held postdoctoral positions at Cambridge and the Institut des Hautes Études Scientifiques (France), and held an EPSRC Advanced Research Fellowship at the University of Glasgow. He is currently a Chancellor's Fellow at the University of Edinburgh. He is also the author of Higher Operads, Higher Categories (Cambridge University Press, 2004), and one of the hosts of the research blog, The n-Category Café.
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.
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.
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.