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[Pierce, Benjamin C.]のBasic Category Theory for Computer Scientists (Foundations of Computing)

Basic Category Theory for Computer Scientists (Foundations of Computing) [プリント・レプリカ] Kindle版


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内容紹介

Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Benjamin C. Pierce received his doctoral degree from Carnegie Mellon University.Contents : Tutorial. Applications. Further Reading.

著者について

Benjamin C. Pierce is Professor of Computer and Information Science at the University of Pennsylvania.


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  • フォーマット: Kindle版
  • ファイルサイズ: 4428 KB
  • 紙の本の長さ: 114 ページ
  • 出版社: The MIT Press; 1版 (1991/8/7)
  • 販売: Amazon Services International, Inc.
  • 言語: 英語
  • ASIN: B00MG7E5WE
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3 人中、3人の方が、「このレビューが参考になった」と投票しています。
5つ星のうち 4.0 Concise and not too difficult 2014/3/21
投稿者 hans - (Amazon.com)
形式: ペーパーバック
Dr. Pierce's style is a little informal compared to pure math books like Mac Lane's "Categories for the Working Mathematician", but I enjoy that more relaxed style of writing when I am first learning a field.

The biggest obstacle for learning category theory is the fact that category theory generalizes a lot of areas of pure mathematics like topology, abstract algebra, and geometry. It's hard to generalize before you have examples to generalize, but the examples being generalized in category theory are mostly from higher level mathematics found in senior level undergraduate and graduate level courses. Pierce ameliorates this problem by introducing some of the most basic categories first: sets, ordered sets, partially ordered sets, groups, monoids, vector spaces, measure spaces, topological spaces, proofs, and a simple functional computer language. He takes the time to explicitly define most of these ideas, so, in theory, you could read this book without a background in theoretical mathematics, but it would be hard.

After defining categories and introducing the most basic categories, Pierce describes and defines the most basic ideas in category theory: subcategories, commutative diagrams, monomorphisms, epimorphisms, isomorphisms, initial/terminal objects, products, coproducts, universal constructions, equalizers, pullbacks, pushouts, limits, cones, colimits, cocones, exponentiation, and closed Cartesian categories. These ideas are spelled out over the thirty pages of chapter one including illuminating homework exercises. The homework exercises varied significantly in difficulty. Many of the exercises were trivial and there are two or three that I am still working on despite investing several hours of thought. Generally, I found the exercises to be a bit harder than those in Mac Lane's book, but Pierce's book required less of a background in mathematics. A couple of the exercises were incorrectly stated or impossible.

Chapter two introduced functors, natural transformations, adjoints, and F-algebras. After reading this chapter, I was finally able to understand the definition of monads which are an important part of the computer language Haskell! Pierce provides many examples of each of these ideas and enjoyable homework exercises to increase understanding. Pierce's definition of adjoints is much easier to understand than the standard definitions using counit adjunction or Hom Sets.

The last major chapter concerns applications of category theory to computer science-specifically lambda-calculus and programming language design.

The first two chapters of the book give a reasonable, condensed introduction to category theory for students that have taken a course in abstract algebra. A course in topology or linear algebra would be another useful prerequisite. I carried around the light 100 page book for a few months so that I could learn something whenever I had some extra time. I had hoped that when I had proven that several functors where monads, I would then really understand monads, but a full understanding still eludes me. Similarly, I had proven that several functor pairs are adjoint, but even after I finished the book, I did not feel as though I understand adjoint functors. I had to read a few other sources and the Wikipedia before I felt comfortable with adjoints.

In summary, it's a nice, concise introduction to category theory that almost does not require a graduate level understanding of mathematics.

A slightly different version of this review appeared in my blog at artent.net.
3 人中、3人の方が、「このレビューが参考になった」と投票しています。
5つ星のうち 4.0 A Good Read 2008/8/25
投稿者 Jason Dusek - (Amazon.com)
形式: ペーパーバック
This book is not exactly what I would call easy going. I've managed to get through half of it in 7 months. However, I can say, with absolute confidence, that if you do the problems you will learn.

Most everything I've seen on category theory is a confusing mixture of different notations with seemingly identical meanings (but in fact the meanings are totally different). This book is no exception. Often, I have resorted to IRC to sort things out when some notation is simply impenetrable to me. My mathematical training stopped at complex calculus, so this may not apply to you if you've had abstract algebra or something a little more 'meta'.

There seems to be one typographical error, but I am not sure. In the example on the adjunction between products and exponentiation, the right adjoint is listed as "(_)^A x A" but in the diagrams it ends up as "(_)^A". This may be a sensible ellision, but it is not explained anywhere in the text and of it's not easy to find these things on the internet.
1 人中、0人の方が、「このレビューが参考になった」と投票しています。
5つ星のうち 5.0 The first 2 chapters give a good overview of the subject 2015/8/29
投稿者 David Arp - (Amazon.com)
形式: ペーパーバック Amazonで購入
The first 2 chapters give a good overview of the subject. The third chapter is on applications of the theory to computer science, esp. the lambda calculus, which was beyond me and not well written. Would recommend this book despite its shortcomings. Make sure to do the exercises in the first 2 chapters. Should supplement book with the wikipedia article on adjoint functors.
3 人中、3人の方が、「このレビューが参考になった」と投票しています。
5つ星のうち 5.0 nice and slim (the text is only ~70 pages)! 2010/10/3
投稿者 King Yin Yan - (Amazon.com)
形式: ペーパーバック
I'm still a beginner at category theory, but I'd like to say this is a nice textbook. The examples are easy to follow (mainly basic set theory), for people with a com sci background.

A later section explains CCCs (Cartesian closed categories) and its isomorphism to typed lambda calculus. I don't fully grasp the details but this is a very important result in higher-order logic, particularly because the substitution mechanism of lambda calculus can be modeled by category theory.
26 人中、24人の方が、「このレビューが参考になった」と投票しています。
5つ星のうち 2.0 Basic crib sheet for category theory 2006/4/3
投稿者 J. Elliott - (Amazon.com)
形式: ペーパーバック
Anyone coming to this book from Pierce's "Types and Programming Languages" will be disappointed. While his "Types ..." book is a model of clear exposition, this book reads like a set of notes jotted down on the back on an envelope. The extensive bibliographic sections are more than fifteen years out of date. Much of the material referenced is no longer in print, and recent developments are, of course, not mentioned. Those seeking a very gentle introduction to category theory would do better with the book by Lawvere and Schanuel, who cover more of category theory than Pierce. Mathematically mature computer science readers will find everything they need to know about the subject in Mac Lane's book.
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