An Introduction to Category Theory

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Cambridge University Press, Sep 22, 2011 - Mathematics
Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.
 

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Category theory is a relatively new abstract-algebra framework (more abstract than linear spaces, groups, ordered sets, etc....) that has gained some popularity in Computer Science (mostly in semantics of programming languages). This has prompted me to want to learn about it. I have tried a couple of books and this is one which I read almost entirely and enjoyed it as well. It has a very friendly style both in language and in organization, typesetting etc. However, it will not fit every reader: I feel that it is written for mathematicians - set theory and abstract algebra are used as a source of "concrete" examples! The book is written so that the elegance of the theory as an abstract framework can be appreciated (by those who appreciate such things). It does not attempt to address Computer Scientists in particular and does not cover all CT concepts that I have encountered in CS contexts.
A few ciritical remarks: the book could do with a better index.
I think that writing composition from left to right (unlike the notation "f o g" which the book uses for applying f after g) would make many equatons easier to follow.
Chapter 4 is somewhat disappointing: it includes several instances of what appears to be essentially the same construction, so I expected it to culminate with a general statement of this construction, as CT usually does, but such a theorem is not given.
 

Contents

1 Categories
1
2 Basic gadgetry
34
3 Functors and natural transformations
72
4 Limits and colimits in general
108
5 Adjunctions
148
6 Posets and monoid sets
190
Bibliography
223
Index
224
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About the author (2011)

Harold Simmons is officially retired but still active in research. He also teaches postgraduate courses in the School of Mathematics at the University of Manchester.

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