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Category Theory (Oxford Logic Guides)
Product Review "This book made me fall in love with category theory all over again. full of bordering-on-literary descriptions which enhance the various ideas and concepts;the author often uses words extremely well."--MAA Online Product Description This text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of Category Theory understandable to this broad readership. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided; a must for computer scientists, logicians and linguists! Reader Reviews Several years ago I came across an on-line .pdf format of Awodey's manuscript while trying to find a text on Category Theory whose content was not as intense as Mac Lane's `Categories for the Working Mathematician' and it is wonderful to see this book come to fruition. Without a doubt it is true that the available array of Category theoretic texts for mathematicians has been confined to the more abstract texts whose readership is limited to those individuals who are either researching topics integral to Category theory or graduate students of, say Algebraic Topology/Geometry, who utilize Categorical constructs and processes within the confines of their respective fields. So where does this text fit in? I believe this text can be quantified as "the glue" between Category theoretic texts written for non-mathematicians and the hardcore texts of Mac Lane, Herrlich or Ademek et al. What features set this text apart from the others? Simple, it is focused. Let me preface my explanation with the following: I firmly believe in the importance of demonstrating or motivating any given subject through the use of concrete examples and, in particular, through the use of several examples that can be built upon throughout the text. Awodey sees the importance of this and focuses on illuminating the abstractness of Category theory by carefully building on or utilizing Monoids and Posets. Such structures may readily seem un-familiar to some readers but, if they pause long enough to compare what they know with the basic axioms for a given set to be a Monoid/Poset, then they will see that the majority of structures in which they have been working are, in fact, specialized Monoids/Posets. Take for example Groups. Any set possessing an associative binary law of composition all of whose objects satisfy the 3-axioms for a group also trivially satisfy the axioms for a Monoid. This is not to say that Awodey has chosen two basic blocks from which all examples are derived, instead, he motivates each topic with a vast assortment of the standard examples taken from a diverse set of available fields. So who should read this text? Anyone who wants to learn Category Theory from the ground up but lacks the standard assumed breadth of knowledge, namely, familiarity with Topology, in particular Algebraic Topology, as well as advanced abstract Algebra (inclusive of Module theory). As in any case of defining the readership one would state that their text is readable by the illusive and readily undefined "mathematically mature" student. Personally I would assume that you know how construct logically sound proofs and that you have taken courses in set theory (never given in America) as well as Algebra at the level of, say Hungerford's undergraduate text. Furthermore, and as is the case with anything mathematical, you must be willing to suffer through abstractness and be diligent as well as disciplined enough to work through the exercises. With respect to this last point, Awodey does a remarkable job providing a well thought out set of exercises ranging from simple applications of the material to more advanced exercises that will cause you to pull out your hair and possibly throw the book across the room in sheer agony. As a final note regarding the overall text, I would even suggest this Awodey's book to more advanced student who lack a firm understanding of Category Theory but who have already suffered through someone else's text. Why? Simple, because Awodey's text will help you `see' and hence understand, at the necessary level, Category Theory. After all, one can not become proficient in anything unless they `see' what it is they are trying to become proficient in. Finally, I would like to personally thank Mr. Awodey for writing this text and for doing such a remarkable job introducing and motivating a miraculous and awe-inspiring subject. Enjoy! Comments (4) | | (Report this)
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