Category Theory (Oxford Logic Guides) (英語) ペーパーバック – 2010/8/13

This book is good for somebody with Mathematical training (group theory, basic topology, etc) but who is not a professional Mathematician. I have enjoyed working through the book and seeing how often abstract structures from different theories that seem vaguely similar can be described as the same structure in different categories.

This book might be tough for a "general audience", and I'm not sure I'm learning anything practical. But it is far more accessible than Saunders and MacLane, and much deeper and more interesting than the typical "Category for Computer Scientists" book. For me, it's just right.

This book is an excellent and kind introduction to category theory. This theory is a very interesting approach to the foundations of mathematics as well as finding the interesting connections between its various branches.

This is just right, and a perfect companion to Mac Lane.

Very solid, goes nicely with the Simmons book.

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!