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Introduction to Categories, Homological Algebra and Sheaf Cohomology

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Before I begin, I would like to disclose that I am not an Algebraist and prior to buying this book had no background in category theory. I bought this book with the intention of learning the basic language of category theory with the ultimately goal in mind of moving toward sheaf co-homology theory. So please consider my review in that context.

This book is very dense.I don't think that I can think of a better way of describing it. Some people like that- others do not. Personally I do like dense books a lot if written well. I think the greatest strength of this book is that I find it very too the point. As a person trying to learn a subject such as category theory for the first time- this can be very good.

The author does provide a healthy collection of examples with many of the definitions although does not actually prove any of them, for example: in the section on contravariant functors (pg. 5), the author asserts that H^n: Top->Ab, the nth co-homology functor mapping from the category of topological spaces into the category of Abelian groups is a contravariant functor- the end. No proof, nothing. I think the idea here is that it is left as an exercise for the reader.

The notation is sometimes clear, and sometimes is just plain horrible!

Here is an example of an example (again from the contravariant functor section, this is for the "hom" function as everyone else calls it, the author does not give it a name, I am posting word for word- using LaTeX script where appropriate- if you are reading this review, I assume you dream in LaTeX):

"An important example is the contravariant functor (for C in \mathbf{C}) h^{C}A=(A,C) for A \in \mathbf{C} and h^{C}f(u)=uf for f \in (A,B)_{\mathbf{C}}, u \in (B,C)_{\mathbf{C}}"

The end.

So, my take, I found this book to have a lot of strength (no fluff), and a fair amount of deficiencies (fluff is good in moderation). I ended up switching to Hilton and Stammbach- and use this one for comparison.

[Edit: Since writing review, I have spent a fair amount of time with Category theory. Particularly, I have studied from a few other books. I still stand by my comments that this book is not the best choice for someone new to Category theory, and I still feel that this book has some strengths as a reference given its conciseness and terseness. If you would like to learn Category theory, and are new to the subject- I would suggest taking a look at "A Course in Homological Algebra" by Hilton and Stammbach, then if you want- come back and take a look at this book again. ]