Categories and Sheaves
Springer Science & Business Media, Oct 20, 2005 - Mathematics - 498 pages
Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.
This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.
The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization. Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.
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In my first attempt in studying category theory I looked for free online resources and did not get to far. Some of it had to do with me not enjoying reading on the computer screen and in first studying category theory i later learned reading pdf files on the computer is not ideal. The other reason why I did not get too far with my first attempt is because I had the idea that category theory was all about drawing pictures and diagrams. When I finally transfered to a four year university I imeadiatly took advantage with the library. I discovered that there are not a lot books dedicated to category theory. I seen plenty of books that use category theory. I ended up stumbling in to "Category Theory" by Steve Awodey. I first thought that I did not know enough math to study Awodey, and to my understanding Awodey is a very simple introduction to category theory. Awodey takes more of a logical formal language approach. I think it is a good method to learn the subject for the fist time for many people. For some reason I found it boring and really wasn't getting that much out of it.
The third attempt includes me actually spending money on "Categories and Sheaves" by Kashiwara and Schapira. This book is the most abstract thing I ever read and understood. If I wasn't so stubborn I probably wouldn't have purchased it after looking at a few sample pages. This is one event that my stubbornness worked out for me. I swear there is more in the first chapter, which is only like 25 pages then some entire standard undergrad math books I stumbled onto.
This is where I learned that category theory is not about diagrams. As I went though the definitions, lemmas, and want not. You constantly have to flip back because the concepts are so intertwined. Eventually you will start coming up with your own conclusions only to find out that it will be covered later or see an exercise problem that asks you to prove what you concluded on your own.
By the way, I'm guessing have little mathematical experience compared to the audience this book is written for. Before Kashiwara and Schapira the most difficult book I went though was "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" by John Hubbard pure awesomeness, which I went through by myself and is some what incomparable to category theory.
I realized that category theory is not learned in a linear way such as calculus. I really don't think a class on the subject will leave you with any thing of substance. This is really a subject you have a self study and come up with your own conclusions that you try to prove. This book assumes that you are willing to spend time trying to figure things out on your own. Example, there are a lot of indirect implications that are not stated. I don't know if the authors think the reader should be able to seem them because to them I could imagine it is an obvious one or if they want to put you on a wild goose chase and figure it out for yourself, or maybe both.
When coming onto a new definition lemma or what not and you can't find connections that are not stated word for word in the book. It's best to go back and read again. You could be able to make connections with everything as you go on.
I will finish this review later