# Categories and Sheaves, Volume 13

Springer Science & Business Media, 2006 - Mathematics - 497 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

### Contents

 I 1 II 9 III 10 IV 11 V 19 VI 23 VII 27 VIII 30
 LXIII 269 LXIV 270 LXV 272 LXVI 278 LXVII 282 LXVIII 285 LXIX 289 LXX 292

 IX 35 X 36 XI 43 XII 50 XIII 54 XIV 57 XV 59 XVI 62 XVII 64 XVIII 70 XIX 78 XX 81 XXI 87 XXII 90 XXIII 93 XXIV 96 XXV 103 XXVI 107 XXVII 112 XXVIII 117 XXIX 121 XXX 128 XXXI 131 XXXII 138 XXXIII 139 XXXIV 142 XXXV 145 XXXVI 149 XXXVII 158 XXXVIII 161 XXXIX 163 XL 166 XLI 169 XLII 175 XLIII 186 XLIV 188 XLV 193 XLVI 197 XLVII 201 XLVIII 215 XLIX 216 L 217 LI 223 LII 228 LIII 231 LIV 235 LV 239 LVI 241 LVII 242 LVIII 248 LIX 253 LX 256 LXI 258 LXII 265
 LXXI 293 LXXII 297 LXXIII 300 LXXIV 302 LXXV 306 LXXVI 313 LXXVII 316 LXXVIII 319 LXXIX 325 LXXX 329 LXXXI 337 LXXXII 340 LXXXIII 346 LXXXIV 353 LXXXV 354 LXXXVI 357 LXXXVII 366 LXXXVIII 369 LXXXIX 372 XC 374 XCI 381 XCII 387 XCIII 389 XCIV 394 XCV 399 XCVI 401 XCVII 404 XCVIII 411 XCIX 414 C 417 CI 423 CII 424 CIII 429 CIV 431 CV 435 CVI 438 CVII 442 CVIII 444 CIX 445 CX 449 CXI 455 CXII 459 CXIII 461 CXIV 466 CXV 467 CXVI 470 CXVII 474 CXVIII 477 CXIX 480 CXX 482 CXXI 487 CXXII 491 Copyright

### Common terms and phrases

Masaki Kashiwara Professor at the Rims, Kyoto University
Plenary speaker ICM 1978
Invited speaker ICM 1990
http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/

Pierre Schapira, Professor at University Pierre et Marie Curie (Paris VI)
Invited speaker ICM 1990
http://www.math.jussieu.fr/~schapira/