Categories and Sheaves, Volume 13

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Springer Science & Business Media, 2006 - Mathematics - 497 pages
1 Review

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

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References to this book

About the author (2006)

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/