How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
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6$\begingroup$ MacLane and Moerdijk : "Sheaves in Geometry and Logic, a first introduction to topos theory." Springer Verlag. The nLab website is an important source of information : ncatlab.org/nlab/show/Sheaves+in+Geometry+and+Logic. $\endgroup$– Philippe GaucherCommented Sep 2, 2015 at 8:22
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3$\begingroup$ If you have only limited understanding of category theory just yet, "Sheaves ..." is quite a tough book to start with. I would suggest also reading "Categories for the working mathematician" by Saunders Mac Lane or use it as a companion book. $\endgroup$– Loreno HeerCommented Sep 2, 2015 at 12:57
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3$\begingroup$ @PhilippeGaucher I think the concentration on the logic side of things makes this perhaps less than ideal for the student of algebraic geometry. It is a great book, but it does not even touch on cohomology of sheaves, for instance. Not sure of a good replacement book though. $\endgroup$– Steven GubkinCommented Sep 2, 2015 at 18:14
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2$\begingroup$ @LorenoHeer For people having a limited understanding of category theory, I can suggest too once again (I already did in another thread) Tom Leinster's book "Basic Category Theory", Cambridge Studies in Advanced Mathematics. $\endgroup$– Philippe GaucherCommented Sep 2, 2015 at 18:51
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2$\begingroup$ Read Grothendieck's Tohoku paper and, then learn something about derived categories. However if you want to know some topos theoretical related things see Monique Hakim's thesis. $\endgroup$– user40276Commented Sep 3, 2015 at 23:43
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3 Answers
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Paolo Aluffi's Algebra Chapter 0 develops abstract algebra using Category theory from the very beginning. The exposition is very clear and teaches upto and including the derived functor approach to cohomology. The category theory developed here should be more than enough to study sheaves and schemes eventually.
In an answer to your earlier question, Julien Puydt points to (and I second his suggestion) the excellent text by Ravi Vakil. The category theory developed in this text is really all you need; in fact, there is no more than what is needed for purposes of getting started with AG.
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7$\begingroup$ Speaking as a category theorist, I think this is sound advice. As much as I like the book by Moerdijk and Mac Lane, I don't think it's required reading by any means; I'd learn category theory just on an as-needed basis (and if you like it and want to learn more, then consider looking into books by category theorists). Experts like Vakil can certainly guide you in how much you realistically need for AG. (There is also, of course, SGA which develops a ton of category theory ab initio, and which you will probably consult somewhere down the road.) $\endgroup$ Commented Sep 2, 2015 at 17:34
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The answer to your first question "how much of category theory..." is partly that it is a matter of taste. Some people use a lot, while others don't. But, if you plan to study schemes, it is necessary to know some. For example, the product of varieties can be treated naively, but for schemes it definitely needs to be understood in a categorical sense. For existence, it helps to the know that the category of affine schemes is equivalent to the opposite of the category of commutative rings. Many schemes can be understood best in terms of the functor it represents etc. etc.
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These are the first and the second of three volumes on algebraic geometry by Kenji Ueno from AMS as Volume 197 in the Translations of Mathematical Monographs series.
(1) Algebraic Geometry 1: From Algebraic Varieties to Schemes
(2) Algebraic Geometry 2: Sheaves and Cohomology
