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I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft introduction to algebraic topology. But I don't know any references in category theory. Can someone give some idea for what book or lecture notes I can get a introduction on this topic?

Thank you so much in any advance!

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    $\begingroup$ "Categories for the working mathematician" may be what you are looking for. $\endgroup$ – user27126 Jul 2 '13 at 14:17
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    $\begingroup$ Here's a good list of references. You may also read the accompanying article. $\endgroup$ – M. Vinay Mar 9 '16 at 8:40
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I found Category Theory by Steve Awodey to be a really good introduction to the subject. It doesn't lead directly into algebraic geometry (or, indeed, in any particular direction), but gives a readable introduction to the subject with plenty of examples and problems. When you've worked through this you could move on to a more advanced text or one which is more specifically geared towards algebraic geometry.

On the side, I'll mention Ravi Vakil's epic notes on algebraic geometry which kick off with category theory before moving on.

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  • $\begingroup$ Thank you !! I think both will help me a lot! $\endgroup$ – User43029 Jul 3 '13 at 12:36
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I think the best way to learn the basics of category theory is to see it in action. For this reason, I recommend Rotman's An Introduction to Homological Algebra, but you have to be algebra-inclined.

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  • $\begingroup$ I've heard about this, Thank You! $\endgroup$ – User43029 Jul 3 '13 at 12:37
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You can also take a look at this wonderful set of video lectures by Eugenia Cheng

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  • $\begingroup$ That link appears to be dead. $\endgroup$ – celtschk Sep 11 '18 at 6:01
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You could try Gelfand and Manin's Methods of Homological Algebra - it introduces a lot of category theory (the whole thing is really an intro to derived categories) from scratch, and homological algebra is a big part of algebraic geometry. Their examples are more topological than geometric though.

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My favorite book is "Categories and Sheaves" by Kashiwara and Schapira. For a more introductory approach you can pick "Abstract and Concrete Categories: The Joy of Cats" by Adamek, Herrlich and Strecker (I think that there is a free online version of this book at the site of some of the authors) .

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The trilogy of Francis Borceux "Handbook of categorical algebra", the first volume is a nice introduction.

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