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I learned Algebraic Geometry in a geometrical viewpoint, e.g. Hartshorne's book. But I want to learn algebraic geometry categorically, for examples,

i) Sheafification $\mathcal{F}^+$ of a presheaf $\mathcal{F}$ can be defined as the left adjoint functor $(-)^+\colon \mathsf{PSh}(X)\to \mathsf{Sh}(X)$ to the forgetful functor $i\colon \mathsf{Sh}(X)\to \mathsf{PSh}(X)$, where $i$ views any sheaf as a presheaf.

ii) A Grothendieck category can be considered as a scheme, and a subcategory with left adjoint can be considered as an open subscheme. (Someone told me that. I don't know why, and I'm not sure this expression is correct.)

So, this kind of algebraic geometry which is based on category theory and homological algebra is I want to learn. Are there some books or notes doing scheme theory via this highly abstract way?

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  • $\begingroup$ I haven't read it, but you might have a look at Categories and Sheaves by Masaki Kashiwara and Pierre Schapira. It seems to contain lots of the stuff you want. $\endgroup$ Jun 4, 2015 at 21:40

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There is only one algebraic geometry !
Category theory and homological algebra are two among the many tools used to understand it, which there are no particular reasons to idolize nor avoid.
And your example of sheafification as a left adjoint is mildly interesting but will in no way help you understand any genuine theorem in algebraic geometry.

That said, I recommend Mac Lane-Moerdijk's Sheaves in Geometry and Logic which pleasantly explains some notions useful in algebraic geometry .
(Mac Lane is the coinventor with Eilenberg of category theory, so that you'll learn from the proverbial horse's mouth).
Another resource is Grothendieck's legendary paper Sur quelques points d'algèbre homologique which for ever changed our vision of homological algebra and its applications to algebraic geometry.

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    $\begingroup$ When you wrote that MacLane was the 'coinventor' I assumed you were setting up a pun. How disappointing! :) $\endgroup$ Jun 4, 2015 at 23:26
  • $\begingroup$ I imagine that he means that, since MacLane co-invented category theory, co-category theory invented co-MacLane or something of the sort. $\endgroup$ Jun 5, 2015 at 4:28
  • $\begingroup$ Yes, exactly. :-) I was just teasing! $\endgroup$ Jun 5, 2015 at 14:41

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