Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce groups and rings using very basic categorical concepts (no general limits or adjoints). Is there a book which teaches a basic course in group theory, and in commutative algebra, from a more advanced categorical point of view, in particular using adjoints, limits, and basic higher categories? Is there, for instance, a nice categorical viewpoint of localization and prime ideals?

  • $\begingroup$ Perhaps you can make your last question more specific and ask it separately? $\endgroup$ – Martin Brandenburg Dec 16 '14 at 20:12
  • $\begingroup$ @MartinBrandenburg Certainly. I will edit the question. $\endgroup$ – user153312 Dec 16 '14 at 20:13
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    $\begingroup$ Many people will probably tell you or at least think that it is misguided to use category theory to introduce such basic concepts (groups, rings,...), but all the arguments build on what they are used to learn and in which order. I think it would be a very fruitful development if category theory (or at least, the conceptual core ideas) would be included more into courses and books on basic algebra, but I guess that it will take several more decades (or in fact forever) till the ignorance for the importance of category theory will stop. $\endgroup$ – Martin Brandenburg Dec 16 '14 at 20:40
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    $\begingroup$ @MartinBrandenburg well, that's what we kids are for, to start new trends :p $\endgroup$ – user153312 Dec 16 '14 at 20:45
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    $\begingroup$ You may want to have a look at math.stackexchange.com/questions/716878/… $\endgroup$ – Jakob Werner Dec 22 '14 at 12:14

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