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This is a crosspost of this question from MSE.

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 only very basic categorical concepts (no general limits or adjoints). Is there a book which teaches algebra from a more advanced categorical point of view, in particular using general limits and adjoints? Is there a book which teaches algebra from the $n$POV?

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    $\begingroup$ How can you teach elementary algebra assuming advanced algebra concepts? $\endgroup$ Dec 17, 2014 at 21:26
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    $\begingroup$ I think many people would say categories are almost independent of algebra. At any rate, of course I do not personally know how this would be done, but examples to this approach would be introducing the free constructions from the free-forgetful adjunction, introducing quotients as duals of subobjects, etc $\endgroup$
    – Exterior
    Dec 17, 2014 at 21:35
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    $\begingroup$ Maybe it's only me, but in my expierence the notion of adjunction is even difficult to starting graduate students. $\endgroup$ Dec 17, 2014 at 21:37
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    $\begingroup$ I'm not advertising a new approach to teaching math or anything. Almost any field of math has books covering it at very different levels, I'm merely hoping for one as in the post. (I'm an electrical engineering undergrad, and though I certainly haven't worked with adjunctions enough, I find the idea and definition very natural) $\endgroup$
    – Exterior
    Dec 17, 2014 at 21:42
  • $\begingroup$ I cannot help but think Bourbaki , last century , did better. $\endgroup$
    – Al-Amrani
    Dec 18, 2014 at 20:16

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The closest I know for what you're looking for is Bergman's An Invitation to General Algebra and Universal Constructions, which can be downloaded here. It is a simultaneous introduction to both universal algebra and universal constructions in category theory. (Universal algebra generalizes both groups and rings, and it has a natural category-theoretic translation.)

Nothing about the nPOV, though.

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