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I'm no expert on homological algebra, but I have often read that there are many issues with triangulated categories, such as certain maps involving mapping cones failing to be unique, and poor behaviour of the triangulated structure under various constructions. The usual explanation for this that I see is that a triangulated category is in general a "squashed down" stable $(\infty,1)$-category, and the issues come from ignoring the higher cells.

Anyway, as a fan of "abstract nonsense", I am curious if there is a reference that develops homological algebra from the point of view of Quillen model structures and/or stable $(\infty,1)$-category theory?

With regards to my background, I'm pretty comfortable with 1-category theory, but don't have any real background in higher category theory. I'm just starting to read Riehl & Verity's Infinity Category Theory from Scratch. My homotopy theory background is essentially the first two chapters of Emily Riehl's Categorical Homotopy Theory (so basically up to derived functors as Kan extensions), but I can pick that up as I go.

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    $\begingroup$ It's a bit unclear what you're looking for. It's essentially tautological that the classical content of homological algebra does not require any higher category theory for its development. So if you read something about stable $\infty$-categories, you'll gain a modern perspective on various classical topics, but you won't tend to see a focus on the latter. In any case, the canonical reference on stable $\infty$-categories is Lurie's Higher Algebra, $\endgroup$ Apr 21, 2017 at 6:47
  • $\begingroup$ Well, I was hoping for something that doesn't focus on triangulated categories and distinguished triangles, but rather takes a homotopy-theoretic (via model structures) or higher categorical approach. $\endgroup$
    – ಠ_ಠ
    Apr 21, 2017 at 7:04
  • $\begingroup$ Right, I just don't think that most references that could be reasonably described as on "homological algebra" even focus on triangulated categories, as opposed to abelian categories and their categories of chain complexes! So maybe you could clear up more specifically some topics you're hoping to learn about. $\endgroup$ Apr 21, 2017 at 7:23
  • $\begingroup$ I figured that triangulated categories and derived categories should be treated in any reasonable textbook on homological algebra. Is there any reason someone would learn about chain complexes unless they were interested in homology and derived functors? $\endgroup$
    – ಠ_ಠ
    Apr 21, 2017 at 7:58
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    $\begingroup$ The thing is that lots of applications of derived functors don't require the full apparatus of derived categories. It's a conceptual improvement, but frequently not a computational one-so for instance plenty of algebraic geometers are perfectly happy to stick with Hartshorne's level for understanding sheaf cohomology. $\endgroup$ Apr 21, 2017 at 23:14

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The only reference I know of is in the $\infty$-categorical (specifically, quasicategorical) setting, which is Higher Algebra by Lurie; but if you know what a stable $\infty$-category is, then you almost certainly know about this reference. Though I do have to say that Lurie does give a complete treatment of the general setup you would find in any traditional text on homological algebra, that is, Lurie defines derived categories and functors, and also discusses spectral sequences etc.; it just doesn't discuss any concrete applications in depth, as say Weibel does with e.g. group cohomology.
To read this text you should of course learn about quasicategories. I have added some references:

In a model categorical setting I don't know of any general treatment of homological algebra; in fact, I only know of one source discussing the stable theory from scratch, which is the book by Hovey.

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    $\begingroup$ The OP asks for "a reference that develops homological algebra" which I would take to mean proving theorems in homological algebra. I don't think any of the references mentioned in this answer do that. However, if the question was asking about references that develop homotopy theory in the context of chain complexes, then these references do, but most of the results in Higher Algebra were already known in the model category context, and there are many, many model category references missing from this answer. Like, everything by Jim Gillespie, cotorsion pairs, Enochs, Christensen-Hovey $\endgroup$ Dec 3, 2023 at 21:33
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    $\begingroup$ arxiv.org/abs/math/0011216, scholar.google.com/…, amazon.com/Mathematics-Edgar-E-Enochs-Science-Math/…, plus Quillen's homotopical algebra, Dwyer-Spalinski, and all the references here: ncatlab.org/nlab/show/model+structure+on+chain+complexes $\endgroup$ Dec 3, 2023 at 21:36

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