Homological algebra (homotopical approach)

Question

I have gone through a couple of courses in homological algebra, in the context of derived functors, abelian categories,... Now I would like to watch it from another perspective: my main interest is homotopy theory, and it is quite obvious that almost the entire theory and its tools have a homotopy-theoretic core (not only from the lexical point of view, which imitates that of algebraic topology using the link offered by singular (co)homology). Here comes the question: will you please suggest me a book which heavily uses model structures, derived functors (hopefully clarifying the link between the "naive" notion in homological algebra and the one used in homotopy theory) and so on? Thank you in advance!

Answer 1

A quick introduction to the model category-theoretic method in homological algebra can be found in Goerss and Schemmerhorn's "Model Categories and Simplicial Methods". They treat resolutions in the nonabelian setting with the language of model categories.

A central tool in abstract homotopy theory is the simplicial set, so a good warmup if you've not seen these before is Chapter 8 of Weibel's book "An Introduction to Homological Algebra". Here, the author uses simplicial methods to derive various constructions in abelian categories, which are concrete and you can make calculations easily with them. Applying simplicial resolutions can also get you derived functors of more general functors than additive and Tierny and Vogel's paper "Simplicial Derived Functors" may be useful.

There are many places you could go after this. I suggest also you have a specific application in mind so you can see much of this theory in concrete examples. To see how simplicial methods are used in a slightly less abstract setting, Waldhausen's "Algebraic K Theory of Spaces" might be up your alley, and this framework was eventually used by Rognes to prove that the finitely generated abelian group $K_4(\mathbb{Z})$ is trivial, which shows that abstract homotopy theory does indeed lead to calculations.

For a more advanced viewpoint, a very elegant book is Riehl's "Categorical Homotopy Theory" available on her website. Clarifying the connection between derived functors and the essential structure on a category (weak equivalences) needed to define them is on theme in this book.