I was recently reading through Dwyer and Spalinski's notes introducing model categories. In it is shown a different way to describe $Ext(A,B)$ in homotopical terms (see proposition 7.3, I am not repeting it here, as it isn't relevant, just to explain where I am coming from).
I am wondering if there are any other applications of homotopical algbera to homological algbera. I know that these two subjects are closely related, perhaps in part via their common ancestry in algebraic topology. Any answer is welcome, book recommendations, your personal favorite results or philosophical rambling.
Note: I have seen some similar questions on this site, but they don't quite satisfy my itch. The big idea I heard is that homotopical algebra generalizes homological algebra. I am looking for theorems or more generally ideas which belong to clasical homological algebra and admit an elegant proof or enlightening reformulation with modern ideas from homotopical algebra.
I understand there is some subjectivity in what is "classical", what is "elegant" or what is "enlightening", hence my usage of the [soft-question] tag.
Thank you for your time.
One example: the classification of thick subcategories of the derived category of a commutative ring, first stated by Hopkins in his paper "Global methods in homotopy theory" and then clarified by Neeman ("The chromatic tower for $D(R)$" and improved by Thomason ("The classification of triangulated subcategories"). The idea of trying to classify these subcategories was motivated by work in homotopy theory, and the idea of the proof, at least for Hopkins, came from the homotopy theoretic version.
Of course even before you get to results like that, there is the very construction of the derived category in the first place. It is obtained by taking a certain category of chain complexes and inverting the quasi-isomorphisms. Quillen's work on homotopical algebra provides a technique for inverting a collection of morphisms in a category in a controlled way, but some early descriptions of the derived category yield hom sets that are not obviously sets — they may be too large — but Quillen's approach avoids this issue.