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.