I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material covered was the bare bone basics of derived functors as universal $\delta$-functors, projective, injective, and acyclic resolutions, and nice applications to algebraic geometry, group cohomology, and sheaf cohomology.
I want to show the relevance of homotopical algebra to homological algebra. However, the lecturer is not too interested in abstract nonsense, so my original idea of showing the general definition of derived functors as 'homotopical approximations' in homotopical categories is not, by itself, a very promising option.
What I'm looking for are simple "applications" of homotopical ideas to homological algebra, which are elementary in the sense of not requiring too many deep results in homotopy theory and algebraic topology (categorical machinery is not constrained though). Things such as spectra, stable/unstable homotopy theory, etc are to be avoided. By applications I do not necessarily mean explicit computations, by the way.
Understandably, people may be tempted to say the homological algebra is just a special case of homotopical algebra, e.g as in Adeel Khan's comment on this question, and also delve into the realm of $\infty$-categories/topoi, but these ideas, which I feel are huge conceptual advances, are far beyond my level and time limit. I'm really looking for "basic" stuff.
Regretably, my question is ill-posed, but I feel this is inevitable.
The application that convinced me I should care about homotopical ideas is an answer to the following question:
My answer is at this MO question.