Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources).
The kind of course I had in mind would be a class for graduate students, with background in standard topology and algebra, with goal to develop homological algebra from the "higher" point of view (but I'd like to know of any attempt at teaching anything related, for example a "derived commutative algebra" or "derived affine schemes" course).
One could start with explaining $\infty$-categories in an axiomatic way (ie one just assumes things like limits, adjunctions etc. exist and behave the way you think they do; intuition for example coming from categories enriched in Top). Then one moves on to stable categories (and hence triangulated categories) and to the examples arising in nature. Finally, functors between these categories include the theory of derived functors and one goes through many examples there too.
One could argue this is the next logical step of a progression. Older books in homological algebra refused to use spectral sequences. Then Weibel's highly praised book does the opposite and introduces them early on, but relegates derived categories to a final chapter. Then Gelfand-Manin take it one step further and start with derived categories. They discuss dg-algebras and model categories at the very end and stop short of discussing non-abelian derived functors. Lurie's higher algebra is the next step but it's also quite big and not meant to be used for lectures (I would argue that the best place for a quick introduction to derived stuff is Lurie's thesis or Toen's notes in the simplicial operads thing).