Recent advances in the theory of smooth mod $p$ representations of a $p$-adic reductive group $G$ involve more and more derived methods.  It becomes increasingly clear that the proper framework to study smooth mod $p$ representations is the derived category $D(G)$.

I will talk about smooth mod $p$ representations and highlight their shortcomings compared to, say, smooth complex representations of $G$.  After explaining how the situation improves in the derived category, I will spend the remaining time on the left adjoint of the derived parabolic induction functor.