Venkatesh and collaborators have introduced a variety of objects -- the local derived Hecke algebra, the global derived Hecke algebra, and the (global) derived Galois deformation ring -- in order to explain algebraic structures in the cohomology of locally symmetric spaces. I will review this story, and then introduce a new object that we call the "spectral Hecke algebra", which is a Hecke algebra that acts on the spectral side of Langlands, i.e. on moduli spaces of Galois representations. This is joint work with Akshay Venkatesh.