Unifying derived deformation theories

We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L_∞-algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toën-Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as André-Quillen cohomology.