Notes characterising higher and derived stacks concretely

Abstract

This is an informal summary of the main concepts in arXiv:0905.4044, based on notes of various seminars. It gives constructions of higher and derived stacks without recourse to the extensive theory developed by Toen, Vezzosi and Lurie. Explicitly, higher stacks are described in terms of simplicial diagrams of affine schemes, which are analogous to atlases for manifolds. We also describe quasi-coherent sheaves and complexes on such objects.