Presenting higher stacks as simplicial schemes

We show that an nn-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin nn-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin nn-stacks, Deligne–Mumford nn-stacks and nn-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived nn-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich’s dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.