Presenting higher stacks as simplicial schemes

TL;DR

This paper demonstrates that higher geometric stacks can be represented as simplicial schemes called Duskin n-hypergroupoids, unifying various notions of stacks and extending to derived contexts, with applications to deformation theory.

Contribution

It introduces a new perspective by representing n-geometric stacks as Duskin n-hypergroupoids, enabling a unified approach across different stack theories and derived settings.

Findings

Higher stacks are equivalent to Duskin n-hypergroupoids in affine schemes.

The formulation applies to derived stacks using simplicial rings.

The cotangent complex governs infinitesimal deformations of higher and derived stacks.

Abstract

We show that an n-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n-stacks, Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived n-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.