Presenting higher stacks as simplicial schemes

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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.
Original languageEnglish
Pages (from-to)184-245
Number of pages62
JournalAdvances in Mathematics
Issue numbern/a
Publication statusPublished - 1 May 2013


  • Derived algebraic geometry
  • Higher stacks
  • Simplicial schemes


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