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Notes characterising higher and derived stacks concretely

J. P. Pridham

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.

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.

Paper Structure

This paper contains 15 sections, 11 theorems, 25 equations.

Table of Contents

  1. Hypergroupoids
  2. Simplicial sets
  3. Hypergroupoids
  4. Higher stacks
  5. Morphisms and equivalences
  6. Trivial relative hypergroupoids
  7. Sheafification and morphisms
  8. Eilenberg--Mac Lane spaces and cohomology
  9. Quasi-coherent sheaves
  10. Derived stacks
  11. Derived hypergroupoids
  12. Derived Artin stacks
  13. Morphisms and equivalences
  14. Quasi-coherent complexes
  15. Derived schemes and algebraic spaces

Key Result

Theorem \oldthetheorem

If $X$ is an Artin $n$-hypergroupoid $X$ over $R$, then its hypersheafification $X^{\sharp}\colon \mathrm{Alg}_R \to s\mathrm{Set}$ is an $n$-geometric Artin stack in the sense of hag2. Every $n$-geometric Artin stack arises in this way.

Theorems & Definitions (35)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • ...and 25 more