Stacks and Algebraic Stacks

Reading Course at Rutgers University, Fall 2018. Fibered categories, stacks and the theory of descent, algebraic spaces and algebraic stacks.

Macaca nigra self-portrait
Danny Krashen
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Grothendieck Topologies

  • Definitions of Grothendieck topologies and sheaves on them (Tamme, Sections I.1, I.2)
  • Universally effective epimorphisms and the canonical topology (Tamme, Section I.3.1)
  • Examples: G-sets, discrete and profinite cases (Tamme, Sections I.3.2, I.1.3.3)
  • Sheafification of presheaves on Grothendieck topologies

Overview of goals

  • Examples of stacks: Topological sheaves and their gluings, bundles, modules
  • The language of pseudo-functors
  • The language of fibered categories
  • (Some) 2-Yoneda lemma(s)
  • Topological descent and topological stacks
  • Grothendieck topologies, sites and topoi
  • Descent and stacks
  • Algebraic geometry: functors of points and the Yoneda lemma
  • Algebraic geometry: etale morphisms, formally smooth morphisms
  • Algebraic geometry: the etale topology
  • Algebraic spaces
  • Algebraic stacks

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