Motivation: Intersection Theory

Why Stacks

What is a moduli problem?

  • Motivation from topology: classifying spaces for vector bundles over a fixed space.
  • What are some common moduli problems?
    • Vector bundles on a fixed space/curve.
    • Complex structures on a Riemann surface \(\Sigma\) up to homeos isotopic to the identity.
      • Equivalently, marked hyperbolic structures on \(\Sigma\) where \(m\in {\operatorname{Homeo}}(\Sigma)\) mod isotopy.
    • Donaldson-Thomas theory: flat connections on a vector bundle up to gauge equivalence.
    • Gromov-Witten theory: all curves of a fixed genus, e.g. moduli of elliptic curves.
    • Floer theory: pseudo-holomorphic discs, e.g. with Lagrangian boundary conditions

What is a stack?

  • What are some examples?
    • \({ \mathcal{M}_{g, n} }(X)\)
    • \({\mathbf{B}}G = K(G, 1)\)
    • \({\operatorname{Pic}}^0(X)\).
    • \({\mathsf{Bun}}_G(X)\)
    • \({ \mathsf{Vect} }_n(X) = {\mathsf{Bun}}_{\operatorname{GL}_n}(X)\)
    • \({\mathbb{P}}^n = [{\mathbb{A}}^{n+1}/ {\mathbb{G}}_m]\).
    • \({\mathsf{Coh}}(X)\) and \({\mathsf{QCoh}}(X)\).
    • \(\mathsf{Loc}\Sys(X) \cong {\mathsf{Rep}}(\pi_1 X) \cong { \mathsf{Vect} }_n^\flat(X)\).

Issues with schemes and sets

  • Why not just schemes?
    • Existence of universal families
    • Quotients of schemes by group actions (orbit spaces)
    • Representability of moduli functors.
  • Why groupoids?

Why Infinity Categories

  • What is an infinity groupoid?
  • What is the homotopy hypothesis?
  • What is a simplicial set?
  • What is the nonabelian derived category?
  • What is the nerve?
  • What is an infinity category?
    • What is an equivalence of infinity categories?

Why “derived”/higher stacks

  • What does “derived” mean?
  • When is a derived stack strictly necessary?
  • What are some examples of higher stacks?

What problems does this solve?

  • Geometric Langlands
  • What tools are available in the theory?