# 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?