Bertrand Toen, <https://arxiv.org/pdf/math/0604504.pdf>

- Examples:
	- $\Mgn(X)$ the stack of stable maps to $X\in \smooth\proj\Var\slice \CC$
		- Can take $\Mgn(X, \beta)$ for $\beta \in H^2_\sing(X; \ZZ)$ for those maps having $\beta$ as a fundamental class, categorifies quantum cohomology (take $\HoH$ or more generall $\THoH$ to recover)
	- $\Coh(X)$ the stack of coherent sheaves on a variety.
	- $K(G, n)$ Eilenberg-MacLane stacks
		- Can generalize to sheaves of groups on a scheme
		- Can define generalized cohomology $H^n(X; A) = [X, K(A, n)] = \pi_0\Map(F, K(A, n))$.
	- $S^1 = K(\ZZ, 1)$ spheres
	- Based loop stacks: for $x: X\to Y$, given by $\Loop_x Y = X \fiberpower{Y}{2}$, naturally forms a group stack.
		- Defines homotopy sheaves by iterating $\pi_n(Y; x) \da \Omega_x^{(n)}(Y)$
	- Stacks of abelian categories: for $B\in \kalg$ (with conditions), the stack $k\dash\Ab\Cat\slice B$ of $k\dash$linear abelian categories equivalent to $\mods{B}$.
	- Betti stacks: for $X\in\smooth\proj\Var\slice\CC$, the constant stack $X_B$ with values in $\Sing(X(\CC))$ (ssets of singular simplicies) and $X_{\dR}$, by Riemann-Hilbert there is an equivalence of analytic stacks $\Map(X_B, \wait)^\an \iso \Map(X_{\dR}, \wait)^\an$.
		- This is the start of nonabelian Hodge theory, the RHS is the "de Rham nonabelian cohomology" of $X$ with coefficients in $(\wait)$ (a "special" Artin stack). Admits Hodge and weight filtrations.
	- $\Vect_\bullet(X)$, the 1-stack of vector bundles over $X\in \smooth\Sch^\prop$
	- $\Pic^0(X)$ the group stack of degree 0 line bundles
- Tools in the theory:
	- Intersection theory
	- Six functor formalism
	- $\ell\dash$adic cohomology
		- Trace formulas
	- Vanishing theorem
	- Motivic cohomology
	- Riemann-Roch
	- Motivic integration
	- Virtual fundamental classes (eg for Gromov-Witten theory)
		- Can use to define Euler-characteristics, numerical invariants e.g. to count the number of (stable) sheaves with fixed data (similar to Casson invariants).
- Moduli problem: a functor $F: \cat{C}\to \Set$ where $F(X)$ should classify families of $C\in \cat{C}$ parameterized by $X$.
	- Problem: elements in a set $F(X)$ have a strict notion of equality, which is too strong. We usually want to classify up to isomorphism, equivalence, etc.
		- Example weak equivalences: quasi-isomorphism of chain complexes, weak equivalence of spaces, categories up to equivalence.
	- Problem: objects having nontrivial automorphisms makes the set of isomorphism classes ill-behaved.
	- Solution: make target a groupoid instead of a set.
- Idea: stacks over a site $\cat{C}$ are simplicial presheaves with descent.
- Artin stack: covered  by a family of affine schemes $\mcu \covers X$ where any $Y\to X$ factors through $\mcu$ (the atlas).
	Equivalently, $\mcu\covers X$ faithfully flat and locally finitely presented representable morphism.
- Application: geometric Langlands classicaly reads as a triangulated equivalence
$$
\der \mods{\mcd}\slice{\Vect_n(X)} \homotopic \der_{\coh} \Loc^n\slice X = \Map(X, \BGL_n)
$$
	- LHS: derived category of D-modules on the stack of degree $n$ vector bundles over $X$, RHS: stack of rank $n$ flat vector bundles (local systems) over $X$.
	- Hot gossip as of 2005, c/o Lafforgue: the RHS needs to be the derived stack $\RR \Loc^n\slice X$ to have a chance of being true.