Source: "What is...a derived stack?", Gabriele Vezzosi

# Higher Stacks

- Motivation: 
	- manifolds are modeled by Euclidean space, 
	- schemes are modeled by commutative rings
	- stacks are modeled by quotients of schemes by group actions
	- derived schemes are modeled by *derived commutative rings*: $\cdga$ (commutative **differential graded** algebras) or $\ss(\calg)$ (simplicial commutative algebras)
		- Idea for $\sset\calg$: a **simplicial set** which at every level is an object of $\calg$ whose face/degeneracy maps are algebra morpisms
- Use: classification problems where we want to weaken the notion of isomorphism, e.g. chain complexes with quasi-isomorphisms.
- Idea: replace a scheme $(X, \OO_X)\in \Top \times \Sh(\wait; \kalg)$ with its functor of points $h_x: \Sch \to \Set$ . Regard a set as a discrete groupoid (no cross-morphisms), and replace the target category to get $h_x: \Sch\to\Grpd$
	- Why: don't mod out by isomorphisms, keep witnesses to the isomorphisms to remember *how* things were isomorphic.
- A word about higher equivalences: filling cells.
- Motivating example: Eilenberg-MacLane stacks.
	- For $G\in\Grp\Sch\slice k$, form $\BG = K(G, 1)$ the stack classifying principal $G\dash$bundles (or $G\dash$torsors).
	- Via the nerve construction $\nerve{\wait}: \Grpd\to\sset$, view $K(G, 1): \cat{C} \to \ssets$
	- If $G\in \Ab\Grp\Sch\slice k$, can iterate this construction to get $\B^2 G = K(G, 2)$, the first higher stack.
	- Yields a notion of spheres: $S^1 = K(\ZZ, 1)$
- General definition: higher stacks form the category $[\comm\kalg, \ssets]$

# Derived Stacks
- Idea for *derived* stacks: enlarge source category
- Motivations from intersection theory:
	- $X\in \smooth\Proj\Var\slice k$ with $Z, T\leq X$, possibly singular, with $\dim Z + \dim T = \dim X$ with $\dim(Z \intersect T) = 0$
	- Recall graded dimension as a Laurent series (motivated e.g. by the Jones polynomial as a way to categorify dimension):
	$$\gr \dim_k V_\bullet = d_{V\slice k}(z) \da \sum_{n\in \ZZ} \dim_k V_n z^n \in \ZZ[z, z\inv]$$
	-  Points $p\in Z\intersect T$ have an intersection multiplicity computed by Serre's intersection formula using the local rings at $p$: 
$$
\mu_p(X, Z, T) \da \gr\dim_C \qty{ \OO_{Z, \, p} \dtensor_{\OO_{X. p}} \OO_{T, \, p} } \evalfrom_{z=1}
$$
	- Idea behind derived tensor product: $A\dtensor_R B$ is a space with $\pi_0(A\dtensor_R B) = A\tensor_R B$, but contains higher homotopical data in $\pi_{\geq 1}$. Note that $\pi_*$ reduces to $H_*$ here.
	(For experts: this is realized by a smash product of spectra.)

	- Unpacking this:
$$
\mu_{p}(X ; Z, T)\da \sum_{i\geq 0} \operatorname{dim}_{C} \operatorname{Tor}_{i}^{R}\left(\mathcal{O}_{Z, \,p}, \mathcal{O}_{T, \,p}\right) \qquad \sim \rmod, \quad R = O_{X, p}
$$
	- This can be computed using standard homological algebra: take an acyclic (flat) resolution, tensor, etc.
	- For *flat intersections*, i.e. if either entry in the Tor is a flat $R\dash$module, this collapses:
$$\mu_p(X, Z, T) = \dim_C \OO_{Z, p}\tensor_{\OO_{X, p}} \OO_{T, p}$$

- Cotangent complex: goes but to Quillen, Grothendieck, Illusie
	- Idea: for $X = \spec A$ and $A\in\calg\slice k$ in characteristic zero, take a simplicial resolution of $A$ by cdgas and apply $\Omega_{\wait\slice k}$ (algebraic/Kahler differentials) to get $\LL_X$
	- Becomes the actual tangent space of a derived scheme
- Definition of derived stacks:
$$
\der\Stacks \leq [\der \calg\slice k, \sset] = [\sset\calg\slice k, \sset] \underset{\characteristic k = 0}= [\cdga\slice k, \sset],
$$
where we take the functors that send quasi-isomorphisms to weak equivalences (preserving homotopy theories on both sides), plus descent (basically making the functors "etale sheaves").
	- Idea for site structure on source: covering families are morpisms $A\to B_i$ where $\pi_0 A\to \pi_0 B_i$ is an etale covering family and $\pi_i A \tensor_{\pi_0 A} \pi_0 B\iso \pi_i B$
	- Think of $\pi_i = H^i$ here.
	- For any derived ring $A$, get a derived spectrum $\RR\spec A$.
	- Derived stacks admit truncations $\tau_0(\RR \spec A) = \spec \pi_0 A$ which are stacks.
- Interesting application:
	- Write $\Vect_n(X)$ as the classifying stack for rank $n$ vector bundles over $X$ a smooth proper scheme.
	Then $\RR \Vect_n(\wait) = \RR\Hom(\wait, \BGL_n)_\bullet$, which recovers $\Vect_n(X)$ in degree zero.