# Derived Algebraic Geometry

> This is a summary of lectures given by Benjamin Antieau.

The goal of this note is to describe the following result: given a differential graded category $\cat C$, one can attempt to construct a moduli space $\mcm_{\cat C}$ of objects in $\cat C$. A result of [?] shows that the corresponding moduli functor can be represented by a higher stack.

Throughout this note, all sheaves will be assumed étale, categories $\cat C$ will all be regarded as $\infty\dash$categories, and $\cat C \homotopic \cat D$ denotes an *equivalence* of $\infty\dash$categories. 

Let $\mcs$ be the category of homotopy types, which we'll refer to as *spaces*, and let $\tau_{\leq n} \mcs$ be the category of $n\dash$truncated spaces -- those $X$ for which $\pi_k(X) = 0$ for $k\geq n+1$.
Note that $\Set \homotopic \tau_{\leq 0} \mcs$, regarding a set as a discrete space, and $\Grpd \homotopic \tau_{\leq 1} \mcs$, since a groupoid has trivial homotopy groups in degrees larger than 1.

We have the following analogies: 

- Schemes and algebraic spaces such as $\PP^n$ corresponds to sheaves of sets, i.e. 0-truncated spaces.
- Classical stacks, in particular Artin and Deligne-Mumford stacks, correspond to sheaves to groupoids, i.e. 1-truncated spaces.
- Higher stacks such as the Eilenberg-Maclane stack $K(\GG_m, n)$, which will be sheaves of $n\dash$truncated spaces.

More generally, a higher stack can be regarded as an etale sheaf taking values in $\mcs$.

Recall that the classifying stack of a group $G$ is given by the stack quotient $\BG = [\pt/G]$, the delooping of $G$.
One can further deloop to obtain $\B^2 G \da \B (\BG) = [\pt / \BG]$, and proceed inductively to define $\B^n G \da [\pt/ \B^{n-1} G]$ for any $n\geq 1$.
In this way, we define the Eilenberg-Maclane higher stacks $K(G, n) \da \B^n G$, which corepresent etale cohomology in the following precise sense: noting that the category $\St$ of higher stacks will be enriched over $\mcs$, we will write $\Map(X, Y) \da \Hom_{\St}(X, Y)$ when we regard this as a space. In particular, these mapping spaces have homotopy groups, and one can show 
\[
\pi_i \Map(K(G, n), X) = 
\begin{cases}
H^{n-i}_{\et}(X; G) & 0\leq i \leq n 
\\
0 & \text{otherwise}.
\end{cases}
\]

To set up this theory, we first describe a category of derived *affine* schemes, $\der\Aff\Sch$. One such example is given by $\spec \qty{ k \tensor_{k[x]}^{\leftderive} k}$, which satisfies
\[
H^*\qty{ \spec \qty{ k \tensor_{k[x]}^{\leftderive}  k}} \cong \Tor_*^{k[x]}(k, k) \cong k[1] \oplus \Tor_1^{k[x]}(k,k)[2]
.\]

To form such a category, the essential idea is to consider commutative algebra objects in the monoidal category $(\D(\zmod), \tensor^{\leftderive})$, the derived category of $\ZZ\dash$modules with the (derived) tensor product.
Generally one has several ways to model such categories: $\EE_\infty\dash$ring spectra, simplicial commutative rings, or differential graded algebras if working over $\QQ$.

Let $\Delta$ be the simplex category and note that the simplices $\Delta^n \da \Hom_\Delta(\wait, [n])$ generate $\Delta$. 
For a category $\cat C$, let $s\cat C \da \Fun(\opcat{\Delta}, \cat C)$, the $\cat C\dash$valued co-presheaves on $\Delta$, be the simplicial objects in $\cat C$. It is a fact that $s\Set \homotopic \Top$ as $\infty\dash$categories, which should be thought of as having the same homotopy theories.
Recall that there is a functor $\Sing(\wait): \Top \to \Ch(\ZZ)$, where $\Sing(X)$ is the chain complex of abelian groups $C^\bullet_{\Sing}(X)$.
One recovers $H^*_\Sing(X; \ZZ) \cong H^*(C^\bullet_{\Sing}(X))$, the usual homology of this chain complex.
The Dold-Kan correspondence gives an equivalence $s\zmod \homotopic \tau_{\geq 0}\derivedcat(\zmod)$, where the latter are the connective chain complexes. Under this correspondence, one has

\[\begin{tikzcd}
	&&&& s\zmod && {\tau_{\geq 0}\derivedcat(\zmod)} \\
	{M_\bullet =} & {M_0} & {M_1} & {M_2} & \cdots & \mapsto & {\tilde M_\bullet = } & {M_0} & {M_1} & {M_2} & \cdots \\
	&&&& {\pi_* (M_\bullet)} & \mapsto & {H^*(\tilde M_\bullet)}
	\arrow[shift right=3, from=2-3, to=2-2]
	\arrow[shift left=3, from=2-3, to=2-2]
	\arrow[shift right=4, from=2-4, to=2-3]
	\arrow[shift left=4, from=2-4, to=2-3]
	\arrow[from=2-4, to=2-3]
	\arrow[from=2-2, to=2-3]
	\arrow[shift right=2, from=2-3, to=2-4]
	\arrow[shift left=2, from=2-3, to=2-4]
	\arrow["{\small\sum  (-1)^i \del_i}"' {yshift=5pt}, from=2-10, to=2-9]
	\arrow["{\small\sum  (-1)^i \del_i}"' {yshift=5pt}, from=2-9, to=2-8]
	\arrow[from=1-5, to=1-7]
	\arrow[from=2-11, to=2-10]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=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)