Spahn the higher derived cahiers topos (Rev #3, changes)

However the $(\infty,1)$-topos arising in this way is (still) a 1-localic (i.e. localic) $(\infty,1)$-topos; in other words this notion of higher cahiers topos is no more intelligible than just the classical Cahiers topos.

Redemption

An object of the symmetric monoidal (by the usual tensor product) category $Chain_k$ of chain complexes over $k$ is called a commutative differential graded algebra over $k$. There are functors $Chain_k\to Mod_k$ and $CAlg(Chain_k)\to CAlg(Mod_k)\simeq CAlg_k$. A quasi-isomorphism in $CAlg_{dg}$ is defined to be a morphism inducing an isomorphism between the underlying chain complexes. There is a notion of smallness for $k$-module spectra and $E_\infty$-algebras over $k$; the corresponding full sub ∞-categories are denoted by ${Mod_k}_sm$ resp. ${CAlg_k}_sm$. A formal moduli problem over $k$ is defined to be a functor $X:{CAlg_k}_{sm}\to \infty Grpd$ such that $X(k)$ is contractible and $X$ preserves pullbacks of maps inducing epimorphisms between the $0$-th homotopy groups.

The (Grothendieck) tangent space of a formal moduli problem $X:{CAlg_k}_{sm}\to \infty Grpd$ is defined to be a map $T_X(0):=X(k[\epsilon]/\epsilon^2)\to X(k)$. $T_X(0)\in \infty Grpd$ is a topological space. Define $T_X(n):=X(k\otimes k[n])$ where $k[n]$ denotes the $n$-fold shift of $k$ (as a $k$-module spectrum). One can elaborate that $T_X(n-1)$ is the loop space of $T_X(n)$; define the tangent complex of the formal moduli problem $X$ to be the sequence $T_X:=(T_X(n))_{n\ge 0}$; $T_X$ is a $k$-module spectrum. The operation $T_{(-)}$ reflects equivalences.

Let $k$ be a field of characteristic zero. A differential graded Lie algebra over $k$ is defined to be a Lie algebra object in $Chain_k$: a chain complex $g$ equipped with a binary operation $[-;-]:g\otimes g\to g$ such that $[x,y]+(-1)^{d(x)d(y)}[y,x]=0$ and $(-1)^{d(z)d(x)}[x,[y,z]]+(-1)^{d(x)d(y)}[y,[z,x]] + (-1)^{d(y)d(x)}[z,[x,y]]=0$ for homogenous elements $x\in g_{d(x)},y\in g_{d(y)},z\in g_{d(z)}$. The category of differential graded Lie algebras over $k$ localized at quasi-isomorphisms is denoted by $Lie_k^{dg}$ and just also called the category of differential graded Lie algebras over $k$.

(Theorem 5.3): Let $k$ be a field of characteristic zero, let $Moduli\hookrightarrow Fun({CAlg_k}_{sm}$ the full subcategory spanned by formal moduli problems over $k$, let $Lie_k^{dg}$ denotes the ∞-category of differential graded Lie algebras over $k$. Then there is an equivalence $Moduli\stackrel{\sim}{\to}Lie_k^{dg}$12