Spahn the higher derived cahiers topos (Rev #2, changes)

Requisites

Let $Sp$ denote the ∞-category of spectra, $E_\infty Ring:=CAlg(Sp)$ the ∞-category of commutative algebra objects in $Sp$, for $R\in E_\infty Ring$ let $Mod_R(Sp)$ denote the category of $R$-module objects in $Sp$. A derived moduli problem is defined to be a functor $X:E_\infty Ring\to \infty Grpd$. For a field $k$ let $CAlg_k$ denote the coslice of $E_\infty Ring$ over $k$ and call it the ∞-category of $E_\infty$-algebras; such a $k$-algebra $A$ is called to be discrete if its homotopy groups vanish for $i\neq 0$. 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 $CAlg(Chain_k)\to CAlg(Mod_k)\simeq CAlg_k$. Aquasi-isomorphism* in $CAlg_{dg}$ is defined to be 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.