Spahn the higher derived cahiers topos (Rev #2)

The Cahiers topos is the sheaf topos on the site ThCartSp of infinitessimally thickened cartesian spaces. More generally the higher cahiers topos is the $(\infty,1)$-sheaf $(\infty,1)$-topos on the $(\infty,1)$-site ThCartSp.

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.

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.

References

• Jacob Lurie, Formal moduli problems, containing: DAGX: Formal Moduli Problems, 2011, (166 p.). And another more condensed (30 p.) version of this text titled “Moduli Problems and DG-Lie Algebras”. In particular Theorem 5.3 in the second version

• Vladimir Hinich, DG coalgebras as formal stacks, (arXiv:math/9812034