Spahn the higher derived cahiers topos (Rev #3)

Apology

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.

Redemption

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 $X:E_\infty Ring\to \infty Grpd$ (There is also a notion of classical moduli problem where an instance is called to be enhanced by an associated derived moduli problem). 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 $CAlg(Chain_k)\to CAlg(Mod_k)\simeq CAlg_k$. 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

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