Spahn the higher derived cahiers topos (Rev #4, changes)

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Apology Confession

The Cahiers topos is being the sheaf topos on thesite ~~ThCartSp~~ SDCartSp of infinitessimally thickened cartesian spaces . is an important example of a differentially cohesive toppos. More generally thehigher cahiers topos is being the $(\infty,1)$ -sheaf $(\infty,1)$ -topos on the $(\infty,1)$ -site ~~ThCartSp~~ SDCartSp . is an important example of a differentially cohesive $(\infty,1)$ -topos.

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. In fact there hasn’t been described any example of a differentially cohesive non-localic $(\infty,1)$ -topos.

Apology and Redemption

What is $S D Cart Sp$ ? The construction is a follows: Let $T:=Cart Sp_{smooth}$ denote the opposite of the category of cartesian spaces of finite dimension (as $\mathbb{R}$ -vector spaces, so these are essentially of the form $\mathbb{R}^n$ ). $T$ is the syntactic category of the Lawvere theory of smooth algebras. Define $Inf Point\hookrightarrow T Alg^{op}$ to be the subcategory of Weil algebras; i.e. the subcategory on those objects having as vector space at least dimension $1$ and which are nilpotent as algebras. Then $S D Cart Sp$ is defined to be the category of objects being of the form a product $\mathbb{R}^n\times D$ with $D\in Inf Point$ and $n\in \mathbb{N}$ .

By substituting into this receipt $\infty Inf Point:={{C Alg_k}_{sm}}^{op}$ (see below) for $Inf Point$ we obtain the notion of higher derived Cahiers topos which is not $n$ -localic for any $n\lt\infty$ . In the following shall be argued that it is differentially cohesive.

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 $Moduli ↪ Fun ({CAlg}_{k}_{sm}, ∞ Grpd)$ Moduli\hookrightarrow ~~Fun({CAlg_k}_{sm}~~ Fun({CAlg_k}_{sm},\infty Grpd) 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

Proof

We summarize the proof for the $1$ -localic case from [Schr11] Proposition 4.5.8: A covering family in

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

Urs Schreiber, Differential cohomology in a cohesive $(\infty,1)$ -topos

Revision on February 13, 2013 at 08:12:25 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.