Spahn the higher derived cahiers topos (Rev #9)

Confession

The Cahiers topos being the sheaf topos on the site SDCartSp of infinitessimally thickened cartesian spaces is an important example of a differentially cohesive toppos. More generally the higher cahiers topos being the $(\infty,1)$-sheaf $(\infty,1)$-topos on the $(\infty,1)$-site 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 $CAlg_k:=k/E_\infty Ring$ 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 $k$ (see below); 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 $Moduli\hookrightarrow Fun({CAlg_k}_{sm},\infty Grpd)$ 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 $Moduli\stackrel{\sim}{\to}Lie_k^{dg}$.

Proof

We first summarize the proof for the $1$-localic case from [Schr11] Proposition 4.5.8: A covering family in $S D Cart Sp$ is define to be of the form $\{U_i\times D\stackrel{(f,id)}{\to}U\times D\}$ where $\{U_i\to U\}$ is a covering family in $Cart Sp_{smooth}$. Hence such a covering family by definition does not depend on the thickening components $D$. ?: Since all $D$ are contractible a morphism $V\to U$ is an epimorphisms iff $D\times V\to D\times U$ is an epimorphism. Thus it suffices to show that $Cart Sp_{top}$ is an $(\infty,1)$-cohesive site: $Cart Sp_{top}$ has finite products given by $\mathbb{R}^m\times \mathbb{R}^n\simeq \mathbb{R}^{m+n}$. Every object has a point $*=\mathbb{R}^0\to \mathbb{R}^n$. Let $\{U_i\to U\}_i$ be a good open covering family. This implies that the Cech nerve $\zeta(\coprod_i U_i\to U)\in [Cart Sp^{op}, s Set]$ is degree-wise a coproduct of representables. Hence the nerve theorem implies $colim \zeta(\coprod_i U_i)\stackrel{\sim}{\to} colim U=*$ is an equivalence (the statement of the nerve theorem is that $colim \zeta(U_i\to U)\simeq Sing U$ is an equivalence, our statement is then implied by the fact that $U$ as a cartesian space is contractible). Finally $lim \zeta(\coprod_i U_i)\stackrel{\sim}{\to} lim U= Cart Sp_{loc}(*,U)$ is an equivalence: The morphism $\sim$ has the right lifting property wrt. all boundary inclusions and hence it is an equivalence.

Now we define the $(\infty,1)$-site $\infty S D Cart Sp$: Let $k$ be a field. Define $E_\infty Ring:=C Alg (Sp)$, $C Alg_k:=k/E_\infty Ring$. For an associative ring $R$ let $Mod_R:=Chain_R/q-i$; i.e. the category of chain complexes of $R$-modules modulo quasi-isomorphisms. (Definition 4.4): An object $V\in Mod_k$ is called to be small if (1) For every integer $n$, the homotopy group $\pi_n(V)$ is a finite dimensional $k$-vector space. (2) $\pi_n(V)$ vanishes for $n\lt 0$ and $n\gt\gt 0$. An object $A\in CAlg_k$ is called to be small if it is small as a $k$-module spectrum and satisfies (3): The commutative ring $\pi_0 A$ has a unique maximal ideal $p$ and the composite map $k\to \pi_0 A\to \pi_0 A/p$ is an isomorphism. The full subcategory of $Mod_k$ spanned by the small $k$-module spectra is denoted by ${Mod_k}_{sm}$. The full subcategory of $C Alg_k$ spanned by the small $E_\infty$-algebras over $k$ is denoted by ${CAlg_k}_{sm}$. And we take

and we could write $\infty Inf Point\hookrightarrow\infty Smooth Alg:={CAlg_k}=k/E_\infty Ring=k/CAlg(Sp)$

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

• David Carchedi, Dmitry Roytenberg, On Theories of Superalgebras of Differentiable Functions, arXiv:1211.6134

• Lawvere et al, algebraic theories, Cambridge University Press 2010

• Martin Hyland, The category theoretic understanding of universal algebra Lawvere theories and monads, pdf