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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.
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.
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 $Moduli\stackrel{\sim}{\to}Lie_k^{dg}$.
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 $\infty Inf Point\hookrightarrow\infty Smooth Alg:={CAlg_k}=k/E_\infty Ring=k/CAlg(Sp)=CAlg(Mod_k)$ (where $CAlg_k=CAlg(Mod_k)$ by Remark 4.3). We define
meaning the semi-direct product of categories: there is a functor $Mod_k\ltimes \infty Inf Point\to Mod_k$ which we identify with the
(…)
Semidirect product of categories are described in (Kock p.12). In more modern terminology and generalized to quasicategories this reads: Let $D$ be a category which is (left) tensored over $C$. Then the category $D\ltimes C$ is defined to have as objects pair $(d,c)\in D\times C$ and a morphism $(d_1,c_1)\to (d_2,c_2)$ is defined to be a pair $(f:d_1\to d_2\coprod c_1,\phi:c_2\to c_1)$. Composition of this morphism with $(g:d_2\to d_3\coprod c_2,\gamma:c_3\to c_2)$ is defined to be the pair
The identity morphism in $D\ltimes C$ is defined to be $(d\simeq I\stackrel{d\coprod I}{\to} d\coprod c, id_c)$ where $I$ denotes the initial object of $C$. There is a (full) embedding $D\hookrightarrow D\ltimes C$ given by $d\mapsto (d,I)$ and $f\mapsto (f,id_I)$. This embedding preserves all limits which are preserved by all ${-}\coprod c$. If $D$ has exponential object which are preserved by all $(-)\coproduct c$ (in that $y^x\coprod c\simeq (y\coprod c)^x$) and if $(-)\coprod c$ preserves finite products, then the embedding preserves exponential objects.
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.
Anders Kock, Convenient vector spaces embed into the Cahiers topos, web