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\begin{document}

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\section*{dg-geometry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_toposes}{The $(\infty,1)$-toposes}\dotfill \pageref*{the_toposes} \linebreak
\noindent\hyperlink{over_formal_duals_of_nonpositively_graded_cdgalgebras}{Over formal duals of non-positively graded cdg-algebras}\dotfill \pageref*{over_formal_duals_of_nonpositively_graded_cdgalgebras} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{over_formal_duals_of_general_cdgalgebras}{Over formal duals of general cdg-algebras}\dotfill \pageref*{over_formal_duals_of_general_cdgalgebras} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

This entry describes the [[higher geometry]]/[[derived geometry]] modeled on [[(∞,1)-site]]s of formal duals of [[dg-algebra]]s, bounded or unbounded, over a [[field]] of [[characteristic]] 0.

The corresponding [[(∞,1)-topos]] is the context for classical [[rational homotopy theory]], which arises by forming [[function algebras on ∞-stacks]] over [[constant ∞-stack]]s. It is also the context in which classical and higher order [[Hochschild homology]] of algebras and dg-algebras arises naturally as the function $\infty$-algebra on [[free loop space object]]s.

\hypertarget{the_toposes}{}\subsection*{{The $(\infty,1)$-toposes}}\label{the_toposes}

We discuss some basic aspects of the [[(∞,1)-topos]]es over [[(∞,1)-site]]s of formal duals of cdg-algebras and of cdg-algebras of functions on its objects.

\hypertarget{over_formal_duals_of_nonpositively_graded_cdgalgebras}{}\subsubsection*{{Over formal duals of non-positively graded cdg-algebras}}\label{over_formal_duals_of_nonpositively_graded_cdgalgebras}

Let $k$ be a [[field]] of [[characteristic]] 0, or more generally a commutative $\mathbb{Q}$-[[associative algebra|algebra]].

\hypertarget{definition}{}\paragraph*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{itemize}%
\item $cdgAlg_k$ for the category of graded-commutative cochain [[dg-algebra]]s (meaning: with [[differential]] of degree +1) in arbitrary degree;


\item $cdgAlk_k^-$ for the full [[subcategory]] on objects with vanishing [[cochain cohomology]] in positive degree, $H^{\bullet \geq 1}(-) = 0$.



\end{itemize}
There are the standard projective [[model structures on dg-algebras]] on these categories, whose weak equivalences are the [[quasi-isomorphism]]s and whose fibrations are the degreewise surjections.

\end{defn}
This is considered in (\hyperlink{ToenVezzosiStacks}{To\"e{}n-Vezzosi, 2.3.1})

\begin{defn}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
C \hookrightarrow ((cdgAlg_k^-)^{op})^\circ
\end{displaymath}
be a [[small (∞,1)-category|small]] full [[sub-(∞,1)-category]] of the [[(∞,1)-category]] [[presentable (∞,1)-category|presented]] by this model structure, and let $C$ be equipped with the structure of a [[subcanonical coverage|subcanonical]] [[(∞,1)-site]].

Write

\begin{displaymath}
\mathbf{H} := Sh_{(\infty,1)}(C)
\end{displaymath}
for the [[(∞,1)-category of (∞,1)-sheaves]] on $C$. We have a derived [[Isbell duality]]

\begin{displaymath}
(\mathcal{O} \dashv j) 
    : 
  (cdgAlg_k^{op})^\circ
   \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}}
  \mathbf{H}
\end{displaymath}
where the [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\mathcal{O}$ is the [[Yoneda extension]] of the inclusion $cdgAlg^+_k \hookrightarrow cdgAlg_k$.

\end{defn}
This is considered in (\hyperlink{Ben-ZviNadler}{Ben-ZviNadler}). See [[function algebras on ∞-stacks]] for details.

\hypertarget{properties}{}\paragraph*{{Properties}}\label{properties}

\begin{uprop}
The inclusion

\begin{displaymath}
cdgAlg^-_k \hookrightarrow cdgAlg_k
\end{displaymath}
is a \emph{homotopical context} in the sense of (\hyperlink{ToenVezzosiStacks}{To\"e{}nVezzosi, def. 1.1.0.11}).

\end{uprop}
This is (\hyperlink{ToenVezzosiStacks}{To\"e{}nVezzosi, lemma 2.3.11}). We record the following implications of this statement

\begin{cor}
\label{}\hypertarget{}{}
$(cdgAlg_k, \otimes_k)$ is a [[symmetric monoidal category|symmetric]] [[monoidal model category]].

\end{cor}
\begin{cor}
\label{}\hypertarget{}{}
For $B \in (dgcAlg_k)_{proj}$ a cofibrant object, the [[tensor product]] with $B$ preserves weak equivalences.

\end{cor}
This follows from (\hyperlink{ToenVezzosiStacks}{To\"e{}nVezzosi, assumption 1.1.0.4}).

\begin{cor}
\label{cdgAlgInclusionPreservesLimits}\hypertarget{cdgAlgInclusionPreservesLimits}{}
The inclusion

\begin{displaymath}
(cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}
\end{displaymath}
preserves [[homotopy limit]]s, hence the induced inclusion

\begin{displaymath}
((cdgAlg_k^-)^{op})^\circ \hookrightarrow ((cdgAlg_k)^{op})^\circ
\end{displaymath}
preserves [[(∞,1)-limit]]s.

\end{cor}
This follows from (\hyperlink{ToenVezzosiStacks}{To\"e{}nVezzosi, assumption 1.1.0.6}).

\begin{defn}
\label{}\hypertarget{}{}
For $A \in cdgAlg_k$ a [[nLab:dg-algebra]], write

\begin{itemize}%
\item $A Mod$ for its [[nLab:model structure on dg-modules|category of dg-modules]];

This is naturally a [[nLab:symmetric monoidal category]].


\item $cdgAlg_A := CMon(A Mod)$ for the [[category of commutative monoids]] in $A Mod$, the category of \textbf{cdg-$A$-algebras}.



\end{itemize}
\end{defn}
\begin{cor}
\label{ModelStructureOndgAAlgebras}\hypertarget{ModelStructureOndgAAlgebras}{}
For any $A\in cdgAlg_k$ say a morphism in $cdgAlg_A$ is

\begin{itemize}%
\item a weak equivalence precisely if it is a [[nLab:quasi-isomorphism]];


\item a fibration precisely if it is degreewise surjective.



\end{itemize}
This makes $cdgAlg_A$ into a [[model category]] that is

\begin{itemize}%
\item [[combinatorial model category|combinatorial]];


\item [[proper model category|proper]].



\end{itemize}
There is an [[equivalence of categories]] with the [[under category]] of cdg-algebras under $A$

\begin{displaymath}
cdgAlg_A \simeq A/cdgAlg_k
\end{displaymath}
which is a [[Quillen equivalence]] with respect to the standard [[model structure on an under category]] on the right.

\end{cor}
This is (\hyperlink{ToenVezzosiStacks}{To\"e{}nVezzosi, assumption 1.1.0.4, remark on p. 18}).

\begin{cor}
\label{}\hypertarget{}{}
For $B \in cdgAlg_A$ cofibrant with respect to the model structure in cor \ref{ModelStructureOndgAAlgebras}, the [[tensor product]] ([[base change]]) functor

\begin{displaymath}
B \otimes_A (-) : A Mod \to B Mod
\end{displaymath}
preserves weak equivalences.

\end{cor}
This is (\hyperlink{ToenVezzosiStacks}{To\"e{}nVezzosi, assumption 1.1.0.4}).

\begin{prop}
\label{}\hypertarget{}{}
The [[monoidal Dold-Kan correspondence]] provides a [[Quillen equivalence]]

\begin{displaymath}
(\Gamma^{cmon} \dashv N_\bullet)
  : 
  cAlg_k^{\Delta^{op}}  
    \stackrel{\overset{\Gamma^{cmon}}{\leftarrow}}{\underset{N_\bullet}{\to}}
  cdgAlg_k^+
\end{displaymath}
(since $k$ is assumed to be of characteristic 0). Under this equivalence we have that $U \in cAlg_k \hookrightarrow cAlg_k^{\Delta^{op}} \hookrightarrow \mathbf{H}$ is $\mathcal{O}$-perfect:

\begin{displaymath}
\mathcal{O} (X^{K}) \simeq K \cdot \mathcal{O}(X)
\end{displaymath}
and this recovers the constructions discussed above in \hyperlink{HochschildChainComplex}{The Hochschild chain complex of an associative algebra}.

\end{prop}
\begin{proof}
Since the [[(∞,1)-Yoneda embedding]] $y$ commutes with [[(∞,1)-limit]]s we have that the powering $(y(U))^{K} \simeq y(U^K)$ is still representable. Therefore

\begin{displaymath}
\mathcal{O} ((y(U))^K) \simeq \mathcal{O}(U^K) \;\; 
  \in (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}
\end{displaymath}
is simply the formal dual of $U^K$, which is $K \cdot \mathcal{O}(U)$ formed in $cdgAlg_k$ by formal duality. By the \href{cdgAlgInclusionPreservesLimits}{above proposition} the inclusion $cdgAlg_k^- \hookrightarrow cdgAlg_k$ preserves this $(\infty,1)$-colimit.

\end{proof}
\hypertarget{over_formal_duals_of_general_cdgalgebras}{}\subsubsection*{{Over formal duals of general cdg-algebras}}\label{over_formal_duals_of_general_cdgalgebras}

(\ldots{})

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item [[dg-manifold]], [[dg-scheme]]


\item [[Hochschild cohomology]]: section ;


\item [[rational homotopy theory]]: [[function algebras on ∞-stacks]]


\item [[perfect ∞-stack]]s and their [[geometric ∞-function theory]]


\item [[BRST-BV complex]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Various [[model category]] presentations of dg-geometry are presented in

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Homotopical Algebraic Geometry II: geometric stacks and applications} (\href{http://arxiv.org/abs/math/0404373}{arXiv})

\end{itemize}
The [[geometric ∞-function theory]] of [[perfect ∞-stack]]s in dg-geometry, and the corresponding [[Hochschild cohomology]] is considered in

\begin{itemize}%
\item [[David Ben-Zvi|Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{Integral transforms and Drinfeld centers in derived algebraic geometry} (\href{http://arxiv.org/abs/0805.0157}{arXiv})

\end{itemize}
The $(\mathcal{O} \dashv Spec)$-adjunction for dg-geometry is studied in

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], \emph{Loop Spaces and Connections} (\href{http://arxiv.org/abs/1002.3636}{arXiv:1002.3636})

\end{itemize}
The basic reference for the [[model structure on dg-algebras]] (see there for more details) for the commutative case over a field of characteristic 0 is

\begin{itemize}%
\item [[Vladimir Hinich]], \emph{Homological algebra of homotopy algebras} Communications in algebra, 25(10). 3291-3323 (1997)(\href{http://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}, \emph{Erratum} (\href{http://arxiv.org/abs/math/0309453}{arXiv:math/0309453}))

\end{itemize}
Details on the use of this [[model category]] structure for modelling dg-spaces are in

\begin{itemize}%
\item [[Kai Behrend]], \emph{Differential graded schemes I: prefect resolving algebras} (\href{http://arxiv.org/abs/math/0212225}{arXiv:0212225})


\item [[Kai Behrend]], \emph{Differential Graded Schemes II: The 2-category of Differential Graded Schemes} (\href{http://arxiv.org/abs/math/0212226}{arXiv:0212226})



\end{itemize}


\end{document}