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\section*{complex analytic ∞-groupoid}

\begin{quote}%
under construction


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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

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

[[!include higher geometry - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{analytic_geometry}{}\paragraph*{{Analytic geometry}}\label{analytic_geometry}

[[!include analytic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Cohesion}{Cohesion}\dotfill \pageref*{Cohesion} \linebreak
\noindent\hyperlink{OkaPrinciple}{Oka principle}\dotfill \pageref*{OkaPrinciple} \linebreak
\noindent\hyperlink{DifferentialCohomology}{Complex analytic differential generalized cohomology}\dotfill \pageref*{DifferentialCohomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{multiplicative_group_and_holomorphic_line_bundles}{Multiplicative group and holomorphic line $n$-bundles}\dotfill \pageref*{multiplicative_group_and_holomorphic_line_bundles} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{complex analytic $\infty$-groupoid} is an [[∞-groupoid]] equipped with [[higher geometry|geometric]] structure in the sense of [[complex analytic geometry]], such that [[complex analytic spaces]] constitute a [[full subcategory]] of the [[n-truncated object in an (∞,1)-topos|0-truncated]] complex analytic $\infty$-groupoids. Hence a complex analytic $\infty$-groupoid is an [[(∞,1)-sheaf]]/[[∞-stack]] on the [[site]] of [[complex manifolds]] (or some of its [[dense subsites]]). This is directly analogous to how [[(∞,1)-sheaves]] over the [[site]] of [[smooth manifolds]] may be regarded as \emph{[[smooth ∞-groupoids]]}.

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

\begin{defn}
\label{TheSites}\hypertarget{TheSites}{}
Write $CplxMfd$ for the [[category]] of [[complex manifolds]], regarded as a [[site]] with the standard [[Grothendieck topology]].

Write

\begin{displaymath}
\mathbb{C}Disc \hookrightarrow SteinSp \hookrightarrow CplxMfd
\end{displaymath}
for the [[full subcategories]] of [[Stein spaces]] and of complex [[polydiscs]], respectively, regarded as [[sites]] by equipping them with the induced [[coverages]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The inclusions in def. \ref{TheSites} exhibit [[dense subsite]] inclusions.

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

\begin{displaymath}
\mathbb{C}Analytic\infty Grpd \coloneqq Sh_\infty(CplxMfd)
\end{displaymath}
for the [[hypercomplete (∞,1)-topos|hypercomplete]] [[(∞,1)-sheaf (∞,1)-topos]] over the sites of complex manifolds def. \ref{TheSites}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[model structure on simplicial presheaves]]} this means that $\mathbb{C}Analytic\infty Grpd$ is equivalently the [[simplicial localization]] of any of the hypercomplete local model structures on simplicial (pre-)sheaves, such as the Joyal [[model structure on simplicial sheaves]].

\end{remark}
This is considered in (\hyperlink{HopkinsQuick12}{Hopkins-Quick 12, section 2.1}).

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

\hypertarget{Cohesion}{}\subsubsection*{{Cohesion}}\label{Cohesion}

\begin{prop}
\label{Cohesion}\hypertarget{Cohesion}{}
The [[global section geometric morphism]]

\begin{displaymath}
\Gamma
  \;\colon\;
  \mathbb{C}Analytic\infty Grpd
  \longrightarrow
  \infty Grpd
\end{displaymath}
exhibits a [[cohesive (∞,1)-topos]].

\end{prop}
\begin{proof}
We discuss the existence of the extra left adjoint $\Pi$ (the [[shape modality]]). This proceeds essentially as in the discussion of the [[cohesion]] of [[Smooth∞Grpd]] (see there) only that where there one may choose [[good open covers]] here we have to choose genuine [[split hypercovers]] by [[polydiscs]]. The rest of the proof is verbatim as for [[Smooth∞Grpd]].

To start with, since the [[hypercompletion]] depends only on the underlying [[sheaf topos]] of a site we may represent

\begin{displaymath}
\mathbb{C}Analytic\infty Grpd
  \simeq
  L_{lwhe} sPSh(\mathbb{C}Discs)_{proj, loc}
\end{displaymath}
by the [[simplicial localization]] of the [[local model structure on simplicial presheaves]] over $\mathbb{C}Disc$, [[Bousfield localization of model categories|localized]] (by the \href{hypercover#DescentFromDescentAlongHypercovers}{theorem of descent recognition along hypercovers}) at the [[hypercovers]] $U_\bullet\to X$ as seen over $CplxMfd$ restricted to $X$ a [[polydisc]] (this roundabout way since $\mathbb{C}Disc$ does not carry a [[Grothendieck topology]] but just a [[coverage]]).

Now before [[Bousfield localization of model categories|Bousfield localization]] we have a simplicially enriched [[Quillen adjunction]]

\begin{displaymath}
sPSh(\mathbb{C}Disc)_{proj, loc}
  \stackrel{\overset{\underset{\to}{\lim}}{\longrightarrow}}{\underset{\Delta}{\longleftarrow}}
  sSet_{Quillen}
\end{displaymath}
which is simplicial-degree wise just the defining [[adjunction]] of the [[colimit]] functor $\underset{\to}{\lim}$ [[left adjoint]] to the [[constant presheaf]] functor.

To see that this descends to a Quillen adjunction on the local model structure, by the \href{Quillen+adjunction#RecognitionOfSimplicialQuillenAdjunctions}{recognition theorem for simplicial Quillen adjunctions} we hence need to check that $\Delta$ preserves fibrant objects with respect to the local model structure, hence that any constant simplicial presheaf $\Delta S$ for $S$ a [[Kan complex]] already satisfies [[descent]] with respect to [[hypercovers]] as seen over $CplxMfd$.

By the \href{hypercover#DescentFromDescentAlongHypercovers}{theorem of descent recognition along hypercovers} $\Delta S$ satisfies descent precisely if for each [[hypercover]] $U_\bullet \to X$ of any [[polydisc]] $X$. the induced morphism of [[derived hom-spaces]]

\begin{displaymath}
RHom(X, \Delta S) \longrightarrow RHom(U_\bullet, \Delta S)
\end{displaymath}
in the global [[model structure on simplicial presheaves]] is a [[weak equivalence]]. This $RHom$ in turn may be computed as the $sSet$-enriched [[hom object]] out of a cofibrant resolution $\hat X \to X$ and $\hat U_\bullet \to U_\bullet$, respectively. By the \href{model%20structure%20on%20simplicial%20presheaves#CofibrantObjects}{recognition of cofibrant objects in the projective model structure} the [[representable functor|representable]] $X$ is already cofibrant and a sufficient condition for a resolution $\hat U_\bullet \to U_\bullet$ to be cofibrant is that it is a [[split hypercover]].

Since $\mathbb{C}Disc \hookrightarrow CplxMfd$ is a [[dense subsite]], we may always choose such a split hypercover such that $\hat U_\bullet$ consists simplicial-degreewise of [[coproducts]] of [[polydiscs]]: by the proposition at \emph{\href{hypercover#ExistenceOfSplitRefinements}{hypercover -- Existence of split refinements}}

Since the [[colimit]] of a [[representable functor]] is the point, this means that with such a choice $\underset{\to}{\lim} \hat U_\bullet$ is the [[simplicial set]] obtained by replacing in the hypercover by polydiscs each polydisc by a point. Forgetting the [[complex structure]] on all manifolds involved, one sees that this is precisely the ``[[etale homotopy type]]'' of $X$ as seen by the site [[CartSp]] of [[Cartesian spaces]], by the discussion at \emph{[[Smooth∞Grpd]]}, and this comes out as the ordinary [[homotopy type]] of the underlying [[topological space]] of $X$. But this is of course [[contractible homotopy type|contractible]].

In conclusion this means that

\begin{displaymath}
\begin{aligned}
     RHom(U_\bullet, \Delta S)
     &\simeq
     sPSh(\mathbb{C}Disc)(\hat U_\bullet, \Delta S)
     \\
     & \simeq 
     sSet(Sing(X), S)
     \\
     & \simeq 
     sSet(\ast, S)
     \\
     &\simeq S
  \end{aligned}
\end{displaymath}
Therefore the [[descent]] condition for $\Delta S$ is satisfied. (This is also the statement of (\hyperlink{HopkinsQuick12}{Hopkins-Quick 12, lemma 2.3, prop. 2.4, lemma 2.5, prop. 2.6})).

From here on the argument for the [[cohesion]] of $\mathbb{C}Analytic \infty Grpd$ proceeds as at \emph{[[∞-cohesive site]]} (which might just as well be adapted to the hyper-discussion here).

\end{proof}
\begin{cor}
\label{ShapeOfComplexManifold}\hypertarget{ShapeOfComplexManifold}{}
For $X\in CplxMfd \hookrightarrow \mathbb{C}Analytic\infty Grpd$ a [[complex manifold]], then the [[shape modality|shape]] $\Pi(X)$ is the [[homotopy type]] of its underlying [[topological space]].

\end{cor}
In (\hyperlink{HopkinsQuick12}{Hopkins-Quick 12})) this is part of prop. 2.6.

\hypertarget{OkaPrinciple}{}\subsubsection*{{Oka principle}}\label{OkaPrinciple}

Discussion of the [[Oka principle]] in terms of $\mathbb{C}Analytic\infty Grpd$ is in (\hyperlink{Larusson01}{Larusson 01}).

\begin{defn}
\label{OkaManifold}\hypertarget{OkaManifold}{}
Say that a [[complex manifold]] $X$ is an \emph{[[Oka manifold]]} if for every [[Stein manifold]] $\Sigma$ the canonical inclusion

\begin{displaymath}
Maps_{hol}(\Sigma, X) \longrightarrow Maps_{top}(\Sigma, X)
\end{displaymath}
from the [[mapping space]] of [[holomorphic functions]] to that of [[continuous functions]] (both equipped with the [[compact-open topology]]) is a [[weak homotopy equivalence]].

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
This is the case precisely if $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies [[descent]] with respect to [[finite set|finite]] [[covers]].

\end{theorem}
(\hyperlink{Larusson01}{Larusson 01, theorem 2.1})

\begin{remark}
\label{}\hypertarget{}{}
By corollary \ref{ShapeOfComplexManifold}, in terms of [[cohesion]], prop. \ref{Cohesion}, definition \ref{OkaManifold} should (\ldots{}check\ldots{}) read

\begin{displaymath}
\Pi[\Sigma,X] \simeq \flat [\Pi(\Sigma), \Pi(X)]
  \,.
\end{displaymath}
\end{remark}
\hypertarget{DifferentialCohomology}{}\subsubsection*{{Complex analytic differential generalized cohomology}}\label{DifferentialCohomology}

By prop. \ref{Cohesion} $\mathbb{C}Analytic \infty Grpd$ is [[cohesive (infinity,1)-topos|cohesive]] and hence by the discussion at \emph{[[differential cohomology hexagon]]} the objects $\hat E \in Stab(\mathbb{C}Analytic \infty Grpd)$ (hence the [[sheaves of spectra]] on $\mathbb{C}Mfd$ ) qualify as [[differential cohomology]] refinements of the [[cohomology theories]] [[Brown representability theorem|represented]] by the [[shape modality|shapes]] $E\coloneqq \Pi \hat E \in$ [[Spectra]].

Discussion of such complex analytic differential generalized cohomology is in (\hyperlink{HopkinsQuick12}{Hopkins-Quick 12, section 4}),.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{multiplicative_group_and_holomorphic_line_bundles}{}\subsubsection*{{Multiplicative group and holomorphic line $n$-bundles}}\label{multiplicative_group_and_holomorphic_line_bundles}

The [[multiplicative group]] is a canonical [[∞-group]] object

\begin{displaymath}
\mathbb{G}_m \in Grp(\mathbb{C}Analytic\infty Grpd)
\end{displaymath}
given as an [[(∞,1)-presheaf]] by the assignment

\begin{displaymath}
\mathbb{G}_m \;\colon\; \Sigma \mapsto \mathcal{O}_\Sigma^\times
\end{displaymath}
that sends a [[Stein manifold]] to the multiplicative [[abelian group]] of non-vanishing [[holomorphic functions]] on it.

The [[delooping]] $\mathbf{B}\mathbb{G}_m$ is the universal [[moduli stack]] for [[holomorphic line bundles]] (the [[Picard stack]]) and the double delooping $\mathbf{B}^2 \mathbb{G}_m$ that for [[holomorphic line 2-bundles]] (the [[Brauer stack]]).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[smooth ∞-groupoid]]


\item [[complex analytic stack]]


\item [[holomorphic line n-bundle]]



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

\begin{itemize}%
\item [[Finnur Lárusson]], \emph{Excision for simplicial sheaves on the Stein site and Gromov's Oka principle} (\href{http://arxiv.org/abs/math/0101103}{arXiv:math/0101103})


\item [[Jacob Lurie]], section 4.4. of \emph{[[Structured Spaces]]}


\item [[Jacob Lurie]], sections 11 and 12 of \emph{[[Closed Immersions]]}


\item [[Michael Hopkins]], [[Gereon Quick]], \emph{Hodge filtered complex bordism} (\href{http://arxiv.org/abs/1212.2173}{arXiv:1212.2173})



\end{itemize}
[[!redirects complex analytic ∞-groupoids]]

[[!redirects complex analytic infinity-groupoid]] [[!redirects complex analytic infinity-groupoids]]

[[!redirects ComplexAnalytic∞Grpd]] [[!redirects ComplexAnalyticInfinityGrpd]]

[[!redirects complex analytic ∞-stack]] [[!redirects complex analytic ∞-stacks]] [[!redirects complex analytic infinity-stack]] [[!redirects complex analytic infinity-stacks]]



\end{document}