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\begin{document}

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\section*{infinity-cohesive site}

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

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{sheaves_on_cohesive_sites}{$\infty$-Sheaves on $\infty$-Cohesive sites}\dotfill \pageref*{sheaves_on_cohesive_sites} \linebreak
\noindent\hyperlink{aufhebung}{Aufhebung}\dotfill \pageref*{aufhebung} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

An \emph{$(\infty,1)$-cohesive site} is a [[site]] such that the [[(∞,1)-category of (∞,1)-sheaves]] over it is a [[cohesive (∞,1)-topos]].

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

\begin{defn}
\label{InfinityCohesiveSite}\hypertarget{InfinityCohesiveSite}{}
A [[site]] $C$ is \textbf{$\infty$-cohesive} over [[∞Grpd]] if it is

\begin{itemize}%
\item a [[strongly ∞-connected site]]


\item and an [[∞-local site]].



\end{itemize}
In detail this means that $C$ is

\begin{itemize}%
\item a [[site]] -- a [[small category]] $C$ equipped with a [[coverage]];


\item with the property that

\begin{itemize}%
\item it has a [[terminal object]] $*$;


\item it is a [[cosifted category]]

(for instance in that it has all [[finite products]], see at \emph{[[categories with finite products are cosifted]]});


\item for every [[covering]] family $\{U_i \to U\}$ in $C$

\begin{itemize}%
\item the [[Cech nerve]] $C(U) \in [C^{op}, sSet]$ is degreewise a [[coproduct]] of [[representable functor|representables]];


\item the [[simplicial set]] obtained by replacing each copy of a representable by a point is [[contractible]] (weakly equivalent to the point in the [[classical model structure on simplicial sets]])

\begin{displaymath}
\underset{\longrightarrow}{\lim} C(U) \stackrel{\simeq}{\to} *
\end{displaymath}

\item the simplicial set of points in $C(U)$ is weakly equivalent to the set of points of $U$:

\begin{displaymath}
\underset{\longleftarrow}{\lim}C(U)
  = Hom_C(*, C(U)) \stackrel{\simeq}{\to} Hom_C(*,U)
  \,.
\end{displaymath}


\end{itemize}


\end{itemize}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
These conditions are stronger than for a [[cohesive site]], as the latter only guarantees cohesiveness of the 1-topos over it.

This definition is supposed to model the following ideas:

\begin{itemize}%
\item every object $U$ has an underlying set of points $Hom_C(*,U)$. We may think of each $U$ as specifying one way in which there can be cohesion on this underlying set of points;


\item in view of the [[nerve theorem]] the condition that $\lim_\to C(U)$ is contractible means that $U$ itself is contractible, as seen by the [[Grothendieck topology]] on $C$. This reflects the \emph{local} aspect of cohesion: we only specify cohesive structure on contractible lumps of points;


\item in view of this, the remaining condition that $Hom_C(*,C(U))$ is contractible is the $\infty$-analog of the condition on a [[concrete site]] that $Hom_C(*,\coprod_i U_i) \to Hom_C(*, U)$ is surjective. This expresses that the notion of topology on $C$ and its concreteness over [[Set]] are consistent.



\end{itemize}
\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{Presheaves}\hypertarget{Presheaves}{}
The site for a [[presheaf topos]], hence with trivial topology, is $\infty$-cohesive, def. \ref{InfinityCohesiveSite}, if it has [[finite products]].

\end{example}
\begin{proof}
All covers $\{U_i \to U\}$ consist of only the [[identity]] morphism $\{U \stackrel{Id}{\to} U\}$. The [[Cech nerve]] $C\{U\}$ is then the [[simplicial object]] constant on $U$ and hence satisfies its two conditions above trivially.

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
The following [[sites]] are $\infty$-cohesive, def. \ref{InfinityCohesiveSite}:

\begin{itemize}%
\item the category [[CartSp]] with covering families given by open covers $\{U_i \hookrightarrow U\}$ by [[geodesically convex|convex]] subsets $U_i$;

we can take the morphisms $\mathbb{R}^k \to \mathbb{R}^l$ in $CartSp$ to be

\begin{itemize}%
\item [[continuous maps]]

-- in which case the sheaf topos over it models generalized [[topological space]]s, the [[2-sheaf]] [[2-topos]] contains for instance [[topological stack]]s;


\item or [[smooth maps]]

-- in which case the sheaf topos models generalized [[smooth space]]s such as [[diffeological space]]s, the [[(∞,1)-sheaf (∞,1)-topos]] is that of [[∞-Lie groupoid]]s;



\end{itemize}

\item the site [[ThCartSp]] $\subset \mathbb{L}$ of [[smooth loci]] consisting of smooth loci of the form $R^n \times D^l_{(k)}$ with the second factor infinitesimal, where covering families are those of the form $\{U_i \times D^l_{(k)} \to U \times D^l_{(k)}\}$ with $\{U_i \to U\}$ a covering family in $CartSp$ as above.

This is a site of definition for the [[Cahiers topos]].



\end{itemize}
More discussion of these two examples is at [[∞-Lie groupoid]] and [[∞-Lie algebroid]].

\end{example}
\begin{proof}
Since every [[star-shaped]] region in $\mathbb{R}^n$ is [[diffeomorphic]] to an [[open ball]] (see there for details) we have that the covers $\{U_i \to U\}$ on [[CartSp]] by convex subsets are [[good open covers]] in the strong sense that any finite non-empty intersection is [[diffeomorphic]] to an [[open ball]] and hence diffeomorphic to a [[Cartesian space]]. Therefore these are [[good open cover]]s in the strong sense of the term and their [[Cech nerve]]s $C(U)$ are degreewise coproducts of representables.

The fact that $\lim_\to C(U) \simeq *$ follows from the [[nerve theorem]], using that a [[Cartesian space]] regarded as a [[topological space]] is [[contractible]].

\end{proof}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{sheaves_on_cohesive_sites}{}\subsubsection*{{$\infty$-Sheaves on $\infty$-Cohesive sites}}\label{sheaves_on_cohesive_sites}

\begin{theorem}
\label{}\hypertarget{}{}
Let $C$ be an $\infty$-cohesive site. Then the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(C)$ over $C$ is a [[cohesive (∞,1)-topos]] that satisfies the axiom ``discrete objects are concrete'' .

If moreover for all objects $U$ of $C$ we have that $C(*,U)$ is [[inhabited set|inhabited]], then the axiom ``pieces have points'' also holds.

\end{theorem}
Since the [[(n,1)-topos]] over a site for any $n \in \mathbb{N}$ arises as the full [[sub-(∞,1)-category]] of the $(\infty,1)$-topos on the $n$-[[truncated]] objects and since the definition of cohesive $(n,1)$-topos is compatible with such truncation, it follows that

\begin{cor}
\label{}\hypertarget{}{}
Let $C$ be an $\infty$-cohesive site. Then for all $n \in \mathbb{N}$ the [[(n,1)-topos]] $Sh_{(n,1)}(C)$ is cohesive.

\end{cor}
To prove this, we need to show that

\begin{enumerate}%
\item $Sh_{(\infty,1)}(C)$ is a [[locally ∞-connected (∞,1)-topos]] and a [[∞-connected (∞,1)-topos]].

This follows with the discussion at [[∞-connected site]].


\item $Sh_{(\infty,1)}(C)$ is a [[local (∞,1)-topos]].

This follows with the discussion at [[∞-local site]].


\item The [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite [[(∞,1)-products]].


\item If $\Gamma(U)$ is not empty for all $U \in C$, then \emph{pieces have points} in $Sh_{(\infty,1)}(C)$.



\end{enumerate}
The last two conditions we demonstrate now.

\begin{prop}
\label{}\hypertarget{}{}
The functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ whose existence is guaranteed by the above proposition preserves [[product]]s:

\begin{displaymath}
\Pi(A \times B) \simeq \Pi(A) \times \Pi(B)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the discussion at [[∞-connected site]] we have that $\Pi$ is given by the [[(∞,1)-colimit]] $\lim_\to : PSh_{(\infty,1)}(C) \to \infty Grpd$. By the assumption that $C$ is a [[cosifted (∞,1)-category]], it follows that this operation preserves finite products.

\end{proof}
Finally we prove that \emph{pieces have points} in $Sh_{(\infty,1)}(C)$ if all objects of $C$ have points.

\begin{proof}
By the above discussion both $\Gamma$ and $\Pi$ are presented by left Quillen functors on the projective model structure $[C^{op}, sSet]_{proj,loc}$. By Dugger's cofibrant replacement theorem (see [[model structure on simplicial presheaves]]) we have for $X$ any simplicial presheaf that a cofibrant replacement is given by an object that in the lowest two degrees is

\begin{displaymath}
\cdots
  \stackrel{\to}{\stackrel{\to}{\to}}
  \coprod_{U_0 \to U_1 \to X_1} U
   \stackrel{\to}{\to}
   \coprod_{U \to X_0} U
  \,,
\end{displaymath}
where the coproduct is over all morphisms out of representable presheaves $U_i$ as indicated.

The model for $\Gamma$ sends this to

\begin{displaymath}
\cdots
  \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{U_0 \to U_1 \to X_0} C(*,U_0)
   \stackrel{\to}{\to}
   \coprod_{U \to X_0} C(*,U)
  \,,
\end{displaymath}
whereas the model for $\Pi$ sends this to

\begin{displaymath}
\cdots
  \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{U_0 \to U_1 \to X_0} *
   \stackrel{\to}{\to}
   \coprod_{U \to X_0} *
  \,.
\end{displaymath}
The morphism from the first to the latter is the evident one that componentwise sends $C(*,U)$ to the point. Since by assumption each $C(*,U)$ is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on $\pi_0$.

\end{proof}
\hypertarget{aufhebung}{}\subsubsection*{{Aufhebung}}\label{aufhebung}

A [[cohesive (∞,1)-topos]] over an $\infty$-cohesive site satisfies [[Aufhebung]] of the [[unity of opposites|moments]] of [[becoming]]. See at \emph{[[Aufhebung]]} the section \emph{\href{http://ncatlab.org/nlab/show/Aufhebung#ExamplesBecomingFormalization}{Aufhebung of becoming -- Over cohesive sites}}.

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

\begin{itemize}%
\item [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

\begin{itemize}%
\item [[connected topos]] / [[∞-connected (∞,1)-topos]]


\item [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]


\item [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]



\end{itemize}

\item [[local topos]] / [[local (∞,1)-topos]].


\item [[cohesive topos]] / [[cohesive (∞,1)-topos]]



\end{itemize}
and

\begin{itemize}%
\item [[locally connected site]] / [[locally ∞-connected site]]

\begin{itemize}%
\item [[connected site]] / [[∞-connected site]]


\item [[strongly connected site]] / [[strongly ∞-connected site]]


\item [[totally connected site]] / [[totally ∞-connected site]]



\end{itemize}

\item [[local site]] / [[∞-local site]]


\item [[cohesive site]], \textbf{∞-cohesive site}



\end{itemize}
[[!redirects ∞-cohesive site]] [[!redirects ∞-cohesive sites]]

[[!redirects (infinity,1)-cohesive site]]

[[!redirects (∞,1)-cohesive site]]

[[!redirects (infinity,1)-cohesive sites]]



\end{document}