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\begin{document}

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\section*{cohesive site}

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

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

[[!include cohesive infinity-toposes - contents]]

\hypertarget{cohesive_site}{}\section*{{Cohesive site}}\label{cohesive_site}

\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{sheaves_on_a_cohesive_site_are_cohesive}{Sheaves on a cohesive site are cohesive}\dotfill \pageref*{sheaves_on_a_cohesive_site_are_cohesive} \linebreak
\noindent\hyperlink{aufhebung}{Aufhebung}\dotfill \pageref*{aufhebung} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{cohesive_presheaf_sites}{Cohesive presheaf sites}\dotfill \pageref*{cohesive_presheaf_sites} \linebreak
\noindent\hyperlink{sites_of_open_balls}{Sites of open balls}\dotfill \pageref*{sites_of_open_balls} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \textbf{cohesive site} is a small [[site]] whose [[topos of sheaves]] is a [[cohesive topos]].

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

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a [[small site|small]] [[site]], i.e. a [[small category]] equipped with a [[coverage]]/[[Grothendieck topology]]. We say that $C$ is a \textbf{cohesive site} if

\begin{enumerate}%
\item $C$ has a [[terminal object]].


\item The coverage on $C$ makes it a [[locally connected site]], i.e. every [[cover|covering]] [[sieve]] on an object $U\in C$ is [[connected category|connected]] as a [[subcategory]] of the [[slice category]] $C/U$.


\item Every object $U\in C$ admits a [[global section]] $*\to U$.


\item $C$ is a [[cosifted category]]

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



\end{enumerate}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{sheaves_on_a_cohesive_site_are_cohesive}{}\subsubsection*{{Sheaves on a cohesive site are cohesive}}\label{sheaves_on_a_cohesive_site_are_cohesive}

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a cohesive site, the [[category of sheaves]] $Sh(C)$ on $C$ is a [[cohesive topos]] over [[Set]] for which \emph{[[pieces have points]]} .

\end{prop}
\begin{proof}
Following the notation at [[cohesive topos]], we write

\begin{displaymath}
(Disc \dashv \Gamma) 
    \coloneqq 
  (L Const \dashv \Gamma) 
  \;\colon\; 
  Sh(C) \to Set
\end{displaymath}
for the [[global section]] [[geometric morphism]], where the [[inverse image]] $Disc$ constructs [[discrete object]]s. We need to exhibit two more adjoints

\begin{displaymath}
(\Pi_0 \dashv Disc \dashv \Gamma \dashv CoDisc) 
  \;\colon\; 
  Sh(C) \to Set
\end{displaymath}
and show that $\Pi_0$ preserves [[finite products]]. Finally we need to show that $\Gamma X \to \Pi_0 X$ is an [[epimorphism]] for all $X$.

Firstly, since $C$ is a [[locally connected site]], any constant presheaf is a sheaf. This implies that the functor $Disc$ has a further [[left adjoint]] given by taking [[colimits]] over $C^{op}$, which we denote $\Pi_0$. Hence $Sh(C)$ is a [[locally connected topos]].

Moreover, since $C$ is [[cosifted category|cosifted]], $\Pi_0$ preserves finite products. In particular, $Sh(C)$ is [[connected topos|connected]] and even \emph{strongly connected}.

Next, we claim that $C$ is a [[local site]]. This means that its [[terminal object]] $*$ is \emph{cover-irreducible}, i.e. any covering [[sieve]] of $*$ must contain its identity map. But since $C$ is a locally connected site, every covering family is inhabited, and since every object has a global section, every covering sieve must include a global section. In the case of $*$, the only global section is an identity map; hence $C$ is a local site, and so $Sh(C)$ is a [[local topos]]. The [[right adjoint]] $Codisc$ of $\Gamma$ is defined by

\begin{displaymath}
CoDisc(A)(U) = A^{C(*,U)} = A^{\Gamma(U)}
  \,.
\end{displaymath}
We now claim that the transformation $Disc(A) \to Codisc(A)$ is [[monic]]. Since sheaves are closed under limits in presheaves, this condition can be checked pointwise at each object $U\in C$. But since constant presheaves are sheaves, the map $Disc(A)(U) \to Codisc(A)(U)$ is just the [[diagonal morphism|diagonal]]

\begin{displaymath}
A \to A^{C(*,U)}
\end{displaymath}
which is monic since $C(*,U)$ is always inhabited (by assumption on $C$).

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

A [[cohesive topos]] over a 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{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{cohesive_presheaf_sites}{}\subsubsection*{{Cohesive presheaf sites}}\label{cohesive_presheaf_sites}

Consider a category $C$ equipped with the trivial coverage/topology. Then the [[category of sheaves]] on $C$ is the [[category of presheaves]] on $C$

\begin{displaymath}
Sh(C) \simeq PSh(C)
\end{displaymath}
and trivially every constant presheaf is a sheaf. So we always have an [[adjoint triple]] of functors

\begin{displaymath}
(\Pi_0 \dashv Disc \dashv \Gamma) : Sh(C) \to Set
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\Pi_0$ is the functor that takes [[colimit]]s of functors $X : C^{op} \to Set$

\begin{displaymath}
\Pi_0 X = {\lim_\to} X
\end{displaymath}

\item $\Gamma$ is the functor that takes [[limit]]s;

\begin{displaymath}
\Gamma X = {\lim_\leftarrow} X    
  \,.
\end{displaymath}


\end{itemize}
The condition that $\Pi_0$ preserves finite products is precisely the condition that $C$ be a [[cosifted category]].

In conclusion we have

\begin{prop}
\label{}\hypertarget{}{}
A small category equipped with the trivial coverage/topology is a cohesive site if

\begin{itemize}%
\item it is [[cosifted category|cosifted]];


\item has a [[terminal object]] $*$.


\item every object $U$ has a [[global element]] $* \to U$.



\end{itemize}
The first two conditions ensure that $Sh(C) = PSh(C)$ is a [[cohesive topos]]. The last condition implies that \emph{cohesive pieces have points} in $PSh(C)$.

\end{prop}
\hypertarget{sites_of_open_balls}{}\subsubsection*{{Sites of open balls}}\label{sites_of_open_balls}

Any full small subcategory of [[Top]] on [[connected]] topological spaces with the canonical induced [[open cover]] [[coverage]] is a cohesive site. If a subcategory on [[contractible]] spaces, then this is also an [[(∞,1)-cohesive site]].

Specifically we have:

\begin{prop}
\label{}\hypertarget{}{}
The categories [[CartSp]] and [[ThCartSp]] equipped with the standard [[open cover]] [[coverage]] are cohesive sites.

\end{prop}
The axioms are readily checked.

Notice that the cohesive topos over $ThCartSp$ is the [[Cahiers topos]].

\begin{prop}
\label{}\hypertarget{}{}
The cohesive concrete objects of the cohesive topos $Sh(CartSp)$ are precisely the [[diffeological space]]s.

\end{prop}
See [[cohesive topos]] for more on this.

\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 \textbf{cohesive site} / [[∞-cohesive site]]



\end{itemize}
[[!redirects cohesive sites]] [[!redirects site of cohesion]] [[!redirects sites of cohesion]]



\end{document}