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\begin{document}

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\section*{dense sub-site}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{notions_of_subcategory}{}\paragraph*{{Notions of subcategory}}\label{notions_of_subcategory}

[[!include notions of subcategory]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{problems_with_another_definition}{Problems with another definition}\dotfill \pageref*{problems_with_another_definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{warning}{Warning}\dotfill \pageref*{warning} \linebreak
\noindent\hyperlink{remark_2}{Remark}\dotfill \pageref*{remark_2} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{dense sub-site} is a [[subcategory]] of a [[site]] such that a natural [[functor]] between the corresponding [[categories of sheaves]] is an [[equivalence|equivalence of categories]].

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

\begin{prop}
\label{InducedCoverage}\hypertarget{InducedCoverage}{}
For $(C,J)$ a [[site]] with [[coverage]] $J$ and $D \to C$ any [[subcategory]], the \textbf{induced coverage} $J_D$ on $D$ has as [[covering]] [[sieve|sieves]] the intersections of the covering sieves of $C$ with the morphisms in $D$.

\end{prop}
\begin{prop}
\label{DenseSubSite}\hypertarget{DenseSubSite}{}
Let $(C,J)$ be a [[site]] (possibly [[large site|large]]). A [[subcategory]] $D \to C$ (not necessarily [[full subcategory|full]]) is called a \textbf{dense sub-site} with the \hyperlink{InducedCoverage}{induced coverage} $J_D$ if

\begin{enumerate}%
\item every object $U \in C$ has a [[covering]] [[sieve]] generated by maps $U_i \to U$ with $U_i \in D$.


\item for every morphism $f : U \to V$ in $C$ with $U, V \in D$ there is a [[covering]] [[sieve]] $\{f_i : U_i \to U\}$ of $U$ in $D$ such that the composites $f \circ f_i$ are in $D$.



\end{enumerate}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
If $D$ is a [[full subcategory]] then the second condition is automatic.

\end{remark}
The following theorem is known as the \textbf{[[comparison lemma]]}.

\begin{theorem}
\label{}\hypertarget{}{}
Let $(C,J)$ be a (possibly [[large site|large]]) [[site]] with $C$ a [[locally small category]] and let $f : D \to C$ be a [[small category|small]] \hyperlink{DenseSubSite}{dense sub-site}. Then the pair of [[adjoint functor]]s

\begin{displaymath}
(f^* \dashv f_*) 
    : 
  PSh(D)
    \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}  
   PSh(C)
\end{displaymath}
with $f^*$ given by precomposition with $f$ and $f_*$ given by right [[Kan extension]] induces an [[equivalence of categories]] between the categories of sheaves

\begin{displaymath}
(f_* \dashv f^*) : 
  Sh_{J_D}(D)
  \underoverset
    {\underset{f_*}{\to}}{\overset{f^*}
    {\leftarrow}}
    {\simeq}
  Sh_J{C}
  \,.
\end{displaymath}
\end{theorem}
This appears as (\hyperlink{JTT02}{Johnstone, theorem C2.2.3}).

\hypertarget{problems_with_another_definition}{}\subsection*{{Problems with another definition}}\label{problems_with_another_definition}

The nLab following Johnstone (\hyperlink{JTT02}{2002}, p.546) had initially the following form of condition 2 in definition \ref{DenseSubSite}:

2'. For every morphism $f : U \to V$ in $C$ with $V \in D$ there is a cover $S\in J(U)$ in $C$ generated by a family of morphisms $\{f_i : U_i \to U\}$ in $C$ such that the composites $f \circ f_i$ are in $D$.

But this is too weak to prove the comparison lemma as the following example shows:

Let $C$ be any groupoid, with the trivial topology (only maximal sieves cover), and let $D$ be the discrete category on the same objects. Then for any morphism $f:U\to V$, its inverse $f^{-1}:V\to U$ generates the maximal sieve on $U$, and the composite $f f^{-1} = 1_V$ is in $D$, so the conditions 1 and 2' of the definition are satisfied. But the restriction $Set^{C^{op}} \to Set^{D^{op}}$ is not generally an equivalence.

See the dicussion \href{https://nforum.ncatlab.org/discussion/7220/dense-subsite/#Item_0}{here}.

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

\begin{itemize}%
\item Let $X$ be a [[locale]] with [[frame]] $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the [[join]] of the $U_i$ is $U$). Let $bOp(X)$ be a [[basis for the topology]] of $X$: a complete join-[[semilattice]] such that every object of $Op(X)$ is the [[join]] of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.

\begin{itemize}%
\item For $X$ a [[locally contractible space]], $Op(X)$ its [[category of open subsets]] and $cOp(X)$ the full subcategory of [[contractible]] open subsets, we have that $cOp(X)$ is a dense sub-site.

\end{itemize}

\item For $C = TopManifold$ the category of all [[topological manifold]]s equipped with the [[open cover]] coverage, the category [[CartSp]]${}_{top}$ is a dense sub-site: every [[topological manifold]] has an [[open cover]] by [[open balls]] [[homeomorphic]] to a [[Cartesian space]].

\begin{itemize}%
\item For $C = PCompTopManifold$ the category of all [[paracompact topological space|paracompact]] [[topological manifold]]s equipped with the [[good open cover]] coverage, the category [[CartSp]]${}_{top}$ is a dense sub-site: every paracompact [[topological manifold]] has an [[good open cover]] by [[open balls]] [[homeomorphic]] to a [[Cartesian space]].


\item Similarly for $C =$ [[Diff]] the category of [[paracompact topological space|paracompact]] [[smooth manifold]]s equipped with the [[good open cover]] [[coverage]], the full subcategory [[CartSp]]${}_{smooth}$ is a dense sub-site: every such smooth manifold has a \emph{differentiably} [[good open cover]] (see there): a good cover by open balls each of which are [[diffeomorphic]] to a [[Cartesian space]].



\end{itemize}


\end{itemize}
\hypertarget{warning}{}\subsection*{{Warning}}\label{warning}

Replacing [[sheaves]] by [[(∞,1)-sheaves]] of [[spaces]] produces a strictly stronger notion. See [[(∞,1)-comparison lemma]] for a sufficient criterion for a dense inclusion of [[(∞,1)-sites]].

\hypertarget{remark_2}{}\subsection*{{Remark}}\label{remark_2}

There is also the notion of \emph{[[dense subcategory]]}, which is however only remotly related to the concept of a \emph{dense sub-site} by both vaguely invoking the topological concept of a [[dense subspace]].

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[comparison lemma]]


\item [[(∞,1)-comparison lemma]]


\item [[covering lifting property]]



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

The comparison lemma originates with the expos\'e{} III by [[J. L. Verdier|Verdier]] in

\begin{itemize}%
\item [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas ([[SGA4]])}, LNM \textbf{269} Springer Heidelberg 1972. (p.288)

\end{itemize}
A more general form is used to give a site characterization for [[étendue|étendue toposes]] in

\begin{itemize}%
\item [[A. Kock]], [[I. Moerdijk]], \emph{Presentations of Etendues} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXXII} 2 (1991) pp.145-164. (\href{http://www.numdam.org/numdam-bin/item?id=CTGDC_1991__32_2_145_0}{numdam}, pp.151f)

\end{itemize}
A proof of the comparison lemma together with a nice list of examples is in

\begin{itemize}%
\item [[Saunders Mac Lane|S. Mac Lane]], [[I. Moerdijk]], \emph{Sheaves in Geometry and Logic} , Springer Heidelberg 1994. (Appendix 4, pp.590ff)

\end{itemize}
See also

\begin{itemize}%
\item [[Olivia Caramello]], \emph{Denseness conditions, morphisms and equivalences of toposes} , arXiv:1906.08737 (2019). (\href{https://arxiv.org/abs/1906.08737}{abstract})


\item [[Peter Johnstone]], \emph{[[Elephant|Sketches of an Elephant]] vol 2} , Oxford UP 2002. (Section C2.2, p.546)


\item [[Mike Shulman]], \emph{Exact Completions and Small Sheaves} , TAC \textbf{27} no.7 (2012) pp.97-173. (\href{http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html}{abstract}; Section 11)



\end{itemize}
[[!redirects induced coverage]] [[!redirects dense sub-site]] [[!redirects dense sub-sites]] [[!redirects dense subsite]] [[!redirects dense subsites]]



\end{document}