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\begin{document}

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\section*{plus construction on presheaves}

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

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

[[!include topos theory - contents]]

This is a subentry of \emph{[[sheaf]]} about the \emph{plus-construction on presheaves}. For other constructions called \emph{[[plus construction]]}, see there.

\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{internal_description}{Internal description}\dotfill \pageref*{internal_description} \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}

The \emph{plus construction} $(-)^+ : PSh(C) \to PSh(C)$ on presheaves over a [[site]] $C$ is an operation that replaces a [[presheaf]] via [[local isomorphisms]] first by a [[separated presheaf]] and then by a [[sheaf]].

\begin{displaymath}
PSh(C) \stackrel{(-)^+}{\to} SepPSh(C)
  \stackrel{(-)^+}{\to} Sh(C)
  \,.
\end{displaymath}
Notice that in terms of [[n-truncated]] morphisms, a presheaf is

\begin{itemize}%
\item separated precisely if every [[descent morphism]] is [[(-1)-truncated]], namely a [[monomorphism]];


\item a sheaf precisely if every [[descent morphism]] is [[(-2)-truncated]], namely an [[equivalence]].



\end{itemize}
In the context of [[(n,1)-topos]] theory, therefore, the plus-construction is applied $(n+1)$-times in a row. The second but last step makes an [[(infinity,1)-presheaf|(n,1)-presheaf]] into a [[prestack|separated infinity-stack]] and then the last step into an actual [[(infinity,1)-sheaf|(n,1)-sheaf]]. (See \hyperlink{Lurie}{Lurie, section 6.5.3}.)

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

\begin{defn}
\label{PlusConstruction}\hypertarget{PlusConstruction}{}
Let $C$ be a small site equipped with a [[Grothendieck topology]] $J$, let $A:C^{op}\to Set$ be a functor. Then the \emph{plus construction (functor)} $(-)^+ : PSh(C) \to PSh(C)$, resp. the \emph{plus construction} $A^+$ \emph{of} $A \in PSh(C)$ is defined by one of following equivalent descriptions:

\begin{enumerate}%
\item $A^+:U\mapsto colim_{(R\to U)\in J(U)}A(R)$ where $J(U)$ denotes the poset of $J$-covering [[sieve|sieves]] on $U$.


\item For $U\in C^{op}$ we define $A^+(U)$ to be an [[equivalence class]] of pairs $(R,s)$ where $R\in J(U)$ and $s=(s_f\in A(dom f)|f\in R)$ is a [[matching family|compatible family]] of elements of $A$ relative to $R$, and $(R,s)\sim (R^\prime,s^\prime)$ iff there is a $J$-covering sieve $\R^{\prime \prime}\subseteq R\cap R^\prime$ on which the restrictions of $s$ and $s^\prime$ agree.


\item $A^+:U\mapsto colim_{(V\hookrightarrow U)\in W}A(V)$ where $W$ denotes the class $W:=(f^*)^{-1}Core(Sh(C)_1)$ of those morphisms in $PSh(C)$ which are sent to isomorphisms by the sheafification functor $f^*$ and the colimit is taken over all [[dense monomorphism|dense monomorphisms]] only.



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

\begin{remark}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item $(-)^+:A\mapsto A^+$ is a functor.


\item $A^+$ is a functor.


\item $A^+$ is a separated presheaf.


\item If $A$ is separated then $A^+$ is a sheaf.



\end{enumerate}
\end{remark}
Note that $(-)^+ : PSh(C) \to SepPSh(C)$ is not left adjoint to the inclusion $\iota : SepPSh(C) \hookrightarrow PSh(C)$ of the full subcategory of separated presheaves. If it were, it would be a [[reflector]] and therefore satisfy $(-)^+ \circ \iota \cong Id$. But this is false, since the plus construction applied to separated presheaves yields their sheafification. See \href{http://mathoverflow.net/questions/49486/the-single-plus-construction-is-not-the-left-adjoint-of-the-inclusion-of-separat}{this MathOverflow question} for details.

\hypertarget{internal_description}{}\subsection*{{Internal description}}\label{internal_description}

The plus construction can be described in the [[internal language]] of the presheaf topos $PSh(C)$. For a presheaf $A$, seen as a set from the internal point of view, the separated presheaf $A^+$ is given by the internal expression

\begin{displaymath}
A^+ = \{ K \subseteq A \,|\, j(K\,\text{ is a singleton}) \}/{\sim},
\end{displaymath}
where $\sim$ is the equivalence relation given by $K \sim L$ if and only if $j(K = L)$ and $j$ is the [[Lawvere-Tierney topology]] describing the subtopos $Sh(C) \hookrightarrow PSh(C)$.

With this internal description, the verification of the properties of the plus construction becomes an exercise with sets and subsets (instead of colimits).

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

\begin{itemize}%
\item [[sheafification]]


\item [[(∞,1)-sheafification]] / [[∞-stackification]]



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

Related entries: [[sheafification]]

A standard textbook reference in the context of 1-[[topos theory]] is:

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an elephant]] - a topos theory compendium}, Section C.2.2, proof of proposition 2.2.6, p.551

\end{itemize}
Remarks on the plus-construction in [[(infinity,1)-topos theory]] is in section 6.5.3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
Plus construction for presheaves in values in abelian categories is also called Heller-Rowe construction:

\begin{itemize}%
\item Alex Heller, K. A. Rowe, \emph{On the category of sheaves} Amer. J. Math. 84 1962 205--216, \href{http://www.ams.org/mathscinet-getitem?mr=144341}{MR144341}, \href{http://dx.doi.org/10.2307/2372759}{doi} [[!redirects plus-construction on presheaves]] [[!redirects plus constructions on presheaves]] [[!redirects plus-constructions on presheaves]]

\end{itemize}


\end{document}