Could not include topos theory - contents
This is a subentry of sheaf about the plus-construction on presheaves. For other constructions called plus construction, see there.
The 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.
Notice that in terms of n-truncated morphisms, a presheaf is
separated precisely if every descent morphism is (-1)-truncated, namely a monomorphism;
a sheaf precisely if every descent morphism is (-2)-truncated, namely an equivalence.
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 (n,1)-presheaf into a separated infinity-stack and then the last step into an actual (n,1)-sheaf. (See Lurie, section 6.5.3.)
Let $C$ be a small site equipped with a Grothendieck topology $J$, let $A:C^{op}\to Set$ be a functor. Then the plus construction (functor) $(-)^+ : PSh(C) \to PSh(C)$, resp. the plus construction $A^+$ of $A \in PSh(C)$ is defined by one of following equivalent descriptions:
$A^+:U\mapsto colim_{(R\to U)\in J(U)}A(R)$ where $J(U)$ denotes the poset of $J$-covering sieves on $U$.
Let $A:C^{op}\to Set$ be a functor. Then 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 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.
$A^+:U\mapsto colim_{(V\hookrightarrow U)\in W}A(V)$ 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 monomorphisms only.
$(-)^+:A\mapsto A^+$ is a functor.
$A^+$ is a functor.
$A^+$ is a separated presheaf.
If $A$ is separated then $A^+$ is a sheaf.
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 this MathOverflow question for details.
Related entries: sheafification
A standard textbook reference in the context of 1-topos theory is:
Remarks on the plus-construction in (infinity,1)-topos theory is in section 6.5.3 of
Plus construction for presheaves in values in abelian categories is also called Heller-Rowe construction: