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 $U$.
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.
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
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).
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:
Last revised on August 2, 2018 at 04:14:34. See the history of this page for a list of all contributions to it.