The plus construction for a presheaf

Let $\mathcal{F}$ be a presheaf on a topological space $X$, and let $U\subseteq X$ be open.

A germ-compatible local presentation of a section over $U$ consists of:

• an open cover $U={\bigcup }_{i\in I}{U}_i$;
• sections $s_i\in \mathcal{F}(U_i)$;

such that for every $x\in U_i\cap U_j$ the germs agree: ${\left(s_i\right)}_x={\left(s_j\right)}_x\text{ in }{\mathcal{F}}_x\mathrm{.}$

Two germ-compatible local presentations $(U_i,s_i)$ and $(V_j,t_j)$ over $U$ are called equivalent if for every $x\in U$ and every choice of indices $i,j$ with $x\in U_i\cap V_j$, one has ${\left(s_i\right)}_x={\left(t_j\right)}_x\text{ in }{\mathcal{F}}_x\mathrm{.}$

The plus construction ${\mathcal{F}}^{+}$ is the presheaf defined by letting ${\mathcal{F}}^{+}(U)$ be the set of equivalence classes of germ-compatible local presentations over $U$. Restriction to an open subset $V\subseteq U$ is obtained by replacing the cover $U_i$ with $U_i\cap V$ and restricting each section $s_i$ to $U_i\cap V$.

There is a canonical morphism of presheaves $\eta :\mathcal{F}⟶{\mathcal{F}}^{+}$ whose component on $U$ sends a section $s\in \mathcal{F}(U)$ to the class of the single-chart presentation $(U,s)$.