nLab plus construction on presheaves

topos theory

Theorems

This is a subentry of sheaf about the plus-construction on presheaves. For other constructions called plus construction, see there.

Contents

Idea

The plus construction $\left(-{\right)}^{+}:\mathrm{PSh}\left(C\right)\to \mathrm{PSh}\left(C\right)$ 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.

$\mathrm{PSh}\left(C\right)\stackrel{\left(-{\right)}^{+}}{\to }\mathrm{SepPSh}\left(C\right)\stackrel{\left(-{\right)}^{+}}{\to }\mathrm{Sh}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$PSh(C) \stackrel{(-)^+}{\to} SepPSh(C) \stackrel{(-)^+}{\to} Sh(C) \,.

Notice that in terms of n-truncated morphisms, if presheaf is

In the context of (n,1)-topos theory, therefore, the plus-construction is applied $\left(n+1\right)$-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.)

Definition

Definition

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

1. ${A}^{+}:U↦{\mathrm{colim}}_{\left(R\to U\right)\in J\left(U\right)}A\left(R\right)$ where $J\left(U\right)$ denotes the poset of $J$-covering sieves on $U$.

2. Let $A:{C}^{\mathrm{op}}\to \mathrm{Set}$ be a functor. Then for $U\in {C}^{\mathrm{op}}$we define ${A}^{+}\left(U\right)$ to be an equivalence class of pairs $\left(R,s\right)$ where $R\in J\left(U\right)$ and $s=\left({s}_{f}\in A\left(\mathrm{dom}f\right)\mid f\in R\right)$ is a compatible family of elements of $A$ relative to $R$, and $\left(R,s\right)\sim \left({R}^{\prime },{s}^{\prime }\right)$ 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.

3. ${A}^{+}:U↦{\mathrm{colim}}_{\left(V↪U\right)\in V}A\left(V\right)$ where $W$ denotes the class $W:=\left({f}^{*}{\right)}^{-1}\mathrm{Core}\left(\mathrm{Sh}\left(C{\right)}_{1}\right)$ of those morphisms in $\mathrm{PSh}\left(C\right)$ which are sent to isomorphisms by the sheafification functor ${f}^{*}$ and the colimit is taken over all dense monomorphisms only.

Properties

Remark
1. $+:A↦{A}^{+}$ is a functor.

2. ${A}^{+}$ is a functor.

3. ${A}^{+}$ is a separated presheaf.

4. If $A$ is separated then ${A}^{+}$ is a sheaf.

References

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:

• Alex Heller, K. A. Rowe, On the category of sheaves Amer. J. Math. 84 1962 205–216, MR144341, doi
Revised on April 30, 2012 16:08:33 by Zoran Škoda (137.44.187.119)