sheafification

Let $T$ be a site. Let $P_{T}$ denote the category of presheaves on $T$ (with values in the category of abelian groups), and $S_{T}$ the category of sheaves on $T$ . There is a natural inclusion functor $\iota\colon S_{T}\to P_{T}$ .

Theorem 1

The functor $\iota$ has a left adjoint $\sharp\colon P_{T}\to S_{T}$ , that is, for any sheaf $F$ and presheaf $G$ , we have

$\operatorname{Hom}_{S_{T}}(G^{\sharp},F)\cong\operatorname{Hom}_{P_{T}}(G,% \iota F).$

This functor $\sharp$ is called sheafification, and $G^{\sharp}$ is called the sheafification of $F$ .

One can readily check that this description in terms of adjoints characterizes $\sharp$ completely, and that this definition reduces to the usual definition of sheafification (http://planetmath.org/Sheafification) when $T$ is the Zariski site. It also allows derivation of various exactness properties of $\sharp$ and $\iota$ .

References

1 Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web at http://www.math.mcgill.ca/ archibal/SGA/SGA.htmlhttp://www.math.mcgill.ca/ archibal/SGA/SGA.html

Title	sheafification
Canonical name	Sheafification1
Date of creation	2013-03-22 14:13:08
Last modified on	2013-03-22 14:13:08
Owner	archibal (4430)
Last modified by	archibal (4430)
Numerical id	4
Author	archibal (4430)
Entry type	Theorem
Classification	msc 14F20
Classification	msc 18F10
Classification	msc 18F20
Related topic	Sheafification
Related topic	Site
Related topic	Sheaf2
Related topic	Sheaf
Defines	sheafification