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 $\iota :{S}_{T}\to {P}_{T}$.
Theorem 1
The functor^{} $\iota $ has a left adjoint $\mathrm{\u266f}\mathrm{:}{P}_{T}\mathrm{\to}{S}_{T}$, that is, for any sheaf $F$ and presheaf^{} $G$, we have
$${\mathrm{Hom}}_{{S}_{T}}({G}^{\mathrm{\u266f}},F)\cong {\mathrm{Hom}}_{{P}_{T}}(G,\iota F).$$ |
This functor $\mathrm{\u266f}$ is called sheafification^{}, and ${G}^{\mathrm{\u266f}}$ is called the sheafification of $F$.
One can readily check that this description in terms of adjoints characterizes $\mathrm{\u266f}$ 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 $\mathrm{\u266f}$ 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 |