# 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