Let T be a site. Let PT denote the category of presheaves on T (with values in the category of abelian groups), and ST the category of sheaves on T. There is a natural inclusion functor ι:STPT.

Theorem 1

The functorMathworldPlanetmath ι has a left adjoint :PTST, that is, for any sheaf F and presheafPlanetmathPlanetmathPlanetmath G, we have


This functor is called sheafificationPlanetmathPlanetmath, and G is called the sheafification of F.

One can readily check that this description in terms of adjoints characterizes completely, and that this definition reduces to the usual definition of sheafification ( when T is the Zariski site. It also allows derivation of various exactness properties of and ι.


  • 1 Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web at archibal/SGA/SGA.html 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