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sheafification
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(Theorem)
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Let  be a site. Let  denote the category of presheaves on  (with values in the category of abelian groups), and  the category of sheaves on  . There is a natural inclusion functor
 .
Theorem 1 The functor  has a left adjoint
 , that is, for any sheaf  and presheaf  , we have
This functor  is called sheafification, and  is called the sheafification of  .
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  is the Zariski site. It also allows derivation of various exactness properties of  and  .
- 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.html
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"sheafification" is owned by archibal.
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(view preamble)
Cross-references: properties, derivation, Zariski site, adjoints, terms, presheaf, left adjoint, functor, inclusion functor, sheaves, abelian groups, presheaves, category, site
There are 3 references to this entry.
This is version 1 of sheafification, born on 2004-02-29.
Object id is 5654, canonical name is Sheafification2.
Accessed 2847 times total.
Classification:
| AMS MSC: | 14F20 (Algebraic geometry :: homology theory :: Étale and other Grothendieck topologies and cohomologies) | | | 18F10 (Category theory; homological algebra :: Categories and geometry :: Grothendieck topologies) | | | 18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves) |
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Pending Errata and Addenda
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