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sheafification
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(Theorem)
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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 $$ \Hom_{S_T}(G^\sharp,F)\cong\Hom_{P_T}(G,\iota F). $$ This functor $\sharp$ is called sheafification, and $G^\sharp$ is called the <</SPAN>#93#>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 when $T$ is the Zariski site. It also allows derivation of various exactness properties of $\sharp$ and $\iota$
- 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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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 3384 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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