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sheafification
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(Definition)
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Let $F$ be a presheaf over a topological space $X$ with values in a category $\A$ for which sheaves are defined. The sheafification of $F$ , if it exists, is a sheaf $F'$ over $X$ together with a morphism $\theta: F
\lra F'$ satisfying the following universal property:
For any sheaf $G$ over $X$ and any morphism of presheaves $\phi: F \lra G$ over $X$ , there exists a unique morphism of sheaves $\psi: F' \lra G$ such that the diagram
commutes.
In light of the universal property, the sheafification of $F$ is uniquely defined up to canonical isomorphism whenever it exists. In the case where $\A$ is a concrete category (one consisting of sets and set functions), the sheafification of any presheaf $F$ can be constructed by taking $F'(U)$ to be the set of all functions $s: U \lra \bigcup_{p \in U} F_p$ such that
- $s(p) \in F_p$ for all $p \in U$
- For all $p \in U$ , there is a neighborhood $V \subset U$ of $p$ and a section $t \in F(V)$ such that, for all $q \in V$ , the induced element $t_q \in F_q$ equals $s(q)$
for all open sets $U \subset X$ . Here $F_p$ denotes the stalk of the presheaf $F$ at the point $p$ .
The following quote, taken from [1], is perhaps the best explanation of sheafification to be found anywhere:
$F'$ is ``the best possible sheaf you can get from $F$ ''. It is easy to imagine how to get it: first identify things which have the same restrictions, and then add in all the things which can be patched together.
- 1
- David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, Springer-Verlag, 1999 (LNM 1358)
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"sheafification" is owned by djao. [ full author list (2) ]
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Cross-references: restrictions, point, stalk, open sets, element, induced, section, neighborhood, functions, set functions, concrete category, isomorphism, canonical, diagram, morphism of sheaves, presheaves, universal property, morphism, sheaf, sheaves, category, topological space, presheaf
There are 8 references to this entry.
This is version 5 of sheafification, born on 2002-05-01, modified 2004-02-26.
Object id is 2889, canonical name is Sheafification.
Accessed 6552 times total.
Classification:
| AMS MSC: | 18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves) | | | 54B40 (General topology :: Basic constructions :: Presheaves and sheaves) | | | 14F05 (Algebraic geometry :: homology theory :: Vector bundles, sheaves, related constructions) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{ F \ar[r]^{\theta} \ar@/_1pc/[rr]_{\phi} & F' \ar[r]^{\psi} & G } $](http://images.planetmath.org:8080/cache/objects/2889/js/img1.png "$\displaystyle \xymatrix{ F \ar[r]^{\theta} \ar@/_1pc/[rr]_{\phi} & F' \ar[r]^{\psi} & G } $")