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sheafification
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(Definition)
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Let  be a presheaf over a topological space  with values in a category
 for which sheaves are defined. The sheafification of  , if it exists, is a sheaf  over  together with a morphism
 satisfying the following universal property:
For any sheaf  over  and any morphism of presheaves
 over  , there exists a unique morphism of sheaves
 such that the diagram
commutes.
In light of the universal property, the sheafification of  is uniquely defined up to canonical isomorphism whenever it exists. In the case where
 is a concrete category (one consisting of sets and set functions), the sheafification of any presheaf  can be constructed by taking  to be the set of all functions
 such that
-
 for all 
- For all
 , there is a neighborhood
 of  and a section
 such that, for all  , the induced element
 equals 
for all open sets
 . Here  denotes the stalk of the presheaf  at the point  .
The following quote, taken from [1], is perhaps the best explanation of sheafification to be found anywhere:
 is “the best possible sheaf you can get from  ”. 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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(view preamble)
Cross-references: restrictions, point, stalk, open sets, induced, section, neighborhood, functions, set functions, concrete category, isomorphism, canonical, 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 4329 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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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ F \ar[r]^{\theta} \ar@/_1pc/[rr]_{\phi} & F' \ar[r]^{\psi} & G } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/2889/l2h/img13.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ F \ar[r]^{\theta} \ar@/_1pc/[rr]_{\phi} & F' \ar[r]^{\psi} & G } } \end{xy}$")