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sheafification
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
1. for all
2. 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 fromΒ [1], is perhaps the best explanation of sheafification to be found anywhere:
is βthe best possible sheaf you can get from β. 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.
References
- 1 David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, SpringerβVerlag, 1999 (LNM 1358)
Mathematics Subject Classification
18F20 Presheaves and sheaves54B40 Presheaves and sheaves
14F05 Sheaves, derived categories of sheaves and related constructions
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