Sheaf (mathematics)

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function.

Generalizations

It is possible to define a cohomology theory for sheaves of abelian groups (sheaf cohomology) that can give much useful, more concrete information. The main issue is the existence of the long exact sequence coming from an exact sequence of sheaves. In applications emphasis was placed on sheaves on spaces that were less well-behaved than finite complexes. For example, in algebraic geometry spaces carrying the Zariski topology are rarely Hausdorff.

Related Topics:
Cohomology - Sheaf cohomology - Long exact sequence - Exact sequence - Well-behaved - Finite complex - Zariski topology - Hausdorff

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The algebraic geometry case was first tackled by Jean-Pierre Serre by developing an analogue of Čech cohomology; this worked, though in general the construction doesn't have such good properties. Then Alexander Grothendieck used derived functors of the global section functor, providing a more definitive solution.

Related Topics:
Jean-Pierre Serre - Čech cohomology - Alexander Grothendieck - Derived functor

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Grothendieck was motivated to develop a cohomology theory for sheaves that would give stronger results, and that would, in particular, allow a proof of the Weil conjectures. By precisely analyzing the properties of X needed to define sheaves, he defined the notion of a Grothendieck topology on a category (this came in a somewhat roundabout fashion — see background and genesis of topos theory).

Related Topics:
Weil conjectures - Grothendieck topology - Background and genesis of topos theory

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A category together with a Grothendieck topology is called a site. It is possible to define the notion of a sheaf on any site. The notion of sites later led Lawvere to develop the notion of an elementary topos.

Related Topics:
Lawvere - Elementary topos

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