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.

The étale space of a sheaf

In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E together with a continuous map from E to X. More precisely: to every sheaf F of sets on X there exists a local homeomorphism

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:π: E → X

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such that F is isomorphic to the sheaf of sections of π that was described in the example section above.

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Furthermore, the space E is determined up to homeomorphism by F. It is the space of stalks of F: each stalk is given the discrete topology, and we take the disjoint union of all the stalks, with π mapping all of the stalks Fx to x. The topology on this space of stalks can be chosen so that the sheaf F can be recovered as the sheaf of sections of π.

Related Topics:
Up to - Homeomorphism - Discrete topology

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At a higher level of abstraction, we can say that the category of sheaves of sets on X is equivalent to the category of local homeomorphisms to X. (One can also consider such a space in the light of the theory of representable functors; the history shows that this theory developed also in the mid-1950s.)

Related Topics:
Equivalent - Representable functor

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The space E was called espace étalé in Godement's influential book about homological algebra and sheaf theory (Topologie Algebrique et Theorie des Faisceaux, R. Godement); in that book, sheaves are in fact defined as coming from sections of local homeomorphisms; the functorial approach we gave above came later and is now more common.

Related Topics:
Homological algebra - R. Godement

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The above considerations remain true for sheaves of C on X: we can still form the space of stalks, each stalk is an object in C, and the sections naturally become objects in C as well.

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Given an arbitrary continuous map g : Z → X, the corresponding sheaf of sections gives rise in the above manner to a space of stalks E and a local homeomorphism π : E → X. In a sense this deals with all the 'ramification' in the map g, in the 'best possible way'. This may be expressed by adjoint functors; but is also important as an intuition about sheaves of sets. This collection of ideas is related to topos theory, but in a sense that more general notion of sheaf moves away from geometric intuition.

Related Topics:
Ramification - Adjoint functor - Topos

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