Karl Jones at 19:53, 22 September 2016

2016-09-22T19:53:22Z

Karl Jones: Created page with "In mathematics, a '''sheaf''' is a tool for systematically tracking locally defined data attached to the open sets of a topological space. == Description..."

2016-09-22T19:47:03Z

Created page with "In mathematics, a '''sheaf''' is a tool for systematically tracking locally defined data attached to the open sets of a topological space. == Description..."

New page

In [[mathematics]], a '''sheaf''' is a tool for systematically tracking locally defined data attached to the [[Open set|open sets]] of a [[topological space]].

== Description ==

The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set.

Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in [[topology]] and especially in [[Algebraic topology|algebraic]] and [[differential geometry]]. First, geometric structures such as that of a [[differentiable manifold]] or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves.

Second, sheaves provide the framework for a very general [[cohomology theory]], which encompasses also the "usual" topological cohomology theories such as singular cohomology.

Especially in [[algebraic geometry]] and the theory of [[Complex manifold|complex manifolds]], sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of [[Differential equation|differential equations]].

In addition, generalisations of sheaves to more general settings than topological spaces, such as [[Grothendieck topology]], have provided applications to [[mathematical logic]] and [[number theory]].

== See also ==

* [[Coherent sheaf]]
* [[Cosheaf]]
* [[Gerbe]]
* [[Presheaf of spaces]]
* [[Stack (mathematics)]]
* [[Sheaf of spectra]]
* [[Topological space]]

== External links ==

* [https://en.wikipedia.org/wiki/Sheaf_(mathematics) Sheaf (mathematics)] @ Wikipedia.org

[[Category:Mathematics]]
[[Category:Topology]]

@@ Line 3: / Line 3: @@
 == Description ==
-The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set.
+The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the [[Ring (mathematics)|rings]] of [[Continuous function|continuous]] or [[Smooth function|smooth]] [[Real number|real]]-valued [[Function (mathematics)|functions]] defined on each open set.
 Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.
-There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.
+There are also [[Map (mathematics)|maps]] (or [[Morphism|morphisms]]) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a [[Category:Mathematics|category]]. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These [[Functor|functors]], and certain variants of them, are essential parts of sheaf theory.
 Due to their general nature and versatility, sheaves have several applications in [[topology]] and especially in [[Algebraic topology|algebraic]] and [[differential geometry]]. First, geometric structures such as that of a [[differentiable manifold]] or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves.

Sheaf (mathematics) - Revision history

Karl Jones at 19:53, 22 September 2016

Karl Jones: Created page with "In mathematics, a '''sheaf''' is a tool for systematically tracking locally defined data attached to the open sets of a topological space. == Description..."