Sheaf (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-07

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sheaf_(mathematics)


00:00:56 1 Overview
00:02:21 2 Formal definitions
00:03:18 2.1 Presheaves
00:04:15 2.2 Sheaves
00:05:12 2.3 Morphisms
00:05:40 3 Examples
00:07:34 3.1 Sheaves on manifolds
00:09:27 3.2 Presheaves that are not sheaves
00:10:24 4 Turning a presheaf into a sheaf
00:11:21 5 Operations
00:12:18 6 Images of sheaves
00:13:14 7 Stalks of a sheaf
00:13:43 8 Ringed spaces and locally ringed spaces
00:14:40 9 Sheaves of modules
00:15:36 9.1 Finiteness conditions for sheaves of modules
00:16:05 10 The étalé space of a sheaf
00:17:02 11 Sheaf cohomology
00:17:58 12 Sites and topoi
00:19:24 13 History
00:21:17 14 See also
00:22:42 15 Notes
00:24:07 16 References
00:25:04 17 External links
00:26:29 Finiteness conditions for sheaves of modules
00:28:23 C[x1, x2, ...] of complex polynomials in countably many indeterminates.  Choose n
00:29:20 The étalé space of a sheaf
00:30:17 (i−1F)(U) agrees with the usual F(U), and if i : {x} → X is the inclusion of a point, then Γ({x})
00:31:13 Sheaf cohomology
00:32:39 Sites and topoi
00:33:35 History
00:34:32 See also



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SUMMARY
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In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. 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 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 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 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.