What is "sheaf"

Sheaf \Sheaf\, n. (Mech.) A sheave. [R.]

Sheaf \Sheaf\, n.; pl. Sheaves. [OE. sheef, shef, schef, AS. sce['a]f; akin to D. schoof, OHG. scoub, G. schaub, Icel. skauf a fox's brush, and E. shove. See Shove.]

1. A quantity of the stalks and ears of wheat, rye, or other grain, bound together; a bundle of grain or straw.

   The reaper fills his greedy hands, And binds the golden sheaves in brittle bands.
   --Dryden.

2. Any collection of things bound together; a bundle; specifically, a bundle of arrows sufficient to fill a quiver, or the allowance of each archer, -- usually twenty-four.

   The sheaf of arrows shook and rattled in the case.
   --Dryden.

Sheaf \Sheaf\, v. t. To gather and bind into a sheaf; to make into sheaves; as, to sheaf wheat.

Sheaf \Sheaf\, v. i. To collect and bind cut grain, or the like; to make sheaves.

They that reap must sheaf and bind.
--Shak.

Old English sceaf (plural sceafas) "large bundle of corn," from Proto-Germanic *skauf- (cognates: Old Saxon scof, Middle Dutch scoof, Dutch schoof, Old High German scoub "sheaf, bundle," German Schaub "sheaf;" Old Norse skauf "fox's tail;" Gothic skuft "hair on the head," German Schopf "tuft"), from PIE root *(s)keup- "cluster, tuft, hair of the head." Extended to bundles of things other than grain by c.1300. Also used in Middle English for "two dozen arrows." General sense of "a collection" is from 1728.

n. 1 A quantity of the stalks and ears of wheat, rye, or other grain, bound together; a bundle of grain or straw. 2 Any collection of things bound together; a bundle. 3 A bundle of arrows sufficient to fill a quiver, or the allowance of each archer. 4 A quantity of arrows, usually twenty-four. 5 (context mechanical English) A sheave. 6 (context mathematics English) An abstract construct in topology that associates data to the open sets of a topological space, together with well-defined restrictions from larger to smaller open sets, subject to the condition that compatible data on overlapping open sets corresponds, via the restrictions, to a unique datum on the union of the open sets. vb. 1 (context transitive English) To gather and bind into a sheaf; to make into sheaves; as, to sheaf wheat. 2 (context intransitive English) To collect and bind cut grain, or the like; to make sheaves.

1. n. a package of several things tied together for carrying or storing [syn: bundle]

2. [also: sheaves (pl)]

Sheaf may refer to:

• Sheaf (agriculture), a large bundle in which cereal plants are bound after reaping. Accounts of two usages derived from this are found at:
  • Sheaf (mathematics)
  • Sheaf toss (a Scottish sport)

• Artillery Sheaf, the beaten zone created by battery fires that generally form a line perpendicular to the direction of fire.
• River Sheaf, a tributary of River Don in England
• The Sheaf, a student-run newspaper serving the University of Saskatchewan
• Aluma, a settlement in Israel whose name translates as Sheaf

• Sceafa, a king of English legend
• Sheave, a wheel or roller with a groove along its edge for holding a belt, rope or cable

cs:Snop eo:Garbo pl:Snop

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.

A sheaf is a bundle of cereal plant stems bound together after harvesting, or reaping. The sheaves are bound using the stem material and are stacked together in a stook with the stems vertical and the seed heads aligned; this keeps the heads off the ground, allows the grain to dry, discourages vermin, makes the crop easier to pick up and is convenient for winnowing or threshing. Reaping is the cutting of grain or pulse for harvest, typically using a scythe or sickle. Grass for hay was also sheaved and stacked to dry in the field in this way but is now usually mechanically windrowed and baled.

The advent of mechanical harvesting has generally rendered the sheaf redundant in developed countries but remains in wide-spread use around the world. Wheat sheaves are sized for picking up and pitching onto a cart with the seed heads aligned using a traditional two-pronged pitchfork.

Traditional mowers using scythes and working as a team would cut a field of grain clockwise, starting from an outside edge and finishing in the middle. The action of scything left a windrow of cut stems to the left of the mower, and if cut skillfully, left the seed heads more or less aligned. These were then picked up and assembled into sheaves by following workers who tied the bundles using spare cut stems. These workers, or a separate following team, would then assemble the sheaves into stooks. The stooks would later be loaded onto a cart and taken to a barn in preparation for winnowing or threshing.