## Sheaf (mathematics) |

In **sheaf** is a tool for systematically tracking locally defined data attached to the

There are also

Due to their general nature and versatility, sheaves have several applications in topology and especially in

- overview
- formal definitions
- examples
- turning a presheaf into a sheaf
- operations
- images of sheaves
- stalks of a sheaf
- ringed spaces and locally ringed spaces
- sheaves of modules
- the étalé space of a sheaf
- sheaf cohomology
- sites and topoi
- history
- see also
- notes
- references
- external links

In *localised* or *restricted* to *n* times

*Presheaves* formalise the situation common to the examples above: a presheaf (of sets) on a topological space is a structure that associates to each open set *U* of the space a set *F*(*U*) of *sections* on *U*, and to each open set *V* included in *U* a map *F*(*U*) → *F*(*V*) giving *restrictions* of sections over *U* to *V*. Each of the examples above defines a presheaf by taking the restriction maps to be the usual restriction of functions, vector fields and sections of a vector bundle. Moreover, in each of these examples the sets of sections have additional

Given a presheaf, a natural question to ask is to what extent its sections over an open set *U* are specified by their restrictions to smaller open sets *V*_{i} of an *U*. A presheaf is *separated* if its sections are "locally determined": whenever two sections over *U* coincide when restricted to each of *V*_{i}, the two sections are identical. All examples of presheaves discussed above are separated, since in each case the sections are specified by their values at the points of the underlying space. Finally, a separated presheaf is a *sheaf* if *compatible sections can be glued together*, i.e., whenever there is a section of the presheaf over each of the covering sets *V*_{i}, chosen so that they match on the overlaps of the covering sets, these sections correspond to a (unique) section on *U*, of which they are restrictions. It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is *local* in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.

On the other hand, a function can be bounded on each set of an (infinite) open cover of a space without being bounded on all of the space; thus bounded functions provide an example of a presheaf that in general fails to be a sheaf. Another example of a presheaf that fails to be a sheaf is the *constant presheaf* that associates the same fixed set (or abelian group, or ring,...) to each open set: it follows from the gluing property of sheaves that the set of sections on a disjoint union of two open sets is the *F _{A}* (associated to for instance a set

Maps between sheaves or presheaves (called

Presheaves and sheaves are typically denoted by capital letters, *F* being particularly common, presumably for the *faisceaux*. Use of calligraphic letters such as is also common.

Other Languages

العربية: حزمة (رياضيات)

asturianu: Teoría de fexes

Deutsch: Garbe (Mathematik)

español: Teoría de haces

فارسی: بافه (ریاضی)

français: Préfaisceau

galego: Teoría de feixes

한국어: 층 (수학)

italiano: Fascio (teoria delle categorie)

עברית: אלומה (מתמטיקה)

Nederlands: Schoventheorie

日本語: 層 (数学)

norsk: Knippe (matematikk)

polski: Snop (matematyka)

português: Teoria dos feixes

русский: Пучок (математика)

suomi: Esilyhde ja lyhde

svenska: Kärve (matematik)

Türkçe: Deste (topoloji)

українська: Пучок (математика)

Tiếng Việt: Không gian Étalé

中文: 层 (数学)