Definitions
1.NounAn abstract mathematical construct which associates data to the open sets of a topological space, generalizing the situation of functions, fiber bundles, manifold structure, etc. on a topological space (but not necessarily in such a way as to make the local and global data compatible, as in a sheaf). Formally, A contravariant functor
F
{\displaystyle {\mathcal {F}}}
whose domain is a category whose objects are open sets of a topological space (called the base space or underlying space) and whose morphisms are inclusion mappings. The image of each open set under
F
{\displaystyle {\mathcal {F}}}
is an object whose elements are called sections, and are which are said to be over the given open set; the image of each inclusion map
A
→
B
{\displaystyle A\to B}
under
F
{\displaystyle {\mathcal {F}}}
is a morphism
F
(
B
)
→
F
(
A
)
{\displaystyle {\mathcal {F}}(B)\to {\mathcal {F}}(A)}
, called the restriction from
B
{\displaystyle B}
to
A
{\displaystyle A}
and denoted
res
B
,
A
{\displaystyle \operatorname {res} _{B,A}}
or
|
B
,
A
{\displaystyle |_{B,A}}
.