Sheaf
A sheaf is a pre-sheaf (cf. also Sheaf theory) on a topological space
such that for every union
of open subsets
of
the following conditions are satisfied:
a) if on every the restrictions of two elements
and
in
coincide, then
;
b) if are such that for any pair of indices
and
the restrictions of
and
to
coincide, then there exists an element
which on each
has restriction coinciding with
.
Every sheaf on is isomorphic to the sheaf of continuous sections of a certain covering space
over
, which is determined uniquely up to an isomorphism (by a covering space one means a continuous mapping from
onto
which is a local homeomorphism), therefore a sheaf is also commonly understood to be the covering space
itself (see Sheaf theory).
Comments
Generalizing the above notion of a sheaf on a topological space, it is also possible to define sheaves on an arbitrary site. Cf. also Topos.
For a more detailed treatment of sheaves, and additional references, see Sheaf theory.
References
[a1] | G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) |
[a2] | R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) |
Sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sheaf&oldid=17372