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) MR0221500 Zbl 0158.20505
[a2]	R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) MR0102797 Zbl 0080.16201