Sheaf Theory

Presheaves
* Idea: A presheaf of rings (groups, algebras, sets, ...) is a contravariant functor from the category of open sets on a topological space (and embeddings) to the category of rings (groups, algebras, sets, ...).
@ References: de Groote mp/01 [on a quantum lattice].

Sheaves > s.a. ring space; sheaf cohomology.
* Idea: A sheaf is a kind of bundle or fiber bundle.
$ Def: A sheaf is a presheaf F such that for all U in T(X) and every covering {U_a} of U, the following sequence is exact,

0 → F(U) →_{i* a} F(U_a) →_{j* a,b} F(U_a  U_b) ,

where i : U_a → U and j is the difference of U_a U_b → U_a and U_b.
* Example: The association of the ring of k-smooth functions on an open set U (which defines a k-differentiable structure on it).
$ Sheaf of K-modules over M: A topological space S with a map : S → M, such that
(1) is a local homeomorphism of S onto M;
(2) ^–1(m) is a K-module for each m in M;
(3) The composition laws are continuous in the topology on S.
$ Sheaf of Abelian groups: A fiber bundle (E, B, , F, G), with fiber F a zero-dimensional abelian Lie group with a G-action of isomorphisms.
$ Structure sheaf: A sheaf of rings on a topological space.
@ References: Godement 58; Swan 64; Bredon 67; Kashiwara & Schapira 90.

And Physics
* Applications: The theory of presheaves has applications in operator algebras and foundations of quantum mechanics [@ de Groote mp/01].
@ Spacetime sheaves: Raptis IJTP(00)gq/01, IJTP(01)gq.
@ Various theories: Raptis gq/01-in [quantum logic]; Mallios 05 [Maxwell fields].

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