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; Conceptually, it is based on the idea of germ of a object in a topological space.
* Hist: The concept of sheaf was was first formulated by Leray and Cartan in the 1950s.
$ Def: A sheaf is a presheaf F such that for all U in T(X) and every covering {Ua} of U, the following sequence is exact,

0 → F(U) →i*a F(Ua) →j*a,b F(Ua ∩ Ub) ,

where i : Ua U and j is the difference of Ua Ub Ua and Ub.
* 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 π: SM, such that
(1) π is a local homeomorphism of S onto M;
(2) π–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, π, 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; Mallios 98 [vector sheaves]; Mallios & Zafiris 15 [differential sheaves].
> Online resources: see Wikipedia page.

And Physics
* Applications: The theory of presheaves has applications in operator algebras and foundations of quantum mechanics [@ de Groote mp/01].
@ General references: Mallios & Zafiris 15 [differential sheaves].
@ Spacetime sheaves: Raptis IJTP(00)gq/01, IJTP(01)gq.
@ Field theories: Mallios 05 [Maxwell fields].
@ Quantum theory: Raptis gq/01-conf [quantum logic]; Constantin a1510-PhD [and information theory]; > s.a. quantum states [generalization to quantum sheaves].


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