Sheaf Theory| 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];
> s.a. contextuality.
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:
S → M, 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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send feedback and suggestions to bombelli at olemiss.edu – modified 22 oct 2019