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 {*U _{a}*} of

0 → *F*(*U*) →_{i*}
∏_{a}*F*(*U _{a}*)
→

where *i *: *U _{a}* →

*

$

(1)

(2)

(3) The composition laws are continuous in the topology on

$

$

@

>

**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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