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 {*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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