# nLab spectral presheaf

Contents

topos theory

## Theorems

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# Contents

## Idea

Given a C*-algebra $A$, not necessarily commutative, write $ComSub(A)$ for its poset of commutative subalgebras, whose morphisms are the inclusion maps. By Gelfand duality, the presheaf topos $PSh(ComSub(A))$ over this contains a canonical object, namely the presheaf

$\Sigma \;\colon\; C \mapsto \Sigma_C$

which maps a commutative C*-algebra $C \hookrightarrow A$ to (the point set underlying) its Gelfand spectrum $\Sigma_C$. This is called the spectral presheaf of $A$ (Isham-Döring 07)

## Properties

The Kochen-Specker theorem of quantum mechanics is equivalent to the statement that for $H$ a (complex) Hilbert space of dimension greater than 2, then the spectral presheaf of the algebra of bounded operators $\mathcal{B}(H)$ (the quantum observables) has no global element (Butterfield-Hamilton-Isham 98). (This observation motivates the topos-theoretic development in (Isham-Döring 07)).

See at Kochen-Specker theorem and at Bohr topos for more on this.

## References

The term “spectral presheaf” was introduced in

Last revised on October 3, 2013 at 10:31:24. See the history of this page for a list of all contributions to it.