nLab spectral presheaf

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.

Related concepts

• Bohr topos

• daseinisation

References

The term “spectral presheaf” was introduced in

• Jeremy Butterfield, John Hamilton, Chris Isham, A topos perspective on the Kochen-Specker theorem, I. quantum states as generalized valuations, Internat. J. Theoret. Phys. 37(11):2669–2733, 1998, MR2000c:81027, doi; II. conceptual aspects and classical analogues Int. J. of Theor. Phys. 38(3):827–859, 1999, MR2000f:81012, doi; III. Von Neumann algebras as the base category, Int. J. of Theor. Phys. 39(6):1413–1436, 2000, arXiv:quant-ph/9911020, MR2001k:81016,doi; IV. Interval valuations, Internat. J. Theoret. Phys. 41 (2002), no. 4, 613–639, MR2003g:81009, doi

• Chris Isham, Andreas Döring, A Topos Foundation for Theories of Physics, (arXiv:quant-ph/0703060, arXiv:quant-ph/0703062, arXiv:quant-ph/0703066)