[Paper Review] A Geometric Bohr topos

Simon Henry|arXiv (Cornell University)|Feb 6, 2015

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper introduces a geometric Bohr topos for C*-algebras by constructing a sublocale of the localic completion's lower power locale that classifies commutative localic sub-C*-algebras, ensuring stability under pullbacks and incorporating topological structure. The key contribution is a topos-theoretic framework that refines the discrete Bohr topos by internalizing commutative C*-algebras as localic objects, enabling a geometric logic-based approach to quantum foundations.

ABSTRACT

In this short note, we construct a variant of the Bohr topos of a C*-algebra which takes into account the topology of the algebra in a finer way and such that this construction is stable under pullback along geometric morphisms. This generalizes a construction for finite dimensional algebras of G.Raynaud. Our idea is to construct the Bohr topos of a C*-algebra A as the sublocale of the lower power locale of the localic completion of A which classifies the commutative localic sub-C*-algebras of A.

Motivation & Objective

• To address the limitations of the discrete Bohr topos, which ignores topological structure and lacks stability under pullbacks.
• To generalize Raynaud’s finite-dimensional construction to arbitrary C*-algebras using geometric logic and locale theory.
• To construct a topos that classifies commutative localic sub-C*-algebras of a C*-algebra A, preserving the Bohr doctrine of classical concepts.
• To ensure the construction is stable under pullback along geometric morphisms, satisfying the tovariance principle in topos-theoretic physics.
• To internalize the Gelfand duality for bundles of commutative localic C*-algebras, enabling a spectrum map to the topos.

Proposed method

• Construct the localic completion of a C*-algebra A, treating its underlying set as a locale.
• Use the lower power locale construction to classify fiberwise closed, locally positive sublocales of A×A.
• Define S(A) as the sublocale of the lower power locale of the localic completion that classifies commutative localic sub-C*-algebras of A.
• Leverage the universal property of classifying toposes to ensure stability under pullbacks along geometric morphisms.
• Apply constructive Gelfand duality to the internal bundle of commutative localic C*-algebras over S(A), yielding a spectrum map ΣA→S(A).
• Ensure the construction classifies generalized points corresponding to commutative localic sub-C*-algebras with non-degenerate characters.

Experimental results

Research questions

• RQ1 How can the Bohr topos construction be refined to incorporate topological structure of commutative sub-C*-algebras in a C*-algebra?
• RQ2 Can a topos-theoretic construction of the Bohr topos be made stable under pullbacks along geometric morphisms?
• RQ3 What is the role of localic C*-algebras in generalizing the Bohr topos beyond the discrete case?
• RQ4 How does the spectrum of a bundle of commutative localic C*-algebras relate to the internal geometry of the topos?
• RQ5 What is the relationship between the new geometric Bohr topos and the original discrete Bohr topos Sd(A)?

Key findings

• The geometric Bohr topos S(A) is constructed as a sublocale of the lower power locale of the localic completion of A, classifying commutative localic sub-C*-algebras.
• The construction is stable under pullbacks along geometric morphisms due to its universal property as a classifying topos.
• S(A) admits a canonical bundle of commutative localic C*-algebras, internal to the topos, which supports constructive Gelfand duality.
• The spectrum object ΣA classifies pairs (B,χ) where B is a commutative localic sub-C*-algebra of A and χ is a non-degenerate character of B.
• In classical logic, the points of ΣA correspond to pairs (B,χ) with B a commutative sub-C*-algebra and χ a non-zero character.
• There exists a canonical geometric morphism from the discrete Bohr topos Sd(A) to the geometric Bohr topos S(A), via which the original bundle and spectrum are recovered as pullbacks.