# Topos quantum theory with short posets

## TL;DR

This paper modifies topos quantum theory by using a smaller poset of abelian subalgebras, simplifying the framework while preserving core results and revealing connections to projective geometry.

## Contribution

It introduces a new topos based on a restricted poset of height at most two, maintaining key results and offering conceptual simplifications.

## Key findings

- Core results (a)--(d) are preserved with the smaller poset
- The smaller poset simplifies the topos structure
- Connections to projective geometry are highlighted

## Abstract

Topos quantum mechanics, developed by Isham et. al., creates a topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N and measures on the spectral presheaf; and (d) a model of dynamics in terms of V(N). We consider a modification to this approach using not the whole of the poset V(N), but only its elements of height at most two. This produces a different topos with different internal logic. However, the core results (a)--(d) established using the full poset V(N) are also established for the topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.