Topos quantum theory with short posets

Topos quantum mechanics, developed by Döring (2008); Döring and Harding Houston J. Math. 42(2), 559–568 (2016); Döring and Isham (2008); Flori 2013)); Flori (2018); Isham and Butterfield J. Theoret. Phys. 37, 2669–2733 (1998); Isham and Butterfield J. Theoret. Phys. 38, 827–859 (1999); Isham et al. J. Theoret. Phys. 39, 1413–1436 (2000); Isham and Butterfield J. Theoret. Phys. 41, 613–639 (2002), 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 V(N)∗ 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.