1 Quantum logic topoi
Definition 1.1. A quantum logic topos (QLT) is defined as an extension of the concept of
topos in which the Heyting logic algebra (or subobject classifier) of the standard elementary
topos is replaced by a quantum logicwhich is axiomatically defined by non-commutative and
non-distributive lattice structures.
1.1 Remark
Quantum logic topoi are thus generalizations of the Birkhoff and von Neumann definition of
quantum state spaces based on their definition of a quantum logic (lattice), as well
as a non-Abelian, higher dimensional extension of the recently proposed concept of a
‘quantum’ topos which employs the (commutative) Heyting logic algebra as a subobject
classifier.
Some specific examples are considered in the following two recent references.
References
[1] Butterfield, J. and C. J. Isham: 2001, Space-time and the philosophical challenges of
quantum gravity., in C. Callender and N. Hugget (eds. ) Physics Meets Philosophy at the
Planck scale., Cambridge University Press,pp.33–89.
[2] Butterfield, J. and C. J. Isham: 1998, 1999, 2000–2002, A topos perspective on the
Kochen–Specker theorem I - IV, Int. J. Theor. Phys, 37 No 11., 2669–2733 38 No 3.,
827–859, 39 No 6., 1413–1436, 41 No 4., 613–639.
