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# quantum logic toposes

This is a topic entry on extensions of standard and elementary toposes to quantum topoi founded upon many-valued logics.

###### Definition 0.1.

*A quantum logic topos* (*QLT*) is defined as an *extension of the concept of a topos* in which the Heyting algebra or subobject classifier of the standard elementary topos is replaced by a *quantum logic* that is axiomatically defined by a *non-commutative* lattice structure such as that of a many valued,
$LM_{n}$-logic algebra, modified to a non-distributive lattice structure corresponding to that of the quantum
physics events.

###### Remark 0.1.

Quantum logics 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.

## Mathematics Subject Classification

81P68*no label found*18B25

*no label found*81P15

*no label found*81P10

*no label found*

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