Preliminary Data.
There are several distinct definitions of quantum topos in the Mathematical Physics literature attempting to redefine the quantum logic that was first introduced by von Neumann and Birkhoff for the foundation of Quantum Mechanics. The definitions of quantum topoi published so far are not, however, those of `quantum' categories (previously introduced as rigid monoidal categories) - with finite limits and power objects.

Heyting Logic Concept and Algebraic Structure

Definition 0.2   A Heyting lattice is a Brouwer-intuitionistic logic lattice with a bottom, or lowest element 0. In the more technical classification it is a commutative lattice which is both `pseudocomplemented and also relatively pseudocomplemented'. The concept of relative pseudocomplementation coincides with the material implication operator, , in symbolic propositional logic based on chryssippian or Boolean logic.

Definition 0.3   A Heyting algebra is a p-algebra (as defined next in Definition 1.3 ) with the relative pseudocomplentation operation (which replaces the propositional implication ).
Given an element in a bounded lattice , a complement of is defined to be an element , if such an element exists, such that

To surmount the non-uniqueness of the complement, an alternative to the latter was defined- the pseudocomplement of an element.

An element in a lattice with 0 is a pseudocomplement of if

1. 
2. for any such that then .

In other words, is the maximal element in the set .

Definition 0.4   A convenient modification of the pseudocomplemented (pc) lattice concept is a p-algebra (or pseudocomplemented algebra) which is a pc-lattice where is regarded as an algebraic operator. Thus, a morphism of pc-lattices is a proper lattice homomorphism, whereas a morphism between two p-algebras is a lattice homomorphism that also preserves the pc-algebraic operation , i.e., . One can therefore define a category of p-algebras by specifying the morphism between any pair of p-algebras (considered as objects of this algebraic logic category) as the -lattice homomorphism, with the following condition being also satisfied.

Remark Unlike the Heyting lattice, an -logic algebra has a non-commutative lattice structure and is therefore considered as a stronger candidate for quantum logics, including those based on the orthomodular lattices of the original quantum logic of Birkhoff and von Neumann. Thus, a generalized topos defined with a subobject classifier based on -logic algebra may provide suitable representations of arbitrary quantum state spaces.

Bibliography

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.

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.