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\begin{document}
\section{Quantum logic topoi}
\begin{definition}
A \emph{\htmladdnormallink{quantum logic}{http://planetphysics.us/encyclopedia/TheoryOfHilbertLattices.html} \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html}} (\emph{QLT}) is defined as an extension of the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of topos in which the Heyting logic algebra (or subobject classifier) of the standard \htmladdnormallink{elementary topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} is replaced by a \emph{quantum logic}which is axiomatically defined by \emph{\htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} and non-distributive} lattice structures.
\end{definition}
\subsection{Remark}
Quantum logic topoi are thus generalizations of the Birkhoff and von Neumann definition of \htmladdnormallink{quantum state spaces}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} based on their definition of a quantum logic (lattice), as well as a \emph{\htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html}}, higher dimensional extension of the recently proposed concept of a `\htmladdnormallink{quantum' topos}{http://planetphysics.us/encyclopedia/QuantumCategories.html} which employs the (\emph{commutative}) \htmladdnormallink{Heyting logic algebra as a subobject classifier}{http://planetphysics.us/encyclopedia/SUSY2.html}.
Some specific examples are considered in the following two recent references.
\begin{thebibliography}{9}
\bibitem{BIsham1}
Butterfield, J. and C. J. Isham: 2001, Space-time and the
philosophical challenges of quantum gravity., in C. Callender and
N. Hugget (eds. ) \emph{Physics Meets Philosophy at the Planck
scale.}, Cambridge University Press,pp.33--89.
\bibitem{BIsham2}
Butterfield, J. and C. J. Isham: 1998, 1999, 2000--2002, A topos
perspective on the Kochen--Specker theorem I - IV, \emph{Int. J.
Theor. Phys}, \textbf{37} No 11., 2669--2733 \textbf{38} No 3.,
827--859, \textbf{39} No 6., 1413--1436, \textbf{41} No 4.,
613--639.
\end{thebibliography}
\end{document}
