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quantum category

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\begin{document}
\begin{definition}
A \emph{quantum category} $\Q$ is defined as the (non-Abelian) category of quantum groupoids, $[Q_{\grp}]_i$, 
and quantum groupoid homomorphisms, $[q_{\grp}]_{ij}$, where $i$ and $j$ are indices in an 
index class, $\mathbf{I}$, all subject to the usual ETAC axioms and their interpretations.
\end{definition}


\begin{remark}
  The category of quantum groupoids, $[Q_{\grp}]_i$, is trivially a subcategory of the groupoid category, that can also
be regarded as a functor category, or $2$-category, if  $\grp$ is small, that is, if $G^0$ is a set rather than a class. 
\end{remark}


\begin{remark}
 One notes that an alternative definition of quantum category has also been reported
in physical mathematics as a rigid monoidal category, or its equivalent. A more 
general definition of {\em quantum category} is however necessary that has both quantum groups
and locally compact quantum groupoids as particular cases. This would require a notion
of quantum compactness in a category as well as the definition of associated Haar systems 
to a category.
\end{remark}

\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{ICB71a}
Baianu, I.C.: 1971a, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), \emph{Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science}, September 1--4, 1971, the University of Bucharest.


\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}


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