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%%% Primary Title: quantum categories
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%%% Owner: bci1
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\begin{document}
\begin{definition}
A \emph{quantum category} $\Q$ is defined as the (\htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}) \htmladdnormallink{category of quantum groupoids}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid3.html}, $[Q_{\grp}]_i$, \emph{and quantum groupoid homomorphisms}, $[q_{\grp}]_{ij}$, where $i$ and $j$ are indices in an index class, $\mathbf{I}$, all subject to the usual \htmladdnormallink{ETAC axioms}{http://planetphysics.us/encyclopedia/Formula.html} and their interpretations.
\end{definition}
\begin{remark}
The category of quantum groupoids, $[Q_{\grp}]_i$, is trivially a subcategory of the \htmladdnormallink{groupoid category}{http://planetphysics.us/encyclopedia/GroupoidCategory3.html}, that can also be regarded as a \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, or $2$-category, if $\grp$ is small, that is, if $G^0$ is a set rather than a class.
\end{remark}
\begin{remark}
A \htmladdnormallink{physical mathematics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} definition of quantum category has also been reported as a rigid monoidal category, or its equivalents.
\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}
\end{document}
