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# quantum category

###### Definition 0.1.

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

###### Remark 0.1.

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

###### Remark 0.2.

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 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.

# 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.33β89. - 2
Baianu, I.C.: 1971a, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.),
*Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science*, September 1β4, 1971, the University of Bucharest. - 3
Butterfield, J. and C. J. Isham: 1998, 1999, 2000β2002, A topos
perspective on the KochenβSpecker theorem I - IV,
*Int. J. Theor. Phys*, 37 No 11., 2669β2733 38 No 3., 827β859, 39 No 6., 1413β1436, 41 No 4., 613β639.

## Mathematics Subject Classification

18-00*no label found*81P05

*no label found*81P10

*no label found*

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