quantum categories

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   A physical mathematics definition of quantum category has also been reported as a rigid monoidal category, or its equivalents.

Bibliography

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.33-89.

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 1-4, 1971, the University of Bucharest.

3
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.



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