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quantum categories (Topic)
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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See Also: category theory

Other names:  quantum topos
Also defines:  quantum category, Q(gpd), quantum topos, quantum groupoid homomorphism, rigid monoidal category, generalized quantum topoi, braided monoidal category
Keywords:  quantum categories, braided monoidal categories

Cross-references: physical mathematics, functor category, groupoid category, ETAC axioms, category of quantum groupoids, non-Abelian
There are 13 references to this object.

This is version 6 of quantum categories, born on 2009-02-17, modified 2009-02-26.
Object id is 536, canonical name is QuantumCategories.
Accessed 2061 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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