PlanetPhysics/Quantum Categories

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A quantum category   $\mathbb {Q}$  is defined as the (non-Abelian) category of quantum groupoids, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle [Q_{\grp}]_i}
, and quantum groupoid homomorphisms , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle [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.

The category of quantum groupoids, Failed to parse (unknown function "\grp"): {\displaystyle [Q_{\grp}]_i} , is trivially a subcategory of the groupoid category, that can also be regarded as a functor category, or $2$ -category, if Failed to parse (unknown function "\grp"): {\displaystyle \grp} is small, that is, if $G^{0}$ is a set rather than a class.

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

All Sources

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References

↑ 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.
↑ 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.
↑ 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}, 37 No 11., 2669--2733 38 No 3., 827--859, 39 No 6., 1413--1436, 41 No 4., 613--639.

[BIsham1-1] 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.

[ICB71a-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.

[BIsham2-3] 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}, 37 No 11., 2669--2733 38 No 3., 827--859, 39 No 6., 1413--1436, 41 No 4., 613--639.

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