# PlanetPhysics/Quantum Categories

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


The category of quantum groupoids, $\displaystyle [Q_{\grp}]_i$ , is trivially a subcategory of the groupoid category, that can also be regarded as a functor category, or ${\displaystyle 2}$-category, if $\displaystyle \grp$ is small, that is, if ${\displaystyle G^{0}}$ 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.

## 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. ) \emph{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, \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.