PlanetPhysics/Quantum Categories

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

A quantum category  $\mathbb{Q}$ is defined as the ([[../AbelianCategory3/|non-Abelian]]) [[../QuantumFundamentalGroupoid3/|category of quantum groupoids]], Failed to parse (unknown function "\grp"): {\displaystyle [Q_{\grp}]_i}
, and quantum groupoid homomorphisms , Failed to parse (unknown function "\grp"): {\displaystyle [q_{\grp}]_{ij}}
, where $i$ and $j$ are indices in an index class, $𝐈$, all subject to the usual [[../Formula/|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 [[../GroupoidCategory3/|groupoid category]], that can also be regarded as a [[../TrivialGroupoid/|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 [[../PhysicalMathematics2/|physical mathematics]] definition of quantum category has also been reported as a rigid monoidal category, or its equivalents.