PlanetPhysics/2C Category

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A ${\displaystyle 2-C^{*}}$ -category , ${\displaystyle {{\mathcal {C}}^{*}}_{2}}$, is defined as a (small) 2-category for which the following conditions hold:

1. for each pair of ${\displaystyle 1}$-arrows ${\displaystyle (\rho ,\sigma )}$ the space ${\displaystyle Hom(\rho ,\sigma )}$ is a complex Banach space.
2. there is an anti-linear involution `${\displaystyle *}$' acting on ${\displaystyle 2}$-arrows, that is,

${\displaystyle *:Hom(\rho ,\sigma )\to Hom(\rho ,\sigma )}$, ${\displaystyle S\mapsto S^{*}}$ , with ${\displaystyle \rho }$ and ${\displaystyle \sigma }$ being ${\displaystyle 2}$-arrows;

1. the Banach norm is sub-multiplicative (that is,

${\displaystyle \left\|T\circ S\right\|\leq \left\|S\right\|\left\|T\right\|}$, when the composition is defined, and satisfies the ${\displaystyle C^{*}}$ -condition: ${\displaystyle \left\|S^{*}\circ S\right\|=\left\|S^{2}\right\|;}$

1. for any 2-arrow ${\displaystyle S\in Hom(\rho ,\sigma )}$, ${\displaystyle S^{*}\circ S}$ is a positive element in

${\displaystyle Hom(\rho ,\rho )}$, (denoted also as ${\displaystyle End(\rho )}$).

Note: The set of ${\displaystyle 2}$-arrows ${\displaystyle End(\iota A)}$ is a commutative monoid, with the identity map ${\displaystyle \iota :{\mathcal {C}}_{0}^{2*}\to {\mathcal {C}}_{1}^{2*}}$ assigning to each object ${\displaystyle A\in {\mathcal {C}}_{0}^{2*}}$ a ${\displaystyle 1}$-arrow ${\displaystyle \iota A}$ such that ${\displaystyle s(\iota A)=t(\iota A)=A.}$