# PlanetPhysics/2C Category

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

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

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

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

$\left\|T\circ S\right\|\leq \left\|S\right\|\left\|T\right\|$ , when the composition is defined, and satisfies the $C^{*}$ -condition: $\left\|S^{*}\circ S\right\|=\left\|S^{2}\right\|;$ 1. for any 2-arrow $S\in Hom(\rho ,\sigma )$ , $S^{*}\circ S$ is a positive element in

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

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