PlanetPhysics/2C Category

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A $2-C^{*}$ -category , ${{𝒞}^{*}}_2$, 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 [[../NormInducedByInnerProduct/|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 [[../NormInducedByInnerProduct/|norm]] is sub-multiplicative (that is,

$\Vert T\circ S\Vert \le \Vert S\Vert \Vert T\Vert$, when the [[../Cod/|composition]] is defined, and satisfies the $C^{*}$ -condition: $\Vert S^{*}\circ S\Vert =\Vert S^2\Vert ;$

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 [[../TrivialGroupoid/|monoid]], with the [[../Cod/|identity]] map $\iota :{𝒞}_0^{2*}\to {𝒞}_1^{2*}$ assigning to each [[../TrivialGroupoid/|object]] $A\in {𝒞}_0^{2*}$ a $1$-arrow $\iota A$ such that $s(\iota A)=t(\iota A)=A.$