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:

$$ * : Hom(\rho, \sigma) \to Hom(\rho, \sigma)$$, $$ S \mapsto S^*$$, with $$\rho$$ and $$\sigma$$ being $$2$$-arrows; $$\left\|T \circ S\right\| \leq \left\|S\right\|\left\|T\right\|$$, when the composition is defined, and satisfies the $$C^*$$ -condition: $$\left\|S^* \circ S\right\| = \left\|S^2\right\|; $$ $$Hom(\rho, \rho)$$, (denoted also as $$End(\rho)$$).
 * 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,
 * 1) the Banach norm is sub-multiplicative (that is,
 * 1) for any 2-arrow $$S \in Hom(\rho, \sigma)$$, $$S^* \circ S$$ is a positive element in

Note: The set of $$2$$-arrows $$End(\iota A)$$ is a commutative monoid, with the identity map $$\iota : \mathcal{C}^{2*}_0 \to \mathcal{C}^{2*}_1$$ assigning to each object $$A \in \mathcal{C}^{2*}_0$$ a $$1$$-arrow $$\iota A$$ such that $$s(\iota A) = t(\iota A) = A.$$