Definition 0.1  

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;
3. 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: $$\left\|S^* \circ S\right\| = \left\|S^2\right\|; $$
4. for any 2-arrow $S \in Hom(\rho, \sigma)$ , $S^* \circ S$ is a positive element in $Hom(\rho, \rho)$ , that is often denoted as $End(\rho)$ .

Remark 0.1   With the above notations, 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 $$s(\iota A) = t(\iota A) = A.$$