Definition 0.1.

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 (ρ,σ) the space Hom(ρ,σ) is a complex Banach space.

2.

there is an anti-linear involution ‘∗’ acting on 2-arrows, that is, ∗ : Hom(ρ,σ) → Hom(ρ,σ), SS^∗ , with ρ and σ being 2-arrows;

3.

the Banach norm is sub-multiplicative (that is,

, when the composition is defined, and satisfies the C^∗ -condition:

4.

for any 2-arrow S ∈ Hom(ρ,σ), S^∗∘ S is a positive element in Hom(ρ,ρ), (denoted also as End(ρ)).