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A -category , , is defined as a (small)
2-category for which the following conditions hold:
- for each pair of -arrows the space is a complex Banach space.
- there is an anti-linear involution `' acting on -arrows, that is,
, , with and being -arrows;
- the Banach norm is sub-multiplicative (that is,
, when the composition is defined,
and satisfies the -condition:
- for any 2-arrow , is a positive element in
, (denoted also as ).
Note:
The set of -arrows is a commutative monoid, with the identity map
assigning to each object a -arrow such that