Categorical supermaps on any generalised theory with channel-state duality are exactly CJ-supermaps, recovering classical, quantum, and NSWSE-Boxworld supermaps.
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Higher order quantum map types are identified with Boolean type functions, with comb types corresponding to chain posets, and type functions decomposed via max/min of basic chains corresponding to affine mixtures and intersections.
Diagonal-unitary-covariant superchannels are exactly characterized by four independent block components A,B,C,D with explicit complete-positivity and trace-preservation conditions.
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On the structure of higher order quantum maps
Higher order quantum map types are identified with Boolean type functions, with comb types corresponding to chain posets, and type functions decomposed via max/min of basic chains corresponding to affine mixtures and intersections.