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.
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Supermaps on generalised theories
Categorical supermaps on any generalised theory with channel-state duality are exactly CJ-supermaps, recovering classical, quantum, and NSWSE-Boxworld supermaps.