-combs. Furthermore, our framework is general enough to accommodate recently-proposed generalisations of classical and quantum theory where processes only need to have a fixed causal ordering locally, but globally allow indefinite causal ordering. To illustrate this point, we show that certain processes of this kind, such as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner, and a classical three-party example due to Baumeler, Feix, and Wolf are all instances of a certain family of processes we refer to as

\textrm{SOC}_n

in the appropriate category of higher-order causal processes. After defining these families of causal structures within our framework, we give derivations of their operational behaviour using simple, diagrammatic axioms. Comment: Extended version of a LICS 2017 paper with the same title

DOI: 10.23638/lmcs-15(3:15)2019

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n

\textrm{SOC}_n

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\textrm{SOC}_n