Definition [lcc-000S]

Fix any Markov category \mathcal {C}

There is a (pseudo) double category where

• The objects are objects of \mathcal {C}
• The vertical category is precisely \mathcal {C}
• The horizontal morphisms X \to Y are tuples (\Omega \in \mathcal {C}, \mathbb {P}: I \to \Omega , f: \Omega \otimes X \to Y), where f is deterministic, with composition given in the obvious way by tensoring and composing
• A 2-cell consists of a map \Omega _1 \to \Omega _2 in \mathsf {Prob}(\mathcal {C}) making the obvious square commute

If \mathcal {C} is causal, there is an extension of this double category with the same 0- and 1-cells, but where the 2-cells (\Omega _1, \mathbb {P}_1, f_1) \to (\Omega _2, \mathbb {P}_2, f_2) are required to make the diagram commute only \mathbb {P}_1-almost certainly