Definition The category of random morphisms [lcc-000N]

Let \mathcal {C} = (\mathcal {C}, \otimes , I) be a Markov category. Then the category of random morphisms in \mathcal {C}, denoted \mathsf {RandMor}(\mathcal {C}) is defined as follows:

• The objects are the objects of \mathcal {C}.
• A morphism X \to Y is given by an object \Omega \in \mathcal {C}, a state I \to \Omega and a deterministic map X \otimes \Omega \to Y, up to the equivalence relation that identifies (\Omega _1, \mu _1, f_1) \sim (\Omega _2, \mu _2, f_2) if there exists a map \phi : \Omega _1 \to \Omega _2 so that \phi \mu _1 = \mu , f_1(X \otimes \phi ) = f_2
• The identity on X is (I, 1_I, \lambda _x: I \otimes X \cong X)
• Composition is given by (\Omega _1, \mu _1, f_1) \circ (\Omega _2, \mu _2, f_2) = (\Omega _1 \otimes \Omega _2, \mu _1 \otimes \mu _2, f_1 \circ (\Omega _1 \otimes f_2)) (the only thing that makes sense)