Definition Representable Markov category [efr-38FR]

A Markov category is representable if the inclusion \mathcal {C}_\mathrm {det} \hookrightarrow \mathcal {C} admits a right adjoint. In this case we denote the right adjoint P and call the object PX for X \in \mathcal {C} a distribution object for X. Observe that \mathcal {C}(X,Y) = \mathcal {C}_\mathrm {det}(X,PY) (by definition,) and hence \mathcal {C} = Kl(P) where we denote the induced monad on \mathcal {C}_\mathrm {det} P by an abuse of notation.