Interpreting De Finetti's Theorem in the Category of Integrable Cones

We establish a connection between two results in the literature on probabilistic semantics: a formulation of De Finetti's theorem in the language of category theory due to Jacobs and Staton, and the generic construction of the free exponential of Linear Logic by Melliès et al, that has been instantiated in the model of probabilistic coherence spaces by Crubillé et al. The structural proximity of these two constructions is manifest, but making this connection formal requires technical developments on the relationship between the category of stochastic kernels and the category of integrable cones, two well-known categories in probabilistic semantics. We then use this connection to give a characterization of the total elements of the probabilistic coherence space !Bool.