Interpreting De Finetti's theorem in the Category of Integrable Cones (long version)
For researchers in probabilistic semantics and categorical logic, this work bridges two previously separate constructions, but the result is incremental as it formalizes an already suspected connection.
The paper formalizes a connection between a categorical formulation of De Finetti's theorem and the free exponential of Linear Logic in probabilistic coherence spaces, using technical developments on the relationship between stochastic kernels and integrable cones. This connection yields a characterization of the total elements of the probabilistic coherence space !Bool.
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.