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.

翻译：我们建立了概率语义学文献中两个结果之间的联系：Jacobs与Staton基于范畴论语言表述的De Finetti定理，以及Melliès等人提出的线性逻辑自由指数构造的通用框架——该框架已由Crubillé等人实例化于概率相干空间模型中。这两种构造在结构上具有显著相似性，但使这种联系形式化需要解决随机核范畴与可积锥范畴（概率语义学中两个著名范畴）之间的关系这一技术性问题。随后，我们利用这一关联对概率相干空间!Bool中的全元素进行了刻画。