Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams ($=$term graphs) as a combinatorial data structure. We formulate the appropriate $2$-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.

翻译：范畴概率论近期通过Markov范畴的形式化取得了显著进展，若干经典定理已在完全抽象的范畴术语中得到证明。与Markov范畴密切相关的是gs-幺半范畴（亦称CD范畴），后者省略了实现概率归一化的条件。基于Corradini与Gadducci的研究工作，我们构造了由任意元数与余元数的态射集合生成的自由gs-幺半范畴和自由Markov范畴。对于自由gs-幺半范畴，我们给出了其态射的显式组合描述，即结构化的带标签超图cospan。这可被视为gs-幺半弦图（=项图）作为组合数据结构的严格形式化。基于Walters的思想，我们构建了相应的2-范畴泛性质，并证明所构造的范畴满足该性质。预期我们的自由范畴在计算机实现中具有应用价值，同时论证了它们可作为泛化贝叶斯网络的统计因果模型。