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：通过马可夫类别的形式主义,最近分类概率有了显著的进步,其中以完全抽象的绝对术语证明了几个古典理论。与马尔科夫类别密切相关的是Gs-monodal类别,也称为CD类别。这些分类省略了一个实现概率正常化的条件。科拉迪尼和加杜奇的工作扩展,我们根据Walters的想法构建了免费的gs-monoidal和free Markov类别。对于自由的Gs-monodal类别,我们期望我们的自由类别与计算机实施相关,我们还认为,这些类别可以用作通用巴伊斯网络的统计因果模型。