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.

分类概率最近通过马尔可夫类别的形式主义取得了重大进展，其中几个经典定理已经用完全抽象的分类术语得到证明。与 Markov 类别密切相关的是 gs-monoidal 类别，也称为 CD 类别。这些省略了实现概率归一化的条件。扩展 Corradini 和 Gadducci 的工作，我们构建了自由 gs-monoidal 和自由马尔可夫范畴，这些范畴由任意元性和余性态射的集合生成。对于自由 gs-monoidal 类别，这以其态射的显式组合描述的形式出现，作为标记超图的结构化 cospans。这些可以被认为是 gs-monoidal 弦图的形式化 ( \(=\)术语图）作为组合数据结构。我们根据 Walters 的想法制定了适当的 2-categorical 通用属性，并证明我们的类别满足它。我们希望我们的自由类别与计算机实现相关，我们还认为它们可以用作概括贝叶斯网络的统计因果模型。