论文标题
免费的GS-Onoidal类别和免费的马尔可夫类别
Free gs-monoidal categories and free Markov categories
论文作者
论文摘要
最近，通过马尔可夫类别的形式主义，分类概率取得了重大进展，其中几种经典定理已被完全抽象的分类术语证明。与Markov类别密切相关的是GS-Onoidal类别，也称为CD类别。这些省略了实现概率归一化的条件。为了扩展Corradini和gadducci的工作，我们构建了由任意敏锐和舒适性的形态集合而产生的自由GS-Onoidal和自由马尔可夫类别。对于自由的GS - 旋律类别，这是针对其形态的明确组合描述的形式，是标记超图的结构化sospans。这些可以将其视为GS-konoidal string图（$ = $ enter图）作为组合数据结构的形式化。我们根据Walters的思想制定了适当的$ 2 $分类性普遍财产，并证明我们的类别满足了它。 我们希望我们的免费类别与计算机实施相关，我们还认为它们可以用作概括贝叶斯网络的统计因果模型。
Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams ($=$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.
