Free gs-Monoidal Categories and Free Markov Categories

IF 0.6 4区数学 Q3 MATHEMATICS

Applied Categorical Structures Pub Date : 2023-04-08 DOI:10.1007/s10485-023-09717-0

Tobias Fritz, Wendong Liang

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引用次数: 19

Abstract

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.

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自由gs-一元范畴和自由马尔可夫范畴

通过马尔可夫范畴的形式主义，分类概率最近取得了重大进展，其中几个经典定理已经用完全抽象的分类术语证明了。与马尔可夫范畴密切相关的是gs-一元范畴，也称为CD范畴。这些省略了实现概率归一化的条件。扩展了Corradini和Gadducci的工作，构造了由任意性和协性态射集合生成的自由gs-一元和自由马尔可夫范畴。对于自由的gs-一元范畴，这是以其态射作为标记超图的结构化共张的显式组合描述的形式出现的。这些可以看作是gs-monoidal string diagrams (\(=\) term graphs)作为组合数据结构的形式化形式。在沃尔特斯思想的基础上给出了合适的二范畴全称性质，并证明了我们的范畴满足这一性质。我们希望我们的自由类别与计算机实现相关，我们也认为它们可以用作推广贝叶斯网络的统计因果模型。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Categorical Structures 数学-数学

CiteScore

1.30

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant. Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.