作为信息Transformers理论背景的Monoidal Kleisli范畴对象上的附加结构方法
Method of Additional Structures on the Objects of a Monoidal Kleisli Category as a Background for Information Transformers Theory
摘要：
范畴论提供了一种以统一的方式编码数学结构的紧凑方法，从而使得能够使用关于等价和通用构造的一般定理。本文发展了Monoidal Kleisli范畴对象的附加结构方法。提出了将任何一致的信息Transformers类看作满足一定公理集的范畴的态射族。这使得有可能以统一的方式研究不同类型的IT，例如，统计的、多值的和模糊的IT。提出的公理定义一个类别的IT作为monoidal类别，其中包含一个子类别（确定性IT）有限的产品。此外，它表明，许多范畴的IT可以构造为Kleisli范畴与额外的结构。
英文摘要：
Category theory provides a compact method of encoding mathematical structures in a uniform way, thereby enabling the use of general theorems on, for example, equivalence and universal constructions. In this article we develop the method of additional structures on the objects of a monoidal Kleisli category. It is proposed to consider any uniform class of information transformers (ITs) as a family of morphisms of a category that satisfy certain set of axioms. This makes it possible to study in a uniform way different types of ITs, e.g., statistical, multivalued, and fuzzy ITs. Proposed axioms define a category of ITs as a monoidal category that contains a subcategory (of deterministic ITs) with finite products. Besides, it is shown that many categories of ITs can be constructed as Kleisli categories with additional structures.
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