Categories and functors in reverse and computable mathematics

IF 0.4 4区数学 Q1 Arts and Humanities

Archive for Mathematical Logic Pub Date : 2024-12-31 DOI:10.1007/s00153-024-00962-1

Huishan Wu

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引用次数: 0

Abstract

This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to \(\mathrm ACA_{0}\) over the base system \(\mathrm RCA_{0}\). The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in \(\mathrm RCA_{0}\); as an application, we formalize an effective version of the Yoneda Embedding in \(\mathrm RCA_{0}\). Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics

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逆数学和可计算数学中的范畴和函子

本文研究了逆数学和可计算数学中的范畴和函子。在普通的逆向数学中，我们只关注对象和态射可以用自然数表示的范畴。我们首先考虑范畴的态射集，并证明了在基系统\(\mathrm RCA_{0}\)上等价于\(\mathrm ACA_{0}\)的几个相关定理。约田引理是范畴论和同调代数中的一个基本结果。然后，我们在\(\mathrm RCA_{0}\)中开发了Yoneda引理的有效版本；作为一个应用程序，我们在\(\mathrm RCA_{0}\)中形式化了Yoneda嵌入的有效版本。积和余积是定义半加性类和加性类等特殊类的基本概念。研究了一类对象的乘积和余积的性质，给出了半加性范畴和加性范畴用乘积和余积的有效刻画。最后，我们用高阶逆数学的方法进一步考虑了本文所研究的范畴论定理的强度

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来源期刊

Archive for Mathematical Logic MATHEMATICS-LOGIC

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.