What should a generic object be?

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2022-10-09 DOI:10.1017/S0960129523000117
Jonathan Sterling
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Abstract

Abstract Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of (split) generic objects, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, $\lambda2$ -fibrations, triposes, and others. We observe that a split generic object need not in particular be a generic object under the given definitions, and that the definitions of polymorphic fibrations, triposes, etc. are strict enough to rule out some fundamental examples: for instance, the fibered preorder induced by a partial combinatory algebra in realizability is not a tripos in this sense. We propose a new alignment of terminology that emphasizes the forms of generic object appearing most commonly in nature, i.e. in the study of internal categories, triposes, and the denotational semantics of polymorphism. In addition, we propose a new class of acyclic generic objects inspired by recent developments in higher category theory and the semantics of homotopy type theory, generalizing the realignment property of universes to the setting of an arbitrary fibration.
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泛型对象应该是什么?
摘要Jacobs提出了纤维范畴的(弱、强、分裂)泛型对象的定义;在他对(分裂)泛型对象的定义的基础上,Jacobs开发了一系列重要的fibrational结构,并将其应用于分类逻辑和计算机科学,包括高阶fibrations、多态fibrations,$\lambda2$-fibrations和triposes等。我们观察到,在给定的定义下,分裂的泛型对象不一定是泛型对象,多态fibrations、tripoles等的定义足够严格,可以排除一些基本的例子:例如,在可实现性中由部分组合代数诱导的fibed预序在这个意义上不是tripols。我们提出了一种新的术语组合,强调在自然界中最常见的一般对象的形式,即在研究内部类别、三元组和多态性的指称语义时。此外,受高等范畴理论和同源类型理论语义的最新发展的启发,我们提出了一类新的非循环泛型对象,将普遍性的重排性质推广到任意fibration的设置。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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