{"title":"The Grothendieck computability model","authors":"Luis Gambarte , Iosif Petrakis","doi":"10.1016/j.tcs.2025.115550","DOIUrl":null,"url":null,"abstract":"<div><div>Translating notions and results from category theory to the theory of computability models of Longley and Normann, we introduce the Grothendieck computability model. We define the first-projection-simulation and prove its basic properties. With the Grothendieck computability model, the category of computability models is shown to be a type-category, in the sense of Pitts, a result that bridges the categorical interpretation of dependent types with the theory of computability models. We also show that the category of computability models is a category with 2-family arrows and a corresponding structure of Sigma-objects. Finally, we introduce the notion of a fibration and opfibration-simulation, and we prove that the first-projection-simulation is a split opfibration-simulation.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115550"},"PeriodicalIF":1.0000,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397525004888","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
Translating notions and results from category theory to the theory of computability models of Longley and Normann, we introduce the Grothendieck computability model. We define the first-projection-simulation and prove its basic properties. With the Grothendieck computability model, the category of computability models is shown to be a type-category, in the sense of Pitts, a result that bridges the categorical interpretation of dependent types with the theory of computability models. We also show that the category of computability models is a category with 2-family arrows and a corresponding structure of Sigma-objects. Finally, we introduce the notion of a fibration and opfibration-simulation, and we prove that the first-projection-simulation is a split opfibration-simulation.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.