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A new approach to defining interval objects in category theory is being explored, focusing on the universal characterization of the Euclidean interval. This research, a collaboration between Martin Escardo and Alex Simpson, aims to establish a definition of interval objects applicable to general categories, capturing both geometrical and computational aspects. The goal is to find a definition that works across diverse categorical settings, allowing for a more abstract and unified understanding of intervals. This work builds upon their previous research, aiming for a broader mathematical foundation for interval objects.
The work by Escardo and Simpson delves into defining arithmetic operations within this abstract framework. Given an interval object [-1,1] in a category with finite products, they demonstrate how to define operations such as negation and multiplication using the universal property of the interval. Negation, denoted as -x, is defined as the unique automorphism that maps -1 to 1 and 1 to -1, ensuring that -(-x) = x. Similarly, multiplication x × (-) is defined as the unique automorphism mapping -1 to -x and 1 to x, resulting in commutative and associative multiplication.
This research has already produced significant results, including two joint papers: "A universal characterization of the closed Euclidean interval (extended abstract)" from LICS 2001 and "Abstract Datatypes for Real Numbers in Type Theory" from RTA/TLCA'2014. A third paper, focused more on the mathematical aspects, is currently in preparation. This work aims to provide a robust and universal characterization of interval objects, impacting both theoretical mathematics and practical applications in computer science and related fields.
References :
The work by Escardo and Simpson delves into defining arithmetic operations within this abstract framework. Given an interval object [-1,1] in a category with finite products, they demonstrate how to define operations such as negation and multiplication using the universal property of the interval. Negation, denoted as -x, is defined as the unique automorphism that maps -1 to 1 and 1 to -1, ensuring that -(-x) = x. Similarly, multiplication x × (-) is defined as the unique automorphism mapping -1 to -x and 1 to x, resulting in commutative and associative multiplication.
This research has already produced significant results, including two joint papers: "A universal characterization of the closed Euclidean interval (extended abstract)" from LICS 2001 and "Abstract Datatypes for Real Numbers in Type Theory" from RTA/TLCA'2014. A third paper, focused more on the mathematical aspects, is currently in preparation. This work aims to provide a robust and universal characterization of interval objects, impacting both theoretical mathematics and practical applications in computer science and related fields.
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References :
- www.johndcook.com: A paper about a notion of interval object in any category with finite products, on joint work with Alex Simpson.
- Martin Escardo: The original post announcing A universal characterization of the closed Euclidean interval.
Classification:
- HashTags: #CategoryTheory #IntervalObject #HoTT
- Company: Mathstodon
- Target: Abstraction
- Product: Topology
- Feature: Characterization
- Type: Research
- Severity: Medium