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NishMath - #abstraction
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@Martin Escardo
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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.
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Michael Weiss@Diagonal Argument
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Research at the intersection of logic, descent theory, and category theory is currently underway, focusing on advanced mathematical concepts. Key areas of exploration include Galois descent in algebraic contexts, a concept with applications in algebraic geometry as detailed in texts by James Milne and lecture notes by Keith Conrad and Joshua Ruiter. Additionally, researchers are investigating equivariant bicategorical shadows and traces, aiming to generalize topological Hochschild homology.
This research extends to first-order categorical logic and quantum observables, contributing to a broader understanding of these abstract mathematical structures. A recent seminar at Vanderbilt University highlighted "Equivariant Bicategorical Shadows and Traces," where the presenter discussed a new framework of equivariant bicategorical shadows and explained why twisted THH is a g-twisted shadow, also exploring g-twisted bicategorical traces. This work builds upon the foundations laid by Ponto in defining bicategorical shadows, offering potential advancements in algebraic K-theory.
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