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19 pages, 24857 KiB  
Article
Color and Timbre Gestures: An Approach with Bicategories and Bigroupoids
by Maria Mannone, Giovanni Santini, Esther Adedoyin and Carmine E. Cella
Mathematics 2022, 10(4), 663; https://doi.org/10.3390/math10040663 - 20 Feb 2022
Cited by 3 | Viewed by 3377
Abstract
White light can be decomposed into different colors, and a complex sound wave can be decomposed into its partials. While the physics behind transverse and longitudinal waves is quite different and several theories have been developed to investigate the complexity of colors and [...] Read more.
White light can be decomposed into different colors, and a complex sound wave can be decomposed into its partials. While the physics behind transverse and longitudinal waves is quite different and several theories have been developed to investigate the complexity of colors and timbres, we can try to model their structural similarities through the language of categories. Then, we consider color mixing and color transition in painting, comparing them with timbre superposition and timbre morphing in orchestration and computer music in light of bicategories and bigroupoids. Colors and timbres can be a probe to investigate some relevant aspects of visual and auditory perception jointly with their connections. Thus, the use of categories proposed here aims to investigate color/timbre perception, influencing the computer science developments in this area. Full article
(This article belongs to the Special Issue Mathematics and Computation in Music)
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33 pages, 285 KiB  
Communication
The Hecke Bicategory
by Alexander E. Hoffnung
Axioms 2012, 1(3), 291-323; https://doi.org/10.3390/axioms1030291 - 9 Oct 2012
Cited by 2 | Viewed by 5092
Abstract
We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest [...] Read more.
We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
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