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Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics

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German Aerospace Center (DLR), Rosa-Luxemburg-Str. 2, 10178 Berlin, Germany
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Copernicus Center for Interdisciplinary Studies ul. Szczepańska 1/5, 31-011 Cracow, Poland
3
Cognitive Science and Mathematical Modelling Chair, University of Information Technology and Management, ul. Sucharskiego 2, 35-225 Rzeszów, Poland
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(12), 1429; https://doi.org/10.3390/sym11121429
Received: 1 October 2019 / Revised: 4 November 2019 / Accepted: 18 November 2019 / Published: 20 November 2019
(This article belongs to the Special Issue Symmetry Principles in Quantum Systems)
Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos T . In the case of the Basel topos and R 4 , the local invariance with respect to the corresponding atlases implies exotic smoothness on R 4 . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic R 4 underlying an exotic S 4 (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local T -invariance breaks tovariance even in the weaker sense. View Full-Text
Keywords: exotic R4; smooth toposes; tovariance; symmetry and categories in physics exotic R4; smooth toposes; tovariance; symmetry and categories in physics
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Asselmeyer-Maluga, T.; Król, J. Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics. Symmetry 2019, 11, 1429.

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