Abstract
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.
Highlights
Symmetry, and patterns of breaking it, are indisputably one of the basic guiding principles in modern natural science
The tovariance principle claims that theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes
Along with clarifying the role of category theory in relation to the foundations of mathematics and quantum physics, researchers became interested in an emerging fundamental symmetry of a new kind, namely the invariance of theories with respect to the choice of new foundations of mathematics given by the broad class of toposes
Summary
Patterns of breaking it, are indisputably one of the basic guiding principles in modern natural science. 20th-century mathematics has given us an alternative competing view as to its foundations This was achieved via the development of category theory, in which we can view the modifications of set theory, logic, and geometry in a unified way. Such as approach shed light on new understanding of symmetry and physics. Along with clarifying the role of category theory in relation to the foundations of mathematics and quantum physics, researchers became interested in an emerging fundamental symmetry of a new kind, namely the invariance of theories with respect to the choice of new foundations of mathematics given by the broad class of toposes.
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