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After embedding the objects quasifolds into the category Diffeology, we associate a -algebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry – beginning with the today classical example of the irrational torus – which associates a Morita class of -algebras with a diffeomorphic class of quasifolds.
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Patrick Iglesias-Zemmour, Elisa Prato, Quasifolds, diffeology and noncommutative geometry. J. Noncommut. Geom. 15 (2021), no. 2 pp. 735–759