  After embedding the objects quasifolds into the category {Diffeology}, we
associate a C*-agebra 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 C*-algebras
with a diffeomorphic class of quasifolds.
