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plural diffeologies
(differential geometry) The set of smooth parametrizations of a given set X ; the set of smooth maps S → X , where S is any open set ⊆ R n ; the theory of such constructs. quotations examples
In this case, A {\displaystyle A} and f ( A ) {\displaystyle f(A)} are both D-open and f : A → f ( A ) {\displaystyle f:A\rightarrow f(A)} is a diffeomorphism where A {\displaystyle A} and f ( A ) {\displaystyle f(A)} are equipped with their subset diffeology.
2007, Jan Kubarski, Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann; on the Occasion of the Hundredth Anniversary of His Birthday, page 358
And, for G ∗ / Γ {\displaystyle {\mathcal {G}}^{*}/\Gamma } and G ′ ∗ / Γ ′ {\displaystyle {\mathcal {G}}'^{*}/\Gamma '} equipped with the quotient diffeologies, h Γ ∗ {\displaystyle h_{\Gamma }^{*}} is naturally smooth.
2010, Patrick Iglesias-Zemmour, The Moment Maps in Diffeology, American Mathematical Society, page 32