Diffeology is a natural extension of differential geometry that covers a wide spectrum of objects, ranging from singular spaces of any kind to infinite dimensional spaces, and sometimes mixing the two. With its developments in higher homotopy theory, fiber bundles, modeling spaces, Cartan-de-Rham calculus, moment map and symplectic constructions, diffeology spans a wide range of traditional fields, treating geometrical objects that are or are not manifolds on an equal footing in a common framework. 
The textbook DIFFEOLOGY establishes the foundations of the theory, and develops the main areas of application. These are illustrated by a series of examples, chosen explicitly because they are not covered by traditional differential geometry. 
The advantage of diffeology comes from the conjunction of two strong properties: first, diffeology is stable by all set-theoretic operations: sum, product, subset and quotient. It is said to be a complete and co-complete category. It is also Cartesian closed, the set of smooth maps having itself a natural diffeology. Second, and perhaps more importantly, diffeology even deals with non-Hausdorff quotient spaces in a non-trivial way, as is the case with irrational tori. This specific property becomes crucial for new constructions and is at the origin of the generalization of theorems that do not exist otherwise.
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September 1, 2005 — September 1, 2022.
The AMS print The BWPC reprint Patrick Iglesias-Zemmour
Jerusalem  — Aix en Provence 2005/2022