AN INTRODUCTION to DIFFEOLOGY I Since Its Creation, In
AN INTRODUCTION TO DIFFEOLOGY PATRICK IGLESIAS-ZEMMOUR Abstract. This text presents the basics of Difeology and the main domains: Homotopy, Fiber Bundles, Quotients, Singularities, Cartan-de Rham Calculus — which form the core of diferential geometry — from the point of view of this theory. We show what makes difeology special and relevant in regard to these traditional subjects. Introduction Since its creation, in the early 1980s, Dieology has become an alternative, or rather a natural extension of traditional diferential geometry. With its developments in higher homotopy theory, ber bundles, modeling spaces, Cartan-de Rham calcu- lus, moment map and symplectic1 program, for examples, difeology now covers a large spectrum of traditional elds and deploys them from singular quotients to innite-dimensional spaces — and mixes the two — treating mathematical objects that are or are not strictly speaking manifolds, and other constructions, on an equal footing in a common framework. We shall see some of its achievements through a series of examples, chosen because they are not covered by the geom- etry of manifolds, because they involve innite-dimensional spaces or singular quotients, or both. The growing interest in difeology comes from the conjunction of two strong properties of the theory: 1. Mainly, the category {Difeology} is stable under all set-theoretic construc- tions: sums, products, subsets, and quotients. One says that it is a complete and co-complete category. It is also Cartesian closed, the space of smooth maps in difeology has itself a natural functional dieology. Date: August 25, 2017 – Revision May 15, 2020. Text of the talk given at the Conference “New Spaces in Mathematics & Physics”, September 28 – October 2, 2015 Institut Henri Poincaré Paris, France.
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