Diffeological Spaces

Abstract

We define diffeological spaces and give some examples. The diffeological category contains the category of smooth manifolds as full subcategory. We prove that diffeological spaces and smooth maps form a cartesian closed category. The concepts of differential form and tangent functor are extended from smooth manifolds to diffeological spaces. Subjclass : 57R55 Differentiable Structures. See www.ams.org/msc Now we proceed to define diffeological spaces and smooth maps between diffeological spaces as in [Sou85]. Definition 1.3 (Diffeology). Given a set X consider a collection P(X) = ∪ n P n (X) of maps into X, such that P n (X) ⊂ L n (X). We call P(X) a diffeology if it satisfies the following axioms: (D1) Every constant map x : U → X is in P(X). (D2) Given a compatible family of maps in P n (X), its smallest extension is again in P n (X). (D3) If h ∈ L pn and α ∈ P n (X) are composable, then α • h ∈ P p (X). The elements of P n (X) are called n-plots or simply plots of the diffeology. The pair (X, P(X)) is called a diffeological space. Definition 1.4 (D-topology). Given a diffeological space (X, P(X)) we define the-topology to be the initial topology with respect to the plots. That is, a subset U ⊂ X is open if and only if for each plot α ∈ P(X), the inverse image α −1 (U) is open in U α. We can immediately give two examples. Example 1.5 (Indiscrete Diffeology). Given any set X, the collection L(X) of all possible maps into X with open domain is a diffeology. The only open sets are X and the empty set, so the-topology is the indiscrete topology. Example 1.6 (Discrete Diffeology). Let P(X) denote the collection of all locally constant maps. This is the smallest possible diffeology on the set X. Note that we cannot restrict to all constant maps! This would violate (D2): If we take two maps that are constant, taking distinct values on disjoint domains, their smallest extension is merely locally constant. Every subset of X is open in the-topology, so X is a discrete topological space.

Diffeological vector pseudo-bundles

“…The quotient diffeology A quotient of a diffeological space is always a diffeological space 7 for a canonical choice of a diffeology on the quotient. Namely, let X be a diffeological space, let ∼ = be an equivalence relation on X, and let π : X → Y := X/ ∼ = be the quotient map; the quotient diffeology on Y is the pushforward of the diffeology of X by the natural projection (which is automatically smooth).…”

Gauge transformations for categorical bundles

“…We will not need any details concerning smooth structures on path spaces but we note here some minimal background. For our purposes it is convenient to use the framework of diffeological spaces, introduced by Souriau [30] and discussed further by several authors [3,15,17]; however, we will use the term "smooth space", which is used by Baez and Hoffnung [3] in a broader sense. There are several other approaches to smooth structures, such as the one by Fröhlicher [12]; Batubenge et al [7] and Stacey [31] provide overviews and comparisons of different approaches to smoothness.…”

TheD-topology for diffeological spaces

“…Given a diffeological space X, the category DS/X can be used to define geometric structures on X. See [I2,So3,La] for a discussion of differential forms and the de Rham cohomology of a diffeological space, and see [He, La] for tangent spaces and tangent bundles.…”