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Diffeological Spaces Proyecciones Journal of Mathematics Vol. 25, No2, pp. 151-178, August 2006. Universidad Cat´olica del Norte Antofagasta - Chile DIFFEOLOGICAL SPACES MARTIN LAUBINGER LOUISIANA STATE UNIVERSITY, LOUISIANA — U.S.A. Received : January 2006. Accepted : April 2006 Abstract We define diffeological spaces and give some examples. The dif- feological 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 152 Martin Laubinger 1. Introduction Smooth n-dimensional manifolds are by definition topological spaces which locallylooklikeRn and exhibit smooth coordinate changes. In contrast, diffeological spaces do not a priori carry a topology or a dimension. The central concept is that of ‘smooth maps’. So we define diffeological spaces via a collection of maps from open sets of Rn for all n into a set, such that this collection is closed under smooth coordinate changes. This will lead to a concept which is better from a categorical point of view: diffeological spaces form a cartesian closed category, which we will prove in Chapter 3. Even though the concept of a diffeological space is very general, we can still extend some concepts from differential geometry to this new setting. For example, in Chapter 4 we define differential forms, and in Chapter 5 we define a tangent functor. Smooth manifolds are an example of diffeological spaces, and in the last chapter we show that the new definitions agree with the classical ones on the subcategory of manifolds. We start by fixing the notation for the category of open subsets of real spaces and smooth maps. Definition 1.1. Let OR be the category whose objects are all open sets of all Rn,n N, and whose morphisms are all smooth maps between these open sets.∈ The collection of smooth maps with domain in Rn and range in Rp will be denoted by np. ThecollectionofmapsfromanopensetU Rn intoasetX is denotedL by n(X), and we will use the letters α,β,γ,...⊂ for n L elements of (X). The corresponding domains are denoted by Uα,Uβ,Uγ respectively.L Lastly, let (X) be the union of all the n(X). L L Definition 1.2. Given maps α np and β p(X), we say that (β,α) ∈ L ∈ L is a composable pair if α(Uα) Uβ. We will also just say that α and β ⊆ n are composable. A collection αi i I (X) is called compatible if { | ∈ } ⊂ L x Ui Uj implies αi(x)=αj(x). Every compatible family has a unique ∈ ∩ smallest extension α which has domain iUi and is defined to be equal to ∪ αi on Ui. Diffeological Spaces 153 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 (X)= n n n P n (X) of maps into X, such that (X) (X). We call (X) a di∪ ffPeology if it satisfies the following axioms:P ⊂ L P (D1) Every constant map x : U X is in (X). → P (D2) Given a compatible family of maps in n(X), its smallest extension is again in n(X). P P (D3) If h pn and α n(X) are composable, then α h p(X). ∈ L ∈ P ◦ ∈ P The elements of n(X) are called n-plots or simply plots of the diffeology. The pair (X, (XP)) is called a diffeological space. P Definition 1.4 ( -topology). Given a diffeological space (X, (X)) we define the -topologyD to be the initial topology with respect toP the plots. That is, a subset U X is open if and only if for each plot α (X), the 1 ⊂ ∈ P inverse image α− (U) is open in Uα. We can immediately give two examples. Example 1.5 (Indiscrete Diffeology). Given any set X, the collection (X) of all possible maps into X with open domain is a diffeology. The onlyL open sets are X and the empty set, so the -topology is the indiscrete topology. Example 1.6 (Discrete Diffeology). Let (X) denote the collection of all locally constant maps. This is the smallestP 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. 154 Martin Laubinger Axiom (D2) has an immediate consequence, which says that being a plot is a local property of a map. n Lemma 1.7. Let (X, (X)) be a diffeological space, Uα R open and P ⊂ α : Uα X a map. If for each x Uα there is a neighborhood V of x in Uα such that→ α restricted to V is a plot,∈ then α is a plot. Proof. Given x Uα, let Vx be a neighborhood of x such that αx := α V ∈ | x is a plot. Then αx x Uα is a compatible family of plots with smallest extension α. So{ it follows| ∈ from} axiom (D2) that α is a plot. 2 Lemma 1.8. The -topology of a diffeological space (X, (X)) is locally arc-connected. P Proof. Given a point x X, let P be the arc-component of x. So for each y P, there is a continuous∈ path from x to y. It suffices to show that P ∈ 1 is an open set. Let α : UαrightarrowX be a plot, and suppose that α− (P ) is nonempty, containing a point p. As Uα is open, we can find an open ball 1 B Uα containing p. We want to show B α− (P ), so let q B. To show that⊂ α(q) P, we have to connect x and α⊂(q) with a continuous∈ path. By choice of p,∈ the image α(p) can be connected to x by a continuous path. Now p and q can be connected in B, and we can compose this path with α to get a path from α(p)toα(q). Then by concatenation, we get a path from x to α(q). Thus α(q) P. This proves B α 1(P ), hence α 1(P )isopen ∈ ⊂ − − in Uα, which concludes the proof. 2 Intersections of diffeologies are again diffeologies, and therefore it is easy to define the diffeology generated by a given family of functions. We will characterize the plots of such diffeologies. Lemma 1.9. Given a family i i I of diffeologies on a set X, the {P | ∈ } intersection i i is again a diffeology. ∩ P Diffeological Spaces 155 Proof. We have to verify the three axioms (D1)-(D3) for := i i. P ∩ P By definition, each of the diffeologies i contains all constant maps, thus the same is true for their intersection.P Every compatible family of n-plots in is also compatible in each of the i. Therefore the smallest common P P extension is element of each i, hence also of . Similarly, if α and h P P ∈ P are composable, then α h i for all i I, and therefore also α h . 2 ◦ ∈ P ∈ ◦ ∈ P Corollary 1.10. If F (X) is a collection of maps into X, then there is a uniquesmallestdiffeology⊂ L on X containing F. We will denote this diffeology by hFi Proof. Note that the indiscrete diffeology contains F. So the collection of all diffeologies on X which contain F is nonempty, and we can take its intersection. 2 Definition 1.11. If (X) is the smallest diffeology containing a collection of maps F, then the collectionP is called a generating family for the diffeology. For a given generating family F, let nF denote the supremum n := sup dim(Uα) α F . F { | ∈ } Then the dimension of a diffeological space (X, (X)) is the infimum P dim(X, (X)) := inf n F generates (X) P { F | P } Lemma 1.12. Let X be a set and F (X). Then a map α : Uα X is in F if and only if it satisfies ⊂ L → h i (G) For each point x Uα there is an open neighborhood V Uα of x such that the restriction∈ of α to V is either constant or of⊂ the form f h for some f F and a smooth map h : V Uf . ◦ ∈ → 156 Martin Laubinger Proof. Let denote the collection of maps α satisfying (G). Clearly, contains F. WeP will first show that is a diffeology. Constant maps certainlyP P satisfy (G), therefore (D1) holds. Now take a compatible family αi with ∈ P smallest extension α. Each x Uα is contained in some dom(αi)=Ui and ∈ has therefore a neighborhood in Ui on which α is constant or of the form f g with smooth g and f F. Therefore α is in , which shows (D2). To ◦ ∈ P verify (D3), let α and h : U Uα be composable, where α is in and h is smooth. Let x U and y = →h(x). Then there is a neighborhood VPof y such that either ∈ α V is constant. Then α h 1 is also constant. • | ◦ |h− (V ) or α V is of the form f g for smooth g and f . Then • | ◦ ∈ P α h 1 = f g h. ◦ |h− (V ) ◦ ◦ 1 As h− (V )isanopenneighborhoodofx, in either case we have that α h .
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