aa r X i v : . [ m a t h . A T ] M a r GENERALIZED MAPS BETWEENDIFFEOLOGICAL SPACES
KAZUHISA SHIMAKAWA
Abstract.
By exploiting the idea of Colombeau’s generalized function,we introduce a notion of asymptotic map between arbitrary diffeologicalspaces. The category consisting of diffeological spaces and asymptoticmaps is enriched over the category of diffeological spaces, and inher-its completeness and cocompleteness. In particular, asymptotic func-tions on a Euclidean open set include Schwartz distributions and form asmooth differential algebra over Robinson’s field of asymptotic numbers.To illustrate the use of our method, we prove that smooth relative cellcomplexes enjoy homotopy extension property with respect to asymp-totic maps and homotopies. Introduction
The category
Diff of diffeological spaces and smooth maps has excellentproperties such as completeness, cocompleteness, and cartesian closedness,and provides a framework for combining analysis with geometry and topol-ogy. But there is one barrier to spread the use of diffeology. Compared withcontinuous maps, smooth maps are far more difficult to handle with andit often occurs that we cannot find an appropriate smooth map that satis-fies the given requirements. Such a situation typically occurs in analysis,e.g. when solving partial differential equations. Distributions (or general-ized functions) make it possible to differentiate functions whose derivativesdo not exist in the classical sense (e.g. locally integrable functions). Theyare widely used in the theory of partial differential equations, where it maybe easier to establish the existence of weak solutions than classical ones, orappropriate classical solutions may not exist. They are also important inphysics and engineering where many problems naturally lead to differentialequations whose solutions or initial conditions are distributions. The aim ofthis paper is to bring in such generalized notion of maps into the frameworkof diffeology, and utilize them to investigate diffeological spaces in a moreflexible manner.We now state the main result of the paper. For diffeological spaces X and Y , we denote by C ∞ ( X, Y ) the set of smooth maps from X to Y equippedwith the functional diffeology (cf. [5]). A diffeological space is called a smoothdifferential algebra if it has a structure of a differential algebra such that itssum, product and partial differential operators are smooth. Given an opensubset U of R n , let D ′ ( U ) denote the locally convex topological vector space Date : March 17, 2020.2010
Mathematics Subject Classification.
Primary 54C35 ; Secondary 46T30, 58D15.This work was supported by JSPS KAKENHI Grant Number JP18K03279. of Schwartz distributions. Then the main theorem of the paper can be statedas follows.
Theorem 1.1.
There is a supercategory d Diff of Diff enjoying the proper-ties below. Let F be either R or C . (1) d Diff has the same objects as
Diff and is enriched over
Diff . (2) d Diff has all small limits and colimits, and the inclusion
Diff → d Diff preserves (hence creates) limits and colimits. (3)
The space b C ∞ ( X, Y ) of morphisms from X to Y in d Diff includesC ∞ ( X, Y ) as a subspace, and there is a natural diffeomorphism b C ∞ ( X × Y, Z ) ∼ = C ∞ ( X, b C ∞ ( Y, Z )) . (4) Let b F = b C ∞ ( R , F ) . Then b R is a non-archimedean real closed fieldand b C is an algebraically closed field of the form b C = b R + √− b R . (5) For every X ∈ Diff , b C ∞ ( X, F ) is an b F -algebra containing C ∞ ( X, F ) as an F -subalgebra. (6) If U is an open subset of R n then b C ∞ ( U, F ) is a smooth differentialalgebra admitting a continuous linear embedding D ′ ( U ) → b C ∞ ( U, F ) preserving partial derivatives. A member of b C ∞ ( X, Y ) is called an asymptotic map from X to Y , andis called an asymptotic function if Y is R or C .The construction of asymptotic functions is based on Colombeau’s theoryof simplified algebra [1] and its variant by Todorov-Vernaeve [7]. We brieflyreview in Section 2 the basics of Colombeau’s simplified algebra and itsreinterpretation as a smooth differential algebra given by Giordano-Wu [2].It is shown that the embedding D ′ ( U ) → G s ( U ) into the smooth version ofColombeau’s simplified algebra is continuous with respect to the D -topologyassociated with the diffeology of G s ( U ).Section 3 provides a step-by-step construction of b C ∞ ( X, Y ). For a Eu-clidean open set U , the space b C ∞ ( U, F ) closely resembles G s ( U ) and thereis a natural homomorphism G s ( U ) → b C ∞ ( U, F ) of smooth differential al-gebras. But there is a significant difference between the two: The set ofscalars of Colombeau’s theory is not a field but a ring with (lots of) zerodivisors, while on the other hand, our b F is isomorphic to the scalar fieldof Todorov-Vernaeve’s theory, and hence to Robinson’s field of asymptoticnumbers. Once b C ∞ ( U, F ) is established as a functor of Euclidean open sets U , it is straightforward, by the nature of diffeological spaces, to construct b C ∞ ( X, Y ) for arbitrary X and Y , and to show that diffeological spaces andasymptotic maps form a category d Diff enriched over
Diff .Section 4 completes the proof of the main theorem by verifying that theremaining properties (2) and (3) of the main theorem really hold.Finally, in Section 5 we prove, as an illustrating example of the useof asymptotic maps, that any smooth relative cell complex (
X, A ) enjoysthe homotopy extension property in d Diff , that is, for any asymptotic map f : X → Y and asymptotic homotopy f | A ≃ g on A , there is an asymptotichomotopy f ≃ ˜ g extending f | A ≃ g . Note that the homotopy extension ENERALIZED MAPS 3 property plays a key role in the development of homotopy theory. But itfails if maps and homotopies are restricted to smooth ones.The author wish to thank Dan Christensen for pointing out a wrongstatement in the initial version of the manuscript that d Diff is cartesianclosed. As the property (4) of the main theorem says, b F = b C ∞ (pt , F ) is notisomorphic to F , meaning that d Diff is not cartesian closed.2.
Colombeau algebra of generalized functions
We briefly recall the basic notations and properties of simplified Colombeaualgebra of generalized functions. Henceforth, Γ denotes the interval (0 , F is either R or C , and N is the set of non-negative integers. If X is atopological or diffeological space then its underlying set is denoted by | X | .2.1. Simplified Colombeau algebra.
Given an open set U in R n , let E F ( U ) = C ∞ ( U, F ) Γ be the set of Γ-nets in the locally convex space ofsmooth functions U → F . Then the simplified Colombeau algebra on U isdefined to be the quotient G s ( U ) = M ( E F ( U )) / N ( E F ( U ))of the subalgebra of moderate nets over that of negligible nets . Here, we saythat ( f ǫ ) ∈ E F ( U ) is moderate if it satisfies the condition ∀ K ⋐ U, ∀ α ∈ N n , ∃ m ∈ N , max x ∈ K | D α f ǫ ( x ) | = O (1 /ǫ m )and is negligible if ∀ K ⋐ U, ∀ α ∈ N n , ∀ p ∈ N , max x ∈ K | D α f ǫ ( x ) | = O ( ǫ p )where the notation K ⋐ U indicates that K is a compact subset of U .As described in [2, Theorem 1.1], the correspondence U
7→ G s ( U ) is a fineand supple sheaf of differential algebras, and there is a sheaf embedding ofvector spaces ι U : D ′ ( U ) → G s ( U ) preserving partial derivatives. Note, how-ever, that the construction is not canonical because it depends on particularchoice of a δ -net.2.2. Colombeau algebra as a diffeological space.
Giordano and Wuintroduced in [2, §
5] a diffeological space structure for G s ( U ) defined asfollows. Instead of the locally convex topology, let us endow C ∞ ( U, F ) withthe functional diffeology (cf. [5]); so that a map ρ : V → C ∞ ( U, F ) is a plotfor C ∞ ( U, F ) if and only if the composition below is smooth. V × U ρ × −−→ C ∞ ( U, F ) × U ev −→ F , ( x, y ) ρ ( x )( y )Let E F ( U ) = C ∞ ( U, F ) Γ , where Γ is regarded as a discrete diffeologicalspace. Then moderate nets and negligible nets form diffeological subspaces M ( E F ( U )) and N ( E F ( U ))), respectively, and hence a diffeological quotient G s ( U ) = M ( E F ( U )) / N ( E F ( U )) . Theorem 2.1 (Giordano-Wu) . The space G s ( U ) is a smooth differentialalgebra, and there is a linear embedding ι U : D ′ ( U ) → G s ( U ) enjoying thefollowing properties: K. SHIMAKAWA (1) ι U is continuous with respect to the weak dual topology on D ′ ( U ) andthe D -topology on G s ( U ) . (2) ι U preserves partial derivatives, i.e. D α ( ι U ( u )) = ι U ( D α u ) holds forall u ∈ D ′ ( U ) and α ∈ N n . (3) ι U restricts to the embedding of smooth differential algebra C ∞ ( U ) → G s ( U ) which takes f to the class of the constant net with value f .Proof. That G s ( U ) is a smooth differential algebra is proved in [2, Theorem5.1]. The properties (2) and (3) follows, respectively, from (vii) and (viii) of[2, Theorem 1.1]. To prove (1), let D ( U ) ′ be the smooth dual of the vectorspace D ( U ) of test functions equipped with the canonical diffeology (cf. [2,Definition 4.1]), and ( T D ( U )) ′ be the continuous dual of T D ( U ). Then wehave |D ′ ( U ) | = | ( T D ( U )) ′ | = | D ( U ) ′ | by [2, Lemma 4.7], and the inclusion D ′ ( U ) → ( T D ( U )) ′ is continuous because the D -topology of D ( U ) is finer than the locallyconvex topology of D ( U ) by [2, Corollary 4.12 (i)]. Also, the embedding ι U : D ( U ) ′ → G s ( U ) is smooth by [2, Theorem 5.1], and hence T ι U : T ( D ( U ) ′ ) → T G s ( U )is continuous. Thus, to prove (1) it suffices to show that the inclusion( T D ( U )) ′ → T ( D ( U ) ′ )is continuous. By definition, ( T D ( U )) ′ has the coresest topology such thatthe evaluation map ( T D ( U )) ′ × T D ( U ) → F is continuous. But this is equiv-alent to say that it has the final topology with respect to those parametriza-tions σ : V → | ( T D ( U )) ′ | such that for every plot ρ : W → D ( U ) the com-position below is continuous.(2.1) V × W σ × ρ −−→ | ( T D ( U )) ′ | × D ( U ) ev −→ F On the other hand, T ( D ( U ) ′ ) has the final topology with respect to those σ : V → | D ( U ) ′ | = | T D ( U ) ′ | such that for every plot ρ : W → D ( U ) thecomposition below is smooth.(2.2) V × W σ × ρ −−→ | D ( U ) ′ | × D ( U ) ev −→ F Since smooth maps are continuous, we see that ( T D ( U )) ′ has finer topologythan T ( D ( U ) ′ ). Hence the inclusion ( T D ( U )) ′ → ( D ( U ) ′ ) is continuous. (cid:3) Note.
The embedding ι U preserves the products of smooth maps, but doesnot preserve the products of continuous maps because if otherwise, it con-tradicts to the Schwartz impossibility result [6].3. The space of asymptotic maps
We assign to any pair of diffeological spaces (
X, Y ) a diffeological space b C ∞ ( X, Y ) consisting of certain generalized maps which we call asymptoticmaps . The construction consists of three steps: First, we introduce asymp-totic functions on Euclidean open sets, then extend them to the ones ongeneral diffeological spaces, and finally, construct asymptotic maps betweenarbitrary diffeological spaces.
ENERALIZED MAPS 5
Asymptotic functions on Euclidean open sets.
We now modifythe definition of G s ( U ) so that the scalars form a true field. For this purpose,we introduce, as in the definition of Todorov-Vernaeve’s algebra [7], a { , } -valued finitely additive measure on Γ given by an ultrafilter U generated bythe collection { (0 , ǫ ) | < ǫ ≤ } . Observe that U contains those subsets A for which there is a subset B such that B has Lebesgue outer measure 0 and A ∪ B ⊃ (0 , ǫ ] for some 0 < ǫ ≤
1. We say that a predicate P ( ǫ ) defined onΓ holds almost everywhere as ǫ → { ǫ ∈ Γ | P ( ǫ ) } ∈ U .As in the previous section, we denote E F ( U ) = C ∞ ( U, F ) Γ and define b C ∞ ( U, F ) = c M ( E F ( U )) / b N ( E F ( U ))where c M ( E F ( U )) consists of moderate nets ( f ǫ ) ∈ E F ( U ) satisfying ∀ K ⋐ U, ∀ α ∈ N n , ∃ m ∈ N , max x ∈ K | D α f ǫ ( x ) | = O (1 /ǫ m ) a.e.and b N ( E F ( U )) consists of negligible nets ( f ǫ ) ∈ E F ( U ) satisfying ∀ K ⋐ U, ∀ α ∈ N n , ∀ p ∈ N , max x ∈ K | D α f ǫ ( x ) | = O ( ǫ p ) a.e.Clearly, b C ∞ ( U, F ) inherits ring operations from C ∞ ( U, F ) and every smoothmap F : W → U between Euclidean open sets induces a functorial homo-morphism of F -algebras F ∗ : b C ∞ ( U, F ) → b C ∞ ( W, F ) , F ∗ ([ f ǫ ]) = [ f ǫ ◦ F ]Moreover, by arguing as in [2, Theorem 5.1] we see that b C ∞ ( U, F ) is asmooth differential algebra with respect to partial differential operators D α : b C ∞ ( U, F ) → b C ∞ ( U, F ) , D α ([ f ǫ ]) = [ D α f ǫ ] ( α ∈ N n )and there is a natural inclusion of smooth differential algebras i U : C ∞ ( U, F ) → b C ∞ ( U, F )which takes a smooth map f to the class of the constant net with value f .In particular, let b F = b C ∞ ( R , F ) for F = R , C . The following result isessentially the same as [7, Theorem 7.3]. Theorem 3.1 (Todorov-Vernaeve) . b C is an algebraically closed field of theform b R + √− b R .Proof. It is clear that b C is a ring and that we have b C = b R + √− b R . To seethat b C is a field, suppose ( a ǫ ) ∈ M ( C Γ ) represents a non-zero class in b C .Then there exist m, p ∈ N such that Φ = { ǫ | ǫ p ≤ | a ǫ | ≤ /ǫ m } ∈ U . Letus define ( b ǫ ) ∈ C Γ by b ǫ = 1 /a ǫ if ǫ ∈ Φ and b ǫ = 1 if otherwise. Then wehave [ b ǫ ] = [ a ǫ ] − in b C because a ǫ b ǫ = 1 holds almost everywhere. Hence b C is a field. To see that b C is algebraically closed, suppose P ( x ) = x p + a x p − + · · · + a p is a polynomial with coefficients in b C , and choose a representative ( a k,ǫ ) ∈M ( C Γ ) for each a k ∈ b C . For 0 < ǫ ≤
1, let P ǫ ( x ) = x p + a ,ǫ x p − + · · · + a p,ǫ K. SHIMAKAWA and take x ǫ ∈ C satisfying P ǫ ( x ǫ ) = 0. Then ( x ǫ ) ∈ C Γ is moderate because | x ǫ | ≤ | a ,ǫ | + · · · + | a p,ǫ | holds, and hence we have P ([ x ǫ ]) = 0. (cid:3) Corollary 3.2. b R is a real closed field and both b R and b C are non-archimedeanfields in the sense that they contain non-zero infinitesimals.Proof. Since we have b C = b R + √− b R and b C is algebraically closed, b R is areal closed field and is totally ordered in such a way that a ≥ a = b holds for some b ∈ b R . It is non-archimedean because the asymptoticnumber ρ represented by the net ( ǫ ) is a non-zero infinitesimal in b R . (cid:3) Note.
The constant ρ is called the canonical infinitesimal in b R . If we regard ρ as a positive infinitesimal in the nonstandard reals ∗ R = R Γ / ∼ U then b C and b R are isomorphic to Robinson’s fields of ρ -asymptotic numbers ρ C and ρ R , respectively. For details, see Todorov-Vernaeve [7, Section 7].By the definition, M ( E F ( U )) and N ( E F ( U )) are subalgebras of c M ( E F ( U ))and b N ( E F ( U )), respectively. Hence we have Proposition 3.3.
The natural map G s ( U ) → b C ∞ ( U, F ) induced by theinclusions M ( E F ( U )) → c M ( E F ( U )) and N ( E F ( U )) → b N ( E F ( U )) is a ho-momorphism of smooth differential algebras over F . By the definition, the composition D ′ ( U ) → G s ( U ) → b C ∞ ( U, F ) is alinear monomorphism. Thus we have the following by Theorem 2.1. Corollary 3.4.
There exists a (non-canonical) continuous linear embedding D ′ ( U ) → b C ∞ ( U, F ) preserving partial derivatives. Asymptotic functions on general diffeological spaces.
Given anarbitrary diffeological space X with diffeology D X , we define b C ∞ ( X, F ) = c M ( E F ( X )) / b N ( E F ( X ))where E F ( X ) = C ∞ ( X, F ) Γ and for L = M , N we put b L ( E F ( X )) = { ( f ǫ ) ∈ E F ( X ) | ( f ǫ ◦ ρ ) ∈ b L ( E F ( U )) for ∀ ρ : U → X ∈ D X } One easily observes that if X is an open set in R n then b C ∞ ( X, F ) is identicalwith the algebra constructed in the previous step. Also, there is a naturalinclusion i X : C ∞ ( X, F ) → b C ∞ ( X, F ) which takes f ∈ C ∞ ( X, F ) to theclass of the constant net with value f . Evidently, we have the following. Proposition 3.5.
For every X ∈ Diff , b C ∞ ( X, F ) is an algebra over thefield b F , and the following hold. (1) Any smooth map F : X → Y induces a functorial homomorphism of b F -algebras F ∗ : b C ∞ ( Y, F ) → b C ∞ ( X, F ) . (2) The natural inclusion i X : C ∞ ( X, F ) → b C ∞ ( X, F ) is a homomor-phism of F -algebras. ENERALIZED MAPS 7
Asymptotic maps between diffeological spaces.
Given diffeolog-ical spaces X and Y , a net ( f ǫ ) ∈ C ∞ ( X, Y ) Γ is said to be moderate if itsatisfies the condition:( u ǫ ◦ f ǫ ) ∈ c M ( E R ( X )) ( ∀ ( u ǫ ) ∈ c M ( E R ( Y )) )Let c M ( C ∞ ( X, Y ) Γ ) be the subspace consisting of moderate nets in C ∞ ( X, Y ). Lemma 3.6. If ( f ǫ ) ∈ c M ( C ∞ ( X, Y ) Γ ) and ( g ǫ ) ∈ c M ( C ∞ ( Y, Z ) Γ ) then wehave ( g ǫ ◦ f ǫ ) ∈ c M ( C ∞ ( X, Z ) Γ ) .Proof. Because ( g ǫ ) is moderate we have ( u ǫ ◦ g ǫ ) ∈ c M ( E R ( Y )) for every( u ǫ ) ∈ c M ( E R ( Z )). But then we have ( u ǫ ◦ g ǫ ◦ f ǫ ) ∈ c M ( E R ( X )), and hence( g ǫ ◦ f ǫ ) ∈ c M ( C ∞ ( X, Z ) Γ ). (cid:3) We define the space of asymptotic maps from X to Y as the quotient b C ∞ ( X, Y ) = c M ( C ∞ ( X, Y ) Γ ) / ∼ where ( f ǫ ) and ( f ′ ǫ ) are equivalent in c M ( C ∞ ( X, Y ) Γ ) if(( u ǫ ◦ f ′ ǫ − u ǫ ◦ f ǫ ) ◦ ρ ǫ ) ∈ b N ( E R ( U ))holds for every ( u ǫ ) ∈ c M ( E R ( Y )) and ( ρ ǫ ) ∈ c M ( C ∞ ( U, X ) Γ ). Note.
Even if we replace E R ( − ) in the definition above with E C ( − ), theresulting quotient space is the same as b C ∞ ( X, Y ). Proposition 3.7.
There exists a supercategory d Diff of Diff which has thesame objects as
Diff and is enriched over
Diff with b C ∞ ( X, Y ) as the spaceof morphisms from X to Y .Proof. For given
X, Y, Z ∈ Diff consider the smooth map C ∞ ( Y, Z ) Γ × C ∞ ( X, Y ) Γ → C ∞ ( X, Z ) Γ which takes a pair (( g ǫ ) , ( f ǫ )) to the levelwise composition ( g ǫ ◦ f ǫ ). ByLemma 3.6, the map above restricts to a smooth map c X,Y,Z : c M ( C ∞ ( Y, Z ) Γ ) × c M ( C ∞ ( X, Y ) Γ ) → c M ( C ∞ ( X, Z ) Γ ) . Moreover, if ( f ǫ ) ∼ ( f ′ ǫ ) and ( g ǫ ) ∼ ( g ′ ǫ ) hold in c M ( C ∞ ( X, Y ) Γ ) and c M ( C ∞ ( Y, Z ) Γ ), respectively, then we have ( g ǫ ◦ f ǫ ) ∼ ( g ′ ǫ ◦ f ′ ǫ ) because(( u ǫ ◦ ( g ′ ǫ ◦ f ′ ǫ ) − u ǫ ◦ ( g ǫ ◦ f ǫ )) ◦ ρ ǫ )= (( u ǫ ◦ ( g ′ ǫ ◦ f ′ ǫ ) − u ǫ ◦ ( g ǫ ◦ f ′ ǫ ) + u ǫ ◦ ( g ǫ ◦ f ′ ǫ ) − u ǫ ◦ ( g ǫ ◦ f ǫ )) ◦ ρ ǫ )= (( u ǫ ◦ g ′ ǫ − u ǫ ◦ g ǫ ) ◦ ( f ′ ǫ ◦ ρ ǫ )) + ((( u ǫ ◦ g ǫ ) ◦ f ′ ǫ − ( u ǫ ◦ g ǫ ) ◦ f ǫ ) ◦ ρ ǫ )is negligible for every ( u ǫ ) ∈ c M ( E R ( Z )) and ( ρ ǫ ) ∈ c M ( C ∞ ( U, X ) Γ ). Hence c X,Y,Z induces a smooth composition b C ∞ ( Y, Z ) × b C ∞ ( X, Y ) → b C ∞ ( X, Z ) . It is now evident that we can define d Diff to be the category enriched over
Diff with diffeological spaces as objects and with b C ∞ ( X, Y ) as the space ofmorphisms from X to Y . (cid:3) K. SHIMAKAWA Proof of the main theorem
We have already shown that d Diff satisfies all the properties stated inTheorem 1.1 except for (2) and (3).To prove that d Diff has all limits, it suffices to show that it has all smallproducts and equalizers. Given a family of diffeological spaces { X j } j ∈ J ,let X = Q j ∈ J X j be their product in Diff . Then X is a product even inthe supercategory d Diff . To see this, suppose { f j : L → X j } is a familyof asymptotic maps and let ( f j,ǫ ) be a representative for each f j . Thenthere exist unique smooth maps u ǫ : L → X satisfying f j,ǫ = p j ◦ u ǫ , where p j : X → X j is the projection. But then, ( u ǫ ) ∈ C ∞ ( L, X ) Γ is moderate andinduces a unique asymptotic map u = [ u ǫ ] : L → X satisfying f j = p j ◦ u .Hence d Diff has all small products. To see that equalizers exist in d Diff ,suppose ( f, g ) is a pair of asymptotic maps from X to Y . Let ( f ǫ ) and ( g ǫ )be representatives for f and g , respectively, and putEq( f, g ) = { x ∈ X | f ǫ ( x ) = g ǫ ( x ) a.e. } . Clearly, this definition does not depend on the choice of representatives for f and g , and the inclusion Eq( f, g ) → X is an equalizer for ( f, g ) because anyasymptotic map k : Z → X satisfying f ◦ k = g ◦ k factors through Eq( f, g ).Therefore, d Diff has all small limits and the inclusion
Diff → d Diff preserveslimits because it preserves products and equalizers.Similarly, we can prove that d Diff has all colimits by showing that it has allcoproducts and coequalizers. Given a family of diffeological spaces { X j } j ∈ J ,their coproduct ` j ∈ J X j in Diff is a coproduct in the supercategory d Diff ,and any pair of asymptotic maps ( f, g ) from X to Y has a coequalizer Y → Coeq( f, g ) defined as the projection of Y onto its quotient space bythe least equivalence relation such that y ∼ y ′ if both y = f ǫ ( x ) a.e. and y ′ = g ǫ ( x ) a.e. hold for some x ∈ X . Again, the inclusion Diff → d Diff preserves colimits because it preserves coproducts and coequalizers. Thuswe see that the property (2) holds.To prove (3), consider the natural isomorphism γ : C ∞ ( X × Y, Z ) Γ → C ∞ ( X, C ∞ ( Y, Z ) Γ )We first show that γ restricts to an isomorphism(4.1) c M ( C ∞ ( X × Y, Z ) Γ ) ∼ = C ∞ ( X, c M ( C ∞ ( Y, Z ) Γ ))Let ( f ǫ ) ∈ C ∞ ( X × Y, Z ) Γ and ( g ǫ ) = γ (( f ǫ )) ∈ C ∞ ( X, C ∞ ( Y, Z ) Γ ). Thenfor any ( u ǫ ) ∈ c M ( E R ( Z )) and ρ = ( ρ ′ , ρ ′′ ) ∈ C ∞ ( U, X × Y ), we have u ǫ ( f ǫ ( ρ ( t ))) = u ǫ ( g ǫ ( ρ ′ ( t ))( ρ ′′ ( t ))) ( t ∈ U )In particular, by taking ρ ′ to be the constant map with value x , we seethat if ( f ǫ ) is moderate then so is ( g ǫ ( x )) for every x ∈ X . Hence γ maps c M ( C ∞ ( X × Y, Z ) Γ ) into C ∞ ( X, c M ( C ∞ ( Y, Z ) Γ )). Conversely, we can showthat ( g ǫ ) ∈ C ∞ ( X, c M ( C ∞ ( Y, Z ) Γ ) implies ( f ǫ ) ∈ c M ( C ∞ ( X × Y, Z ) Γ ) asfollows. Let G ǫ ( s, t ) = u ǫ ( g ǫ ( ρ ′ ( s ))( ρ ′′ ( t ))) ( s, t ∈ U ) ENERALIZED MAPS 9
Then, for any K ⋐ U , α ∈ N dim U , and s ∈ K there exists an m ∈ N suchthat the following holds for s = s :(4.2) max t ∈ K | D α G ǫ ( s, t ) | = O (1 /ǫ m ) a.e.But then, (4.2) holds for all s sufficiently near s because ( D α G ǫ ( s, t )) issmooth with respect to s . It follows, by the compactness of K , that (4.2)holds for all s ∈ K with sufficiently large m . In particular, as we have u ǫ ( f ǫ ( ρ ( t ))) = G ǫ ( t, t ), the composite u ǫ ◦ f ǫ ◦ ρ is moderate for all u ǫ and ρ .Hence ( f ǫ ) is moderate and γ restricts to an isomorphism (4.1).Now, let ( f ǫ ) , ( f ′ ǫ ) ∈ c M ( C ∞ ( X × Y, Z ) Γ ) and ( g ǫ ) = γ (( f ǫ )), ( g ′ ǫ ) = γ (( f ′ ǫ )). Then we have u ǫ ( f ′ ǫ ( ρ ǫ ( t ))) − u ǫ ( f ǫ ( ρ ǫ ( t ))) = u ǫ ( g ′ ǫ ( ρ ′ ǫ ( t ))( ρ ′′ ǫ ( t ))) − u ǫ ( g ǫ ( ρ ′ ǫ ( t ))( ρ ′′ ǫ ( t )))for every ( u ǫ ) ∈ c M ( E R ( Z )) and ( ρ ǫ ) = (( ρ ′ ǫ , ρ ′′ ǫ )) ∈ c M ( C ∞ ( U, X × Y ) Γ ).From this, we see that ( f ǫ ) ∼ ( f ′ ǫ ) holds if and only if ( g ǫ ( x )) ∼ ( g ′ ǫ ( x )) holdsfor every x ∈ X . Therefore, γ induces a natural isomorphism b C ∞ ( X × Y, Z ) ∼ = C ∞ ( X, b C ∞ ( Y, Z )) Note.
Let b X = b C ∞ ( R , X ) for any X ∈ Diff . Then, by taking Y = R inthe isomorphism above, we obtain b C ∞ ( X, Z ) ∼ = C ∞ ( X, b Z ). In particular,we have b C ∞ ( X, F ) ∼ = C ∞ ( X, b F ), meaning that F -valued asymptotic function is the same thing as b F -valued smooth function .5. Smooth cell complexes and asymptotic homotopy
Asymptotic maps can be expected to provide “weak solutions” to suchproblems that are hard to solve within the usual framework of smooth maps.In this section, we present a modest application related to homotopy theoryof diffeological spaces.Let I be the interval [0 ,
1] equipped with standard diffeology. As in thecase of
Top or Diff , there is a notion of homotopy in the category d Diff :a homotopy between two asymptotic maps f, g : X → Y is an asymptoticmap H : X × I → Y satisfying H ◦ i = f and H ◦ i = g , where i α is theinclusion X → X × { α } ⊂ X × I for α = 0 ,
1. Denote by L n the subspace ∂I n × I ∪ I n × { } of I n +1 . Lemma 5.1. L n is a deformation retract of I n +1 in d Diff .Proof.
It is easy to construct a piecewise linear retraction r : I n +1 → L n byusing suitable polyhedral decomposition of I n +1 . Let r i be the i -th compo-nent function of r . Then there exists for each i a moderate net ( R i,ǫ ) whichrepresents the image of r i under the embedding of continuous functions intoasymptotic functions. Now, let R ǫ = ( R ,ǫ , · · · , R n +1 ,ǫ ). Then the net ( R ǫ )is moderate, and represents an asymptotic retraction R : I n +1 → L n . More-over, there is a retracting homotopy H : I n +1 × I → I n +1 between R andthe identity represented by a net ( h ǫ ), where h ǫ ( u, t ) = (1 − t ) R ( u ) + tu, ( u, t ) ∈ I n +1 × I showing that L n is a deformation retract of I n +1 . (cid:3) Corollary 5.2.
Any asymptotic map L n → X can be extended to an as-ymptotic map I n +1 → X . Definition 5.3.
Let (
X, A ) be a pair of diffeological spaces. We say that(
X, A ) is a smooth relative cell complex if there is an ordinal δ and a δ -sequence Z : δ → Diff such that the inclusion i : A → X coincides with thecomposition Z → colim Z and for each successor ordinal β < δ , there existsa smooth map φ β : ∂I n → Z β − (called an attaching map ) such that Z β isdiffeomorphic to the adjunction space Z β − ∪ φ β I n i.e. a pushout of Z β − φ β ←− ∂I n ֒ → I n In particular, if A = ∅ then X is called a smooth cell complex .The next theorem says that every smooth relative cell complex enjoyshomotopy extension property in the extended category d Diff . Theorem 5.4.
Let ( X, A ) be a smooth relative cell complex, and f : X → Y be an asymptotic map. Suppose there is an asymptotic homotopy h : A × I → Y satisfying h = f | A . Then there exists an asymptotic homotopy H : X × I → Y satisfying H = f and H | A × I = h . To prove this, we need the lemma below (cf. [4, Lemma 4.8]).
Lemma 5.5.
Let φ : ∂I n → Y be a smooth map, and Z = Y ∪ φ I n be theadjunction space given by φ and the inclusion ∂I n → I n . Then the map i × S Φ × Y × I ` I n × I → Z × I induced by the natural maps i : Y → Z and Φ : I n → Z is a subduction.Proof. Let P : U → Z × I be a plot of Z × I given by P ( r ) = ( σ ( r ) , σ ′ ( r )),and let r ∈ U . Since σ is a plot of Z , there exists a plot Q : V → Y suchthat σ | V = i ◦ Q holds, or a plot Q : V → I n such that σ | V = Φ ◦ Q holds. In either case, we have P | V = ( i × S Φ × ◦ ( Q α , σ ′ ) | V, where α is either 1 or 2. This means that i × S Φ × (cid:3) Proof of Theorem . Let Z : δ → Diff be a δ -sequence such that the com-position Z → colim Z is the inclusion A → X . We construct an asymptotichomotopy H : X × I → Y by transfinite induction on β < δ . Suppose β is asuccessor ordinal, and there is an asymptotic homotopy H β − : Z β − × I → Y such that the following hold. H β − | X × { } = f | Z β − , H β − | A × I = h Let i β : Z β − → Z β be the inclusion. Then we have a commutative diagram ∂I n × I ∪ I n × { } ( φ β × ∪ Φ β / / (cid:15) (cid:15) Z β − × I ∪ Φ β ( I n ) × { } H β − ∪ f / / (cid:15) (cid:15) YI n × I Φ β × / / Z β × I ENERALIZED MAPS 11
By Corollary 5.2 the composition of upper arrows can be extended to anasymptotic map K β : I n × I → Y . In order to further extend K β over Z β × I , let us take representatives ( H β − ,ǫ ) for H β − and ( K β,ǫ ) for K β .Then for each ǫ >
0, there is a unique map H β,ǫ : Z β × I → Y such that Z β − × I ` I n × I H β − ,ǫ S K β,ǫ / / i β × S Φ β × (cid:15) (cid:15) YZ β × I H β,ǫ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ commutes. But then H β,ǫ is smooth because i β × S Φ β × H β,ǫ ) is moderate and defines anasymptotic map H β : Z β × I → Y satisfying H β ◦ (Φ β ×
1) = K β . Thus, byapplying transfinite induction on the δ -sequence Z , we obtain an asymptotichomotopy H : X × I → Y extending h : A × I → Y . (cid:3) References [1] J. F. Colombeau,
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Graduate School of Natural Science and Technology, Okayama Univer-sity, Okayama 700-8530, Japan
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