April 10, For Ivan Kolář, on the occasion of his 60th birthday.

Transcription

1 Archivm Mathematicm (Brno) 32,4 (1996) PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS Andreas Kriegl Peter W. Michor Institt für Mathematik, Universität Wien, Strdlhofgasse 4, A-1090 Wien, Astria. Erin Schrödinger Institt für Mathematische Physik, Boltzmanngasse 9, A-1090 Wien, Astria April 10, 2008 For Ivan Kolář, on the occasion of his 60th birthday. Abstract. The theory of prodct preserving fnctors and Weil fnctors is partly extended to infinite dimensional manifolds, sing the theory of C -algebras. Table of contents 1. Introdction Infinite dimensional manifolds Weil fnctors on infinite dimensional manifolds Prodct preserving fnctors from finite dimensional manifolds to infinite dimensional ones Introdction In competition to the theory of jets of Ehresmann André Weil in [21] explained a constrction hich is on the one hand more restricted than that of jets since it allos only for covariant constrctions in the sense of category theory (contravariant in the sense of differential geometry), bt is more flexible since it ses more inpt: a finite dimensional formally real algebra. Later it as realized that Weil s constrction describes all prodct preserving bndle fnctors on the category of finite dimension manifolds. This as developed independently by [1], [4], less completely by [13], and exposed in detail in chapter VIII of [6]. A jet-like approach to Weil s constrction as given in [18] and sed by Kolář in [5] to discss natral transformations. The prpose of this paper is to present some versions of this theory in the realm of 1991 Mathematics Sbject Classification. 58B99. Key ords and phrases. Prodct preserving fnctors, convenient vector spaces, C -algebras. Spported by Fonds zr Förderng der issenscahftlichen Forschng, Projekt P PHY. 1 Typeset by AMS-TEX

2 2 ANDREAS KRIEGL PETER W. MICHOR infinite dimensional manifolds, in the setting of convenient calcls as in [2], [10], [8]. 2. Infinite dimensional manifolds 2.1. Calcls in infinite dimensions. A locally convex vector space E is called convenient if for each smooth crve the Riemann integrals over compact intervals exist (this is a eak completeness condition). The final topology on E ith respect to all smooth crves is called the c -topology. A mapping beteen (c -) open sbsets of convenient vector spaces is called smooth if it maps smooth crves to smooth crves. Mltilinear mappings are smooth if and only if they are bonded. This gives a meaningfl theory hich p to Fréchet spaces coincides ith any reasonable theory of smooth mappings. The main additional property is cartesian closedness: If U, V are c -open sbsets in and if G is a convenient vector space, then C (V, G) is again a convenient vector space (ith the locally convex topology of convergence of compositions ith smooth crves in V, niformly on compact intervals, in all derivatives separately), and e have C (U, C (V, G)) = C (U V, G). Expositions of this theory can be fond in [2], [10], [8], e.g. Real analytic and holomorphic versions of this theory are also available, [11], [7], [10] Manifolds. A chart (U, ) on a set M is a bijection : U (U) E U from a sbset U M onto a c -open sbset of a convenient vector space E U. For to charts (U α, α ) and (U β, β ) on M the mapping αβ := α 1 β : β (U αβ ) α (U αβ ) for α, β A is called the chart changing, here U αβ := U α U β. A family (U α, α ) α A of charts on M is called an atlas for M, if the U α form a cover of M and all chart changings αβ are defined on c -open sbsets. An atlas (U α, α ) α A for M is said to be a C -atlas, if all chart changings αβ : β (U αβ ) α (U αβ ) are smooth. To C -atlases are called C -eqivalent, if their nion is again a C -atlas for M. An eqivalence class of C -atlases is called a C -strctre on M. The nion of all atlases in an eqivalence class is again an atlas, the maximal atlas for this C -strctre. A C -manifold M is a set together ith a C -strctre on it A mapping f : M N beteen manifolds is called smooth if for each x M and each chart (V, v) on N ith f(x) V there is a chart (U, ) on M ith x U, f(u) V, sch that v f 1 is smooth. This is the case if and only if f c is smooth for each smooth crve c : R M. We ill denote by C (M, N) the space of all C -mappings from M to N The topology of a manifold. The natral topology on a manifold M is the identification topology ith respect to some (smooth) atlas ( α : M U α α (U α ) E α ), here a sbset W M is open if and only if α (U α W ) is c -open in E α for all α. This topology depends only on the strctre, since diffeomorphisms are homeomorphisms for the c -topologies. It is also the final topology ith respect to all inverses of chart mappings in one atlas. It is also the final topology ith

3 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS 3 respect to all smooth crves. For a (smooth) manifold e ill reqire certain properties for the natral topology, hich ill be specified hen needed, like: (1) Smoothly Hasdorff: The smooth fnctions in C (M, R) separate points in M. (2) Smoothly reglar: For each neighborhood U of a point x M there exists a smooth fnction f : M R ith f(x) = 1 and carrier f 1 (R \ {0}) contained in U; eqivalently the initial topology ith respect to C (M, R) eqals the natral topology. (3) Smoothly real compact: Any bonded algebra homomorphism C (M, R) R is given by a point evalation; eqivalently, the natral mapping M Hom(C (M, R), R) is srjective Sbmanifolds. A sbset N of a manifold M is called a sbmanifold, if for each x N there is a chart (U, ) of M sch that (U N) = (U) F U, here F U is a c -closed linear sbspace of the convenient model space E U. Then clearly N is itself a manifold ith (U N, U N) as charts, here (U, ) rns throgh all sbmanifold charts as above. A sbmanifold N of M is called a splitting sbmanifold if there is a cover of N by sbmanifold charts (U, ) as above sch that the F U E U are complemented (i.e. splitting) linear sbspaces. Then obviosly every sbmanifold chart is splitting Prodcts. Let M and N be smooth manifolds described by smooth atlases (U α, α ) α A and (V β, v β ) β B, respectively. Then the family (U α V β, α v β : U α V β E α F β ) (α,β) A B is a smooth atlas for the cartesian prodct M N. Beare, hoever, the manifold topology of M N may be finer than the prodct topology, see [10]. If M and N are metrizable, then it is the prodct topology, by [10] again. Clearly the projections M pr1 M N pr2 N are also smooth. The prodct (M N, pr 1, pr 2 ) has the folloing niversal property: For any smooth manifold P and smooth mappings f : P M and g : P N the mapping (f, g) : P M N, (f, g)(x) = (f(x), g(x)), is the niqe smooth mapping ith pr 1 (f, g) = f, pr 2 (f, g) = g Lemma. [10] For a convenient vector space E and any smooth manifold M the set C (M, E) of smooth E-valed fnctions on M is again a convenient vector space ith the locally convex topology of niform convergence on compact sbsets of compositions ith smooth crves in M, in all derivatives separately. Moreover, ith this strctre, for to manifolds M, N, the exponential la holds: C (M, C (N, E)) = C (M N, E). 3. Weil fnctors on infinite dimensional manifolds 3.1. A real commtative algebra A ith nit 1 is called formally real if for any a 1,..., a n A the element 1 + a a 2 n is invertible in A. Let E = {e A : e 2 = e, e 0} A be the set of all nonzero idempotent elements in A. It is not empty since 1 E. An idempotent e E is said to be minimal if for any e E e have ee = e or ee = 0.

4 4 ANDREAS KRIEGL PETER W. MICHOR Lemma. Let A be a real commtative algebra ith nit hich is formally real and finite dimensional as a real vector space. Then there is a decomposition 1 = e e k into all minimal idempotents. Frthermore A = A 1 A k, here A i = e i A = R e i N i, and N i is a nilpotent ideal. This is standard, see [6], 35.1, for a proof Definition. A Weil algebra A is a real commtative algebra ith nit hich is of the form A = R 1 N, here N is a finite dimensional ideal of nilpotent elements. So by lemma 3.1 a formally real and finite dimensional nital commtative algebra is the direct sm of finitely many Weil algebras Chart description of Weil fnctors. Let A = R 1 N be a Weil algebra. We ant to associate to it a fnctor T A : Mf Mf from the category Mf of all smooth manifolds modelled on convenient vector spaces into itself. Step 1. If f C (R, R) and λ1 + n R 1 N = A, e consider the Taylor expansion j f(λ)(t) = f (j) (λ) j=0 j! t j of f at λ and e pt T A (f)(λ1 + n) := f(λ)1 + j=1 f (j) (λ) n j, j! hich is finite sm, since n is nilpotent. Then T A (f) : A A is smooth and e get T A (f g) = T A (f) T A (g) and T A (Id R ) = Id A. Step 2. If f C (R, F ) for a convenient vector space F and λ1+n R 1 N = A, e consider the Taylor expansion j f(λ)(t) = f (j) (λ) j=0 j! t j of f at λ and e pt T A (f)(λ1 + n) := 1 f(λ) + j=1 n j f (j) (λ), j! hich is finite sm, since n is nilpotent. Then T A (f) : A A F =: T A F is smooth. Step 3. For f C (E, F ), here E, F are convenient vector spaces, e ant to define the vale of T A (f) at an element of the convenient vector space T A E = A E. Sch an element may be niqely ritten as 1 x 1 + j n j x j, here 1 and the n j N form a fixed finite linear basis of A, and here the x i E. Let again j f(x 1 )(y) = k 0 1 k! dk f(x 1 )(y k ) be the Taylor expansion of f at x 1 E for y E. Then e pt T A (f)(1 x 1 + j n j x j ) := = 1 f(x 1 ) + k 0 1 n j1... n jk d k f(x 1 )(x j1,..., x jk ) k! j 1,...,j k

5 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS 5 hich is again a finite sm. A change of basis in N indces the transposed change in the x i, namely i ( j aj i n j) x i = j n j ( i aj i x i), so the vale of T A (f) is independent of the choice of the basis of N. Since the Taylor expansion of a composition is the composition of the Taylor expansions e have T A (f g) = T A (f) T A (g) and T A (Id E ) = Id TA E. If ϕ : A B is a homomorphism beteen to Weil algebras e have (ϕ F ) T A f = T B f (ϕ E) for f C (E, F ). Step 4. Let π = π A : A A/N = R be the projection onto the qotient field of the Weil algebra A. This is a srjective algebra homomorphism, so by step 3 the folloing diagram commtes for f C (E, F ): π E A E E T A f f A F π F F If U E is a c -open sbset e pt T A (U) := (π E) 1 (U) = (1 U) (N E), hich is an c -open sbset in T A (E) := A E. If f : U V is a smooth mapping beteen c -open sbsets U and V of E and F, respectively, then the constrction of step 3, applied to the Taylor expansion of f at points in U, prodces a smooth mapping T A f : T A U T A V, hich fits into the folloing commtative diagram: U (N E) T [ T A U A f [ pr 1 [] π E π F f U T A V V V (N F ) pr 1 We have T A (f g) = T A f T A g and T A (Id U ) = Id TA U, so T A is no a covariant fnctor on the category of c -open sbsets of convenient vector spaces and smooth mappings beteen them. Step 5. Let M be a smooth manifold, let (U α, α : U α α (U α ) E α ) be a smooth atlas of M ith chart changings αβ := α 1 β : β (U αβ ) α (U αβ ). Then the smooth mappings T A ( β (U αβ )) T A ( αβ ) T A ( α (U αβ )) π E β β (U αβ ) π E α αβ α (U αβ ) form again a cocycle of chart changings and e may se them to gle the c - open sets T A ( α (U α )) = α (U α ) (N E α ) T A E α in order to obtain a smooth manifold hich e denote by T A M. By the diagram above e see that T A M ill be the total space of a fiber bndle T (π A, M) = π A,M : T A M M, since the atlas (T A (U α ), T A ( α )) constrcted jst no is already a fiber bndle atlas. So if M is Hasdorff then also T A M is Hasdorff, since to points x i can be separated in one

6 6 ANDREAS KRIEGL PETER W. MICHOR chart if they are in the same fiber, or they can be separated by inverse images nder π A,M of open sets in M separating their projections. This constrction does not depend on the choice of the atlas. For to atlases have a common refinement and one may pass to this. If f C (M, M ) for to manifolds M, M, e apply the fnctor T A to the local representatives of f ith respect to sitable atlases. This gives local representatives hich fit together to form a smooth mapping T A f : T A M T A M. Clearly e again have T A (f g) = T A f T A g and T A (Id M ) = Id TA M, so that T A : Mf Mf is a covariant fnctor Remark. If e apply the constrction of 3.3, step 5 to the algebra A = 0, hich e did not allo (1 0 A), then T 0 M depends on the choice of the atlas. If each chart is connected, then T 0 M = π 0 (M), compting the connected components of M. If each chart meets each connected component of M, then T 0 M is one point Theorem. Main properties of Weil fnctors. Let A = R 1 N be a Weil algebra, here N is the maximal ideal of nilpotents. Then e have: 1. The constrction of 3.3 defines a covariant fnctor T A : Mf Mf sch that (T A M, π A,M, M) is a smooth fiber bndle ith standard fiber N E if M is modelled on the convenient space E. For any f C (M, M ) e have a commtative diagram T T A M A f T A M π A,M M f π A,M M. So (T A, π A ) is a bndle fnctor on Mf, hich gives a vector bndle on Mf if and only if N is nilpotent of order The fnctor T A : Mf Mf is mltiplicative: it respects prodcts. It maps the folloing classes of mappings into itself: immersions, splitting immersions, embeddings, splitting embeddings, closed embeddings, sbmersions, splitting sbmersions, srjective sbmersions, fiber bndle projections. It also respects transversal pllbacks. For fixed manifolds M and M the mapping T A : C (M, M ) C (T A M, T A M ) is smooth, so it maps smoothly parametrized families to smoothly parametrized families. 3. If (U α ) is an open cover of M then T A (U α ) is also an open cover of T A M. 4. Any algebra homomorphism ϕ : A B beteen Weil algebras indces a natral transformation T (ϕ, ) = T ϕ : T A T B. If ϕ is injective, then T (ϕ, M) : T A M T B M is a closed embedding for each manifold M. If ϕ is srjective, then T (ϕ, M) is a fiber bndle projection for each M. So e may vie T as a co-covariant bifnctor from the category of Weil algebras times Mf to Mf. Proof. 1. The main assertion is clear from 3.3. The fiber bndle π A,M : T A M M is a vector bndle if and only if the transition fnctions T A ( αβ ) are fiber linear N E α N E β. So only the first derivatives of αβ shold act on N, so any prodct of to elements in N mst be 0, ths N has to be nilpotent of order The fnctor T A respects finite prodcts in the category of c -open sbsets of convenient vector spaces by 3.3, step 3 and 5. All the other assertions follo by

7 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS 7 looking again at the chart strctre of T A M and by taking into accont that f is part of T A f (as the base mapping). 3. This is obvios from the chart strctre. 4. We define T (ϕ, E) := ϕ E : A E B E. By 3.3, step 3, this restricts to a natral transformation T A T B on the category of c -open sbsets of convenient vector spaces and by gling also on the category Mf. Obviosly T is a co-covariant bifnctor on the indicated categories. Since π B ϕ = π A (ϕ respects the identity), e have T (π B, M) T (ϕ, M) = T (π A, M), so T (ϕ, M) : T A M T B M is fiber respecting for each manifold M. In each fiber chart it is a linear mapping on the typical fiber N A E N B E. So if ϕ is injective, T (ϕ, M) is fiberise injective and linear in each canonical fiber chart, so it is a closed embedding. If ϕ is srjective, let N 1 := ker ϕ N A, and let V N A be a linear complement to N 1. Then if M is modeled on convenient vector spaces E α and for the canonical charts e have the commtative diagram: T A M T (ϕ, M) T B M T A (U α ) T A ( α ) α (U α ) (N A E α ) T (ϕ, U α ) Id (ϕ N A E α ) T B (U α ) T B ( α ) α (U α ) (N B E α ) α (U α ) (N 1 E α ) (V E α ) Id 0 Iso α (U α ) 0 (N B E α ) So T (ϕ, M) is a fiber bndle projection ith standard fiber E α ker ϕ Theorem. Let A and B be Weil algebras. Then e have: (1) We get the algebra A back from the Weil fnctor T A by T A (R) = A ith addition + A = T A (+ R ), mltiplication m A = T A (m R ) and scalar mltiplication m t = T A (m t ) : A A. (2) The natral transformations T A T B correspond exactly to the algebra homomorphisms A B Proof. (1) is obvios. (2) For a natral transformation ϕ : T A T B its vale ϕ R : T A (R) = A T B (R) = B is an algebra homomorphisms. The inverse of this mapping is already described in theorem Proposition. For to manifolds M 1 and M 2, ith M 2 smoothly real compact and smoothly reglar, the mapping is bijective. C (M 1, M 2 ) Hom(C (M 2, R), C (M 1, R)) f (f : g g f)

8 8 ANDREAS KRIEGL PETER W. MICHOR Proof. Let x 1 M 1 and ϕ Hom(C (M 2, R), C (M 1, R)). Then ev x1 ϕ is in Hom(C (M 2, R), R), so by 2.4 there is a x 2 M 2 sch that ev x1 ϕ = ev x2 since M 2 is smoothly real compact, and x 2 is niqe since M 2 is smoothly Hasdorff. If e rite x 2 = f(x 1 ), then f : M 1 M 2 and ϕ(g) = g f for all g C (M 2, R). This implies that f is smooth, since M 2 is smoothly reglar Remark. If M is a smoothly real compact and smoothly reglar manifold e consider the set D A (M) := Hom(C (M, R), A) of all bonded homomorphisms from the convenient algebra of smooth fnctions on M into a Weil algebra A. Obviosly e have a natral mapping T A M D A M hich is given by X (f T A (f).x), sing 3.5 and 3.6. Let D be the algebra of Stdy nmbers R.1 R.δ ith δ 2 = 0. Then T D M = T M, the tangent bndle, and D D (M) is the smooth bndle of all operational tangent vectors, i.e. bonded derivations at a point x of the algebra of germs Cx (M, R) see [10]. We ant to point ot that even on Hilbert spaces there exist derivations hich are differential operators of order 2 and 3, respectively, see [10]. It old be nice if D A (M) ere a smooth manifold, not only for A = D. We do not kno hether this is tre. The obvios method of proof hits severe obstacles, hich e no explain. Let A = R.1 N for a nilpotent finite dimensional ideal N, let π : A R be the corresponding projection. Then for ϕ D A (M) = Hom(C (M, R), A) the character π ϕ = ev x for a niqe x M, since M is smoothly real compact. Moreover X := ϕ ev x.1 : C (M, R) N satisfies the expansion property at x: (1) X(fg) = X(f).g(x) + f(x).x(g) + X(f).X(g). Conversely a bonded linear mapping X : C (M, R) N ith property (1) is called an expansion at x. Clearly each expansion at x defines a bonded homomorphism ϕ ith π ϕ = ev x. So e vie D A (M) x as the set of all expansions at x. Note first that for an expansion X D A (M) x the vale X(f) depends only on the germ of f at x: If f U = 0 for a neighborhood U of x, choose a smooth fnction h ith h = 1 off U and h(x) = 0. Then h k f = f and X(f) = X(h k f) = X(h k )X(f) = = X(h) k X(f) hich is 0 for k larger than the nilpotence index of N. Sppose no that M = U is a c -open sbset of a convenient vector space E. We can ask hether D A (U) x is a smooth manifold. We have no proof of this. Let s sketch the difficlty. A natral ay to prove that old be by indction on the nilpotence index of N. Let N 0 := {n N : n.n = 0}, hich is an ideal in A. Consider the short exact seqence 0 N 0 N p N/N 0 0 and a linear section s : N/N 0 N. For X : C (U, R) N e consider X := p X and X 0 := X s X. Then X is an expansion at x U if and only if X is an expansion at x ith vales in N/N 0 and X 0 satisfies (2) X 0 (fg) = X 0 (f)g(x) + f(x)x 0 (g) + s( X(f)).s( X(g)) s( X(f). X(g)).

9 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS 9 Note that (2) is an affine eqation in X 0 for fixed X. By indction the X D A/N0 (U) x form a smooth manifold, and the fiber over a fixed X consists of all X = X 0 + s X ith X 0 in the closed affine sbspace described by (2), hose model vector space is the space of all derivations at x. If e ere able to find a (local) section D A/N0 (U) D A (U) and if these sections old fit together nicely e cold then conclde that D A (U) ere the total space of a smooth affine bndle over D A/N0 (U), so it old be smooth. Bt this translates to a lifting problem as follos: A homomorphism C (U, R) A/N 0 has to be lifted in a natral ay to C (U, R) A. Bt e kno that in general C (U, R) is not a free C -algebra, see 4.4 for comparison The basic facts from the theory of Weil fnctors are completed by the folloing assertion. Proposition. Given to Weil algebras A and B, the composed fnctor T A T B is a Weil fnctor generated by the tensor prodct A B. Proof. For a convenient vector space E e have T A (T B E) = A B E and this is compatible ith the action of smooth mappings, by 3.3. Corollary. There is a canonical natral eqivalence T A T B = TB T A generated by the exchange algebra isomorphism A B = B A Weil fnctors and Lie grops. We shall se the notion of a reglar infinite dimensional Lie grop, modelled on convenient vector spaces, as laid ot in [9], folloing the lead of Omori et. al. [20] and Milnor [15]. We jst remark that they have niqe smooth exponential mappings, and that no smooth Lie grop is knon hich is not reglar. We shall se the notation µ : G G G for the mltiplication and ν : G G for the inversion. The tangent bndle T G of a reglar Lie grop G is again a Lie grop, the semidirect prodct g G of G ith its Lie algebra g. No let A be a Weil algebra and let T A be its Weil fnctor. Then the space T A (G) is again a Lie grop ith mltiplication T A (µ) and inversion T A (ν). By the properties 3.5 of the Weil fnctor T A e have a srjective homomorphism π A : T A G G of Lie grops. Folloing the analogy ith the tangent bndle, for a G e ill denote its fiber over a by (T A ) a G T A G, likeise for mappings. With this notation e have the folloing commtative diagram, here e assme that G is a reglar Lie grop: g N g A 0 (T A ) 0 g T A g g 0 (T A ) 0 exp G T A exp G e (T A ) e G T A G exp G π A G e The strctral mappings (Lie bracket, exponential mapping, evoltion operator, adjoint action) are determined by mltiplication and inversion. Ths their images

10 10 ANDREAS KRIEGL PETER W. MICHOR nder the Weil fnctor T A are again the same strctral mappings. Bt note that the canonical flip mappings have to be inserted like follos. So for example g A = T A g = T A (T e G) κ T e (T A G) is the Lie algebra of T A G and the Lie bracket is jst T A ([, ]). Since the bracket is bilinear, the description of 3.3 implies that [X a, Y b] TA g = [X, Y ] g ab. Also T A exp G = exp T AG. If exp G is a diffeomorphism near 0, (T A ) 0 (exp G ) : (T A ) 0 g (T A ) e G is also a diffeomorphism near 0, since T A is local. The natral transformation 0 G : G T A G is a homomorphism hich splits the bottom ro of the diagram, so T A G is the semidirect prodct (T A ) 0 g G via the mapping T A ρ : (, g) T A (ρ g )(). So from [9], theorem 5.5, e may conclde that T A G is again a reglar Lie grop, if G is reglar. If ω G : T G T e G is the Marer Cartan form of G (i.e. the left logarithmic derivative of Id G ) then κ 0 T A ω G κ : T T A G = T A T G T A T e G = T e T A G is the Marer Cartan form of T A G. 4. Prodct preserving fnctors from finite dimensional manifolds to infinite dimensional ones 4.1. Prodct preserving fnctors. Let Mf fin denote the category of all finite dimensional separable Hasdorff smooth manifolds, ith smooth mappings as morphisms. Let F : Mf fin Mf be a fnctor hich preserves prodcts in the folloing sense: The diagram F (M 1 ) F (pr1) F (M 1 M 2 ) F (pr2) F (M 2 ) is alays a prodct diagram. Then F (point) = point, by the folloing argment: F (pr 1 ) F (point) = f 1 F (point point) f point 4 44 F (pr 2 ) = 4 44 f 2 46 F (point) Each of f 1, f, and f 2 determines each other niqely, ths there is only one mapping f 1 : point F (point), so the space F (point) is single pointed. We also reqire that F has the folloing to properties: (1) The map on morphisms F : C (R n, R) C (F (R n ), F (R)) is smooth, here e regard C (F (R n ), F (R)) as smooth space, see [2] or [10]. Eqivalently the associated mapping C (R n, R) F (R n ) F (R) is smooth. (2) There is a natral transformation π : F Id sch that for each M the mapping π M : F (M) M is a fiber bndle, and for an open sbmanifold U M the mapping F (incl) : F (U) F (M) is a pllback.

11 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS C -algebras. An R-algebra is a commtative ring A ith nit together ith a ring homomorphism R A. Then every map p : R n R m hich is given by an m-tple of real polynomials (p 1,..., p m ) can be interpreted as a mapping A(p) : A n A m in sch a ay that projections, composition, and identity are preserved, by jst evalating each polynomial p i on an n-tple (a 1,..., a n ) A n. A C -algebra A is a real algebra in hich e can moreover interpret all smooth mappings f : R n R m. There is a corresponding map A(f) : A n A m, and again projections, composition, and the identity mapping are preserved. More precisely, a C -algebra A is a prodct preserving fnctor from the category C to the category of sets, here C has as objects all spaces R n, n 0, and all smooth mappings beteen them as arros. Morphisms beteen C -algebras are then natral transformations: they correspond to those algebra homomorphisms hich preserve the interpretation of smooth mappings. Let s explain ho one gets the algebra strctre from this interpretation. Since A is prodct preserving, e have A(point) = point. All the las for a commtative ring ith nit can be formlated by commtative diagrams of mappings beteen prodcts of the ring and the point. We do this for the ring R and apply the prodct preserving fnctor A to all these diagrams, so e get the las for the commtative ring A(R) ith nit A(1) ith the exception of A(0) A(1) hich e ill check later for the case A(R) point. Addition is given by A(+) and mltiplication by A(m). For λ R the mapping A(m λ ) : A(R) A(R) eqals mltiplication ith the element A(λ) A(R), since the folloing diagram commtes: A(R) = A(R) point = A(R point) A AAAA Id A(λ) A(m λ ) A AAAAAC A(Id λ) A(R) A(R) A(R R) ' '' ' ') A(m) A(R) We may investigate no the difference beteen A(R) = point and A(R) point. In the latter case for λ 0 e have A(λ) A(0) since mltiplication by A(λ) eqals A(m λ ) hich is a diffeomorphism for λ 0 and factors over a one pointed space for λ = 0. So for A(R) point hich e assme from no on, the grop homomorphism λ A(λ) from R into A(R) is actally injective. This definition of C -algebras is de to Lavere [12], for a thorogh accont see Moerdijk-Reyes [16], for a discssion from the point of vie of fnctional analysis see [3]. In particlar there on a C -algebra A the natral topology is defined as the finest locally convex topology on A sch that for all a = (a 1,..., a n ) A n the evalation mappings ε a : C (R n, R) A are continos. In [3], 6.6 one finds a method to recognize C -algebras among locally-m-convex algebras. In [14] one finds a characterization of the algebras of smooth fnctions on finite dimensional algebras among all C -algebras Theorem. Let F : Mf fin Mf be a prodct preserving fnctor. Then either F (R) is a point or F (R) is a C -algebra. If ϕ : F 1 F 2 is a natral transformation beteen to sch fnctors, then ϕ R : F 1 (R) F 2 (R) is an algebra homomorphism.

12 12 ANDREAS KRIEGL PETER W. MICHOR If F has property (1) then the natral topology on F (R) is finer than the given manifold topology and ths is Hasdorff if the latter is it. If F has property (2) then F (R) is a local algebra ith an algebra homomorphism π = π R : F (R) R hose kernel is the maximal ideal. Proof. By definition F restricts to a prodct preserving fnctor from the category of all R n s and smooth mappings beteen them, ths it is a C -algebra. If F has property (1) then for all a = (a 1,..., a n ) F (R) n the evalation mappings are given by ε a = ev a F : C (R n, R) C (F (R) n, F (R)) F (R) and ths are even smooth. If F has property (2) then obviosly π R = π : F (R) R is an algebra homomorphism. It remains to sho that the kernel of π is the largest ideal. So if a A has π(a) 0 R then e have to sho that a is invertible in A. Since the folloing diagram is a pllback, F (i) F (R \ {0}) F (R) π π R \ {0} i e may assme that a = F (i)(b) for a niqe b F (R \ {0}). Bt then 1/i : R \ {0} R is smooth, and F (1/i)(b) = a 1, since F (1/i)(b).a = F (1/i)(b).F (i)(b) = F (m)f (1/i, i)(b) = F (1)(b) = 1, compare Examples. Let A be an agmented local C -algebra ith maximal ideal N. Then A is qotient of a free C -algebra C fin (RΛ ) of smooth fnctions on some large prodct R Λ, hich depend globally only on finitely many coordinates, see [16] or [3]. So e have a short exact seqence R 0 I C fin(r Λ ) ϕ A 0. Then I is contained in the codimension 1 maximal ideal ϕ 1 (N), hich is easily seen to be {f Cfin (Rλ ) : f(x 0 ) = 0} for some x 0 R Λ. Then clearly ϕ factors over the qotient of germs at x 0. If A has Hasdorff natral topology, then ϕ even factors over the Taylor expansion mapping, by the argment in [3], 6.1, as follos. Namely, let f Cfin (RΛ ) be infinitely flat at x 0. We shall sho that f is in the closre of the set of all fnctions ith germ 0 at x 0. Let x 0 = 0 ithot loss. Note first that f factors over some qotient R Λ R N, and e may replace R Λ by R N ithot loss. Define g : R N R N R N, { 0 if x y, g(x, y) = (1 y / x )x if x > y. Since f is flat at 0, the mapping y (x f y (x) := f(g(x, y)) is a continos mapping R N C (R N, R) ith the property that f 0 = f and f y has germ 0 at 0 for all y 0. Ths the agmented local C -algebras hose natral topology is Hasdorff are exactly the qotients of algebras of Taylor series at 0 of fnctions in C fin (RΛ ). It seems that local implies agmented: one has to sho that a C -algebra hich is a field is 1-dimensional. Is this tre?

13 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS Chart description of fnctors indced by C -algebras. Let A = R 1 N be an agmented local C -algebra hich carries a compatible convenient strctre, i.e. A is a convenient vector space and each mapping A : C (R n, R m ) C (A n, A m ) is a ell defined smooth mapping. As in the proof of 4.3 one sees that the natral topology on A is then finer than the given convenient one, ths is Hasdorff. Let s call this an agmented local convenient C -algebra. We ant to associate to A a fnctor T A : Mf fin Mf from the category Mf fin of all finite dimensional separable smooth manifolds to the category of smooth manifolds modelled on convenient vector spaces. Step 1. Let π = π A : A A/N = R be the agmentation mapping. This is a srjective homomorphism of C -algebras, so the folloing diagram commtes for f C (R n, R m ): A n π n R n T A f f m A R m π m If U R n is an open sbset e pt T A (U) := (π n ) 1 (U) = U N n, hich is open sbset in T A (R n ) := A n. Step 2. No sppose that f : R n R m vanishes on some open set V R n. We claim that then T A f vanishes on the open set T A (V ) = (π n ) 1 (V ). To see this let x V, and choose a smooth fnction g C (R n, R) ith g(x) = 1 and spport in V. Then g.f = 0 ths e have also 0 = A(g.f) = A(m) A(g, f) = A(g).A(f), here the last mltiplication is pointise diagonal mltiplication beteen A and A m. For a A n ith (π n )(a) = x e get π(a(g)(a)) = g(π(a)) = g(x) = 1, ths A(g)(a) is invertible in the algebra A, and from A(g)(a).A(f)(a) = 0 e may conclde that A(f)(a) = 0 A m. Step 3. No let f : U W be a smooth mapping beteen open sets U R n and W R m. Then e can define T A (f) : T A (U) T A (W ) in the folloing ay. For x U let f x : R n R m be a smooth mapping hich coincides ith f in a neighborhood V of x in U. Then by step 2 the restriction of A(f x ) to T A (V ) does not depend on the choice of the extension f x, and by a standard argment one can define niqely a smooth mapping T A (f) : T A (U) T A (V ). Clearly this gives s an extension of the fnctor A from the category of all R n s and smooth mappings into convenient vector spaces to a fnctor from open sbsets of R n s and smooth mappings into the category of c -open (indeed open) sbsets of convenient vector spaces. Step 4. Let M be a smooth finite dimensional manifold, let (U α, α : U α α (U α ) R m ) be a smooth atlas of M ith chart changings αβ := α 1 β : β (U αβ ) α (U αβ ). Then by step 3 e get smooth mappings beteen c -open

14 14 ANDREAS KRIEGL PETER W. MICHOR sbsets of convenient vector spaces T A ( β (U αβ )) T A ( αβ ) T A ( α (U αβ )) π β (U αβ ) π αβ α (U αβ ) form again a cocycle of chart changings and e may se them to gle the c -open sets T A ( α (U α )) = π 1 R m( α(u α )) A m in order to obtain a smooth manifold hich e denote by T A M. By the diagram above e see that T A M ill be the total space of a fiber bndle T (π A, M) = π A,M : T A M M, since the atlas (T A (U α ), T A ( α )) constrcted jst no is already a fiber bndle atlas. So if M is Hasdorff then also T A M is Hasdorff, since to points x i can be separated in one chart if they are in the same fiber, or they can be separated by inverse images nder π A,M of open sets in M separating their projections. This constrction does not depend on the choice of the atlas. For to atlases have a common refinement and one may pass to this. If f C (M, M ) for to manifolds M, M, e apply the fnctor T A to the local representatives of f ith respect to sitable atlases. This gives local representatives hich fit together to form a smooth mapping T A f : T A M T A M. Clearly e again have T A (f g) = T A f T A g and T A (Id M ) = Id TA M, so that T A : Mf Mf is a covariant fnctor Theorem. Main properties. Let A = R 1 N be a local agmented convenient C -algebra. Then e have: 1. The constrction of 4.5 defines a covariant fnctor T A : Mf fin Mf sch that (T A M, π A,M, M) is a smooth fiber bndle ith standard fiber N m if dim M = m. For any f C (M, M ) e have a commtative diagram T A M T A f T A M π A,M M f π A,M M. 2. The fnctor T A : Mf Mf is mltiplicative: it respects prodcts. It respects immersions, embeddings, etc, similarly as in 3.5. It also respects transversal pllbacks. For fixed manifolds M and M the mapping T A : C (M, M ) C (T A M, T A M ) is smooth. 3. Any bonded algebra homomorphism ϕ : A B beteen agmented convenient C -algebras indces a natral transformation T (ϕ, ) = T ϕ : T A T B. If ϕ is split injective, then T (ϕ, M) : T A M T B M is a split embedding for each manifold M. If ϕ is split srjective, then T (ϕ, M) is a fiber bndle projection for each M. So e may vie T as a co-covariant bifnctor from the category of agmented convenient C -algebras algebras times Mf fin to Mf. Proof. 1. The main assertion is clear from 4.5. The fiber bndle π A,M : T A M M is a vector bndle if and only if the transition fnctions T A ( αβ ) are fiber linear

15 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS 15 N E α N E β. So only the first derivatives of αβ shold act on N, so any prodct of to elements in N mst be 0, ths N has to be nilpotent of order The fnctor T A respects finite prodcts in the category of c -open sbsets of convenient vector spaces by 3.3, step 3 and 5. All the other assertions follo by looking again at the chart strctre of T A M and by taking into accont that f is part of T A f (as the base mapping). 3. We define T (ϕ, R n ) := ϕ n : A n B n. By 4.5, step 3, this restricts to a natral transformation T A T B on the category of open sbsets of R n s by gling also on the category Mf. Obviosly T is a co-covariant bifnctor on the indicated categories. Since π B ϕ = π A (ϕ respects the identity), e have T (π B, M) T (ϕ, M) = T (π A, M), so T (ϕ, M) : T A M T B M is fiber respecting for each manifold M. In each fiber chart it is a linear mapping on the typical fiber NA m NB m. So if ϕ is split injective, T (ϕ, M) is fiberise split injective and linear in each canonical fiber chart, so it is a split embedding. If ϕ is split srjective, let N 1 := ker ϕ N A, and let V N A be a topological linear complement to N 1. Then for m = dim M and for the canonical charts e have the commtative diagram: T (ϕ, M) T A M T B M T A (U α ) T A ( α ) α (U α ) NA m T (ϕ, U α ) Id ϕ N m A T B (U α ) T B ( α ) α (U α ) NB m α (U α ) N1 m V m Id 0 Iso α (U α ) 0 N m B So T (ϕ, M) is a fiber bndle projection ith standard fiber E α ker ϕ Theorem. Let A and B be agmented convenient C -algebras. Then e have: (1) We get the convenient C -algebra A back from the fnctor T A by restricting to the sbcategory of R n s. (2) The natral transformations T A T B correspond exactly to the bonded C -algebra homomorphisms A B. Proof. (1) is obvios. (2) For a natral transformation ϕ : T A T B (hich is smooth) its vale ϕ R : T A (R) = A T B (R) = B is a C -algebra homomorphism hich is smooth and ths bonded. The inverse of this mapping is already described in theorem Proposition. Let A = R 1 N be a local agmented convenient C -algebra and let M be a smooth finite dimensional manifold. Then there exists a bijection ε : T A (M) Hom(C (M, R), A)

16 16 ANDREAS KRIEGL PETER W. MICHOR onto the space of bonded algebra homomorphisms, hich is natral in A and M. Via ε the expression Hom(C (, R), A) describes the fnctor T A in a coordinate free manner. Proof. Step 1. Let M = R n, so T A (R n ) = A n. Then for a = (a 1,..., a n ) A n e have ε(a)(f) = A(f)(a 1,..., a n ), hich gives a bonded algebra homomorphism C (R n, R) A. Conversely, for ϕ Hom(C (R n, R), A) consider a = (ϕ(pr 1 ),..., ϕ(pr n )) A n. Since polynomials are dense in C (R n, R), ϕ is bonded, and A is Hasdorff, ϕ is niqely determined by its vales on the coordinate fnctions pr i (compare [3], 2.4.(3)), ths ϕ(f) = A(f)(a) and ε is bijective. Obviosly ε is natral in A and R n. Step 2. No let i : U R n be an embedding of an open sbset. Then the image of the mapping Hom(C (U, R), A) (i ) Hom(C (R n, R), A) ε 1 R n,a A n is the set π 1 A,R n(u) = T A(U) A n, and (i ) is injective. To see this let ϕ Hom(C (U, R), A). Then ϕ 1 (N) is the maximal ideal in C (U, R) consisting of all smooth fnctions vanishing at a point x U, and x = π(ε 1 (ϕ i )) = π(ϕ(pr 1 i),..., ϕ(pr n i)), so that ε 1 ((i ) (ϕ)) T A (U) = π 1 (U) A n. Conversely for a T A (U) the homomorphism ε a : C (R n, R) A factors over i : C (R n, R) C (U, R), by steps 2 and 3 of 4.5. Step 3. The to fnctors Hom(C (, R), A) and T A : Mf Set coincide on all open sbsets of R n s, so they have to coincide on all manifolds, since smooth manifolds are exactly the retracts of open sbsets of R n s see e.g. [6], Alternatively one may check that the gling process described in 4.5, step 4, orks also for the fnctor Hom(C (, R), A) and gives a niqe manifold strctre on it hich is compatible to T A M. References [1] Eck, D. J., Prodct preserving fnctors on smooth manifolds, J. Pre and Applied Algebra 42 (1986), [2] Frölicher, Alfred; Kriegl, Andreas, Linear spaces and differentiation theory, Pre and Applied Mathematics, J. Wiley, Chichester, [3] Kainz, G.; Kriegl, A.; Michor, P. W., C -algebras from the fnctional analytic viepoint, J. pre appl. Algebra 46 (1987), [4] Kainz, G.; Michor, P. W., Natral transformations in differential geometry, Czechoslovak Math. J. 37 (1987), [5] Kolář, I., Covariant approach to natral transformations of Weil fnctors, Comment. Math. Univ. Carolin. 27 (1986), [6] Kolář, Ivan; Michor, Peter W.; Slovák, Jan, Natral operations in differential geometry, Springer-Verlag, Berlin, Heidelberg, Ne York, 1993, pp. vi+434. [7] Kriegl, Andreas; Michor, Peter W., A convenient setting for real analytic mappings, Acta Mathematica 165 (1990), [8] Kriegl, Andreas; Michor, Peter W., Aspects of the theory of infinite dimensional manifolds, Differential Geometry and Applications 1 (1991), [9] Kriegl, Andreas; Michor, Peter W., Reglar infinite dimensional Lie grops, to appear, J. Lie Theory.

17 PRODUCT PRESERVING FUNCTORS OF INFINITE DIMENSIONAL MANIFOLDS 17 [10] Kriegl, Andreas; Michor, Peter W., The Convenient Setting for Global Analysis, to appear, Srveys and Monographs, AMS, Providence, [11] Kriegl, Andreas; Nel, Lois D., A convenient setting for holomorphy, Cahiers Top. Géo. Diff. 26 (1985), [12] Lavere, F. W., Categorical dynamics, Lectres given 1967 at the University of Chicago, reprinted in, Topos Theoretical Methods in Geometry (A. Kock, ed.), Aarhs Math. Inst. Var. Pbl. Series 30, Aarhs Universitet, [13] Lciano, O. O., Categories of mltiplicative fnctors and Weil s infinitely near points, Nagoya Math. J. 109 (1988), [14] Michor, Peter W.; Vanžra, Jiři, Characterizing algebras of smooth fnctions on manifolds, to appear, Comm. Math. Univ. Carolinae (Prage). [15] Milnor, John, Remarks on infinite dimensional Lie grops, Relativity, Grops, and Topology II, Les Hoches, 1983, B.S. DeWitt, R. Stora, Eds., Elsevier, Amsterdam, [16] Moerdijk, I.; Reyes G. E., Models for smooth infinitesimal analysis, Springer-Verlag, Heidelberg Berlin, [17] Moerdijk, I.; Reyes G. E., Rings of smooth fncions and their localizations, I, J. Algebra 99 (1986), [18] Morimoto, A., Prolongations of connections to bndles of infinitely near points, J. Diff. Geom. 11 (1976), [19] Moerdijk, I.; Ngo Van Qe; Reyes G. E., Rings of smooth fncions and their localizations, II, Mathematical logic and theoretical compter science (D.W. Keker, E.G.K. Lopez- Escobar, C.H. Smith, eds.), Marcel Dekker, Ne York, Basel, [20] Omori, H.; Maeda, Y.; Yoshioka, A., On reglar Fréchet Lie grops IV. Definitions and fndamental theorems, Tokyo J. Math. 5 (1982), [21] Weil, André, Théorie des points proches sr les variétés differentielles, Colloqe de topologie et géométrie différentielle, Strasborg, 1953, pp Institt für Mathematik, Universität Wien, Strdlhofgasse 4, A-1090 Wien, Astria address: kriegl@pap.nivie.ac.at, Peter.Michor@esi.ac.at