31.5

308 Chapter VI. In¬nite dimensional manifolds 31.5

w A—F

TA f

A—E

u u

π—E π—F

wF

f

E

If U ‚ E is a c∞ -open subset we put TA (U ) := (π — E)’1 (U ) = (1 — U ) — (N — E),

which is a c∞ -open subset in TA (E) := A — E. If f : U ’ V is a smooth mapping

between c∞ -open subsets U and V of E and F , respectively, then the construction

of step 3 applied to the Taylor expansion of f at points in U , produces a smooth

mapping TA f : TA U ’ TA V , which ¬ts into the following commutative diagram:

wT

‘ TA f

U — (N — E) V — (N — F )

TA U AV

‘“ &

pr ‘ &pr

u u)

&

π—E π—F

1 1

wV

f

U

We have TA (f —¦ g) = TA f —¦ TA g and TA (IdU ) = IdTA U , so TA is now a covariant

functor on the category of c∞ -open subsets of convenient vector spaces and smooth

mappings between them.

Step 5. Let M be a smooth manifold, let (U± , u± : U± ’ u± (U± ) ‚ E± ) be a

smooth atlas of M with chart changings u±β := u± —¦ u’1 : uβ (U±β ) ’ u± (U±β ).

β

Then the smooth mappings

wT

TA (u±β )

TA (uβ (U±β )) A (u± (U±β ))

u u

π — Eβ π — E±

w u (U

u±β

uβ (U±β ) ±β )

±

form likewise a cocycle of chart changings, and we may use them to glue the c∞ -

open sets TA (u± (U± )) = u± (U± ) — (N — E± ) ‚ TA E± together in order to obtain

a smooth manifold which we denote by TA M . By the diagram above, we see that

TA M will be the total space of a ¬ber bundle T (πA , M ) = πA,M : TA M ’ M , since

the atlas (TA (U± ), TA (u± )) constructed just now is already a ¬ber bundle atlas, see

(37.1) below. So if M is Hausdor¬ then also TA M is Hausdor¬, since two points xi

can be separated in one chart if they are in the same ¬ber, or they can be separated

by inverse images under πA,M of open sets in M separating their projections. This

construction does not depend on the choice of the atlas, because two atlas have a

common re¬nement and one may pass to this.

If f ∈ C ∞ (M, M ) for two manifolds M , M , we apply the functor TA to the local

representatives of f with respect to suitable atlas. This gives local representatives

which ¬t together to form a smooth mapping TA f : TA M ’ TA M . Clearly, we

again have TA (f —¦ g) = TA f —¦ TA g and TA (IdM ) = IdTA M , so that TA : Mf ’ Mf

is a covariant functor.

31.5

31.7 31. Product preserving functors on manifolds 309

31.6. Remark. If we apply the construction of (31.5), step 5 to the algebra A = 0,

which we did not allow (1 = 0 ∈ A), then T0 M depends on the choice of the atlas. If

each chart is connected, then T0 M = π0 (M ), computing the connected components

of M . If each chart meets each connected component of M , then T0 M is one point.

31.7. Theorem. Main properties of Weil functors. Let A = R · 1 • N be a

Weil algebra, where N is the maximal ideal of nilpotents. Then we have:

(1) The construction of (31.5) de¬nes a covariant functor TA : Mf ’ Mf

such that πA:TA M ’M , M is a smooth ¬ber bundle with standard ¬ber N — E

if M is modeled on the convenient space E. For any f ∈ C ∞ (M, M ) we

have a commutative diagram

wT

TA f

TA M AM

πA,M πA,M

u u

wM.

f

M

So (TA , πA ) is a bundle functor on Mf , which gives a vector bundle functor

on Mf if and only if N is nilpotent of order 2.

(2) The functor TA : Mf ’ Mf is multiplicative: it respects products. It

maps the following classes of mappings into itself: embeddings of (split-

ting) submanifolds, surjective smooth mappings admitting local smooth sec-

tions, ¬ber bundle projections. For ¬xed manifolds M and M the mapping

TA : C ∞ (M, M ) ’ C ∞ (TA M, TA M ) is smooth, so it maps smoothly pa-

rameterized families to smoothly parameterized families.

(3) If (U± ) is an open cover of M then TA (U± ) is an open cover of TA M .

(4) Any algebra homomorphism • : A ’ B between Weil algebras induces a

natural transformation T (•, ) = T• : TA ’ TB . If • is injective, then

T (•, M ) : TA M ’ TB M is a closed embedding for each manifold M . If •

is surjective, then T (•, M ) is a ¬ber bundle projection for each M . So we

may view T as a co-covariant bifunctor from the category of Weil algebras

times Mf to Mf .

Proof. (1) The main assertion is clear from (31.5). The ¬ber bundle πA,M :

TA M ’ M is a vector bundle if and only if the transition functions TA (u±β ) are

¬ber linear N — E± ’ N — Eβ . So only the ¬rst derivatives of u±β should act on

N , hence any product of two elements in N must be 0, thus N has to be nilpotent

of order 2.

(2) The functor TA respects ¬nite products in the category of c∞ -open subsets of

convenient vector spaces by (31.5), step 3 and 5. All the other assertions follow by

looking again at the chart structure of TA M and by taking into account that f is

part of TA f (as the base mapping).

(3) This is obvious from the chart structure.

(4) We de¬ne T (•, E) := •—E : A—E ’ B—E. By (31.5), step 3, this restricts to a

natural transformation TA ’ TB on the category of c∞ -open subsets of convenient

31.7

310 Chapter VI. In¬nite dimensional manifolds 31.9

vector spaces, and “ by gluing “ also on the category Mf . Obviously, T is a co-

covariant bifunctor on the indicated categories. Since πB —¦ • = πA (• respects the

identity), we have T (πB , M ) —¦ T (•, M ) = T (πA , M ), so T (•, M ) : TA M ’ TB M is

¬ber respecting for each manifold M . In each ¬ber chart it is a linear mapping on

the typical ¬ber NA — E ’ NB — E.

So if • is injective, T (•, M ) is ¬berwise injective and linear in each canonical ¬ber

chart, so it is a closed embedding.

If • is surjective, let N1 := ker • ⊆ NA , and let V ‚ NA be a linear complement

to N1 . Then if M is modeled on convenient vector spaces E± and for the canonical

charts we have the commutative diagram:

w T uM

u

T (•, M )

TA M B

wT

T (•, U± )

TA (U± ) B (U± )

u u

TA (u± ) TB (u± )

w u (U ) — (N

Id —((•|NA ) — E± )

u± (U± ) — (NA — E± ) — E± )

± ± B

w u (U ) — 0 — (N

Id —0 — iso

u± (U± ) — (N1 — E± ) — (V — E± ) — E± )

± ± B

Hence T (•, M ) is a ¬ber bundle projection with standard ¬ber E± — ker •.

31.8. Theorem. Let A and B be Weil algebras. Then we have:

(1) We get the algebra A back from the Weil functor TA by TA (R) = A with

addition +A = TA (+R ), multiplication mA = TA (mR ) and scalar multipli-

cation mt = TA (mt ) : A ’ A.

(2) The natural transformations TA ’ TB correspond exactly to the algebra

homomorphisms A ’ B.

Proof. (1) is obvious. (2) For a natural transformation • : TA ’ TB its value

•R : TA (R) = A ’ TB (R) = B is an algebra homomorphisms. The inverse of this

mapping has already been described in theorem (31.7.4).

31.9. Remark. If M is a smoothly real compact and smoothly regular manifold

we consider the set DA (M ) := Hom(C ∞ (M, R), A) of all bounded homomorphisms

from the convenient algebra of smooth functions on M into a Weil algebra A.

Obviously we have a natural mapping TA M ’ DA M which is given by X ’ (f ’

TA (f ).X), using (3.5) and (3.6).

Let D be the algebra of Study numbers R.1 • R.δ with δ 2 = 0. Then TD M = T M ,

the tangent bundle, and DD (M ) is the smooth bundle of all operational tangent

vectors, i.e. bounded derivations at a point x of the algebra of germs C ∞ (M ⊇

{x}, R) see (28.12).

31.9

31.10 31. Product preserving functors on manifolds 311

It would be nice if DA (M ) were a smooth manifold not only for A = D. We do

not know whether this is true. The obvious method of proof hits severe obstacles,

which we now explain.

Let A = R.1 • N be a Weil algebra and let π : A ’ R be the corresponding

projection. Then for • ∈ DA (M ) = Hom(C ∞ (M, R), A) the character π —¦ • equals

evx for a unique x ∈ M , since M is smoothly real compact. Moreover, X :=

• ’ evx .1 : C ∞ (M, R) ’ N satis¬es the expansion property at x:

(1) X(f g) = X(f ).g(x) + f (x).X(g) + X(f ).X(g).

Conversely, a bounded linear mapping X : C ∞ (M, R) ’ N with property (1)

is called an expansion at x. Clearly each expansion at x de¬nes a bounded ho-

momorphism • with π —¦ • = evx . So we view DA (M )x as the set of all ex-

pansions at x. Note ¬rst that for an expansion X ∈ DA (M )x the value X(f )

depends only on the germ of f at x: If f |U = 0 for a neighborhood U of x,

choose a smooth function h with h = 1 o¬ U and h(x) = 0. Then hk f = f and

X(f ) = X(hk f ) = 0 + 0 + X(hk )X(f ) = · · · = X(h)k X(f ), which is 0 for k larger

than the nilpotence index of N .

Suppose now that M = U is a c∞ -open subset of a convenient vector space E. We

can ask whether DA (U )x is a smooth manifold. Let us sketch the di¬culty. A

natural way to proceed would be to apply by induction on the nilpotence index of

N . Let N0 := {n ∈ N : n.N = 0}, which is an ideal in A. Consider the short exact

sequence

p

0 ’ N0 ’ N ’ N/N0 ’ 0

’

¯

and a linear section s : N/N0 ’ N . For X : C ∞ (U, R) ’ N we consider X := p —¦ X

¯

and X0 := X ’ s —¦ X. Then X is an expansion at x ∈ U if and only if

¯

(2) X is an expansion at x with values in N/N0 , and X0 satis¬es

¯ ¯ ¯ ¯

X0 (f g) = X0 (f )g(x) + f (x)X0 (g) + s(X(f )).s(X(g)) ’ s(X(f ).X(g)).

¯ ¯

Note that (2) is an a¬ne equation in X0 for ¬xed X. By induction, the X ∈

¯

DA/N0 (U )x form a smooth manifold, and the ¬ber over a ¬xed X consists of all

¯

X = X0 + s —¦ X with X0 in the closed a¬ne subspace described by (2), whose

model vector space is the space of all derivations at x. If we were able to ¬nd

a (local) section DA/N0 (U ) ’ DA (U ) and if these sections ¬tted together nicely

we could then conclude that DA (U ) was the total space of a smooth a¬ne bundle

over DA/N0 (U ), so it would be smooth. But this translates to a lifting problem as

follows: A homomorphism C ∞ (U, R) ’ A/N0 has to be lifted in a ˜natural way™ to

C ∞ (U, R) ’ A. But we know that in general C ∞ (U, R) is not a free C ∞ -algebra,

see (31.16) for comparison.

31.10. The basic facts from the theory of Weil functors are completed by the

following assertion.

31.10

312 Chapter VI. In¬nite dimensional manifolds 31.12

Proposition. Given two Weil algebras A and B, the composed functor TA —¦ TB is

a Weil functor generated by the tensor product A — B.

Proof. For a convenient vector space E we have TA (TB E) = A — B — E, and this

is compatible with the action of smooth mappings, by (31.5).

Corollary. There is a canonical natural equivalence TA —¦ TB ∼ TB —¦ TA generated

=

by the exchange algebra isomorphism A — B ∼ B — A.

=

31.11. Examples. Let A be the algebra R.1+R.δ with δ 2 = 0. Then TA M = T M ,

the tangent bundle, and consequently we get TA—A M = T 2 M , the second tangent

bundle.

31.12. Weil functors and Lie groups. We have (compare (38.10)) that the

tangent bundle T G of a Lie group G is again a Lie group, the semidirect product

g G of G with its Lie algebra g.

Now let A be a Weil algebra, and let TA be its Weil functor. Then in the notation of

(36.1) the space TA (G) is also a Lie group with multiplication TA (µ) and inversion

TA (ν). By the properties (31.7), of the Weil functor TA we have a surjective homo-

morphism πA : TA G ’ G of Lie groups. Following the analogy with the tangent

bundle, for a ∈ G we will denote its ¬ber over a by (TA )a G ‚ TA G, likewise for

mappings. With this notation we have the following commutative diagram, where

we assume that G is a regular Lie group (38.4):

w g—A

g—N

w (T wT wg w0

0 A )0 g Ag

u u u

TA expG expG

(TA )0 expG

w (T wT wG we

πA

e A )e G AG

The structural mappings (Lie bracket, exponential mapping, evolution operator,

adjoint action) are determined by multiplication and inversion. Thus, their images

under the Weil functor TA are the same structural mappings. But note that the

canonical ¬‚ip mappings have to be inserted like follows. So for example

κ

g — A ∼ TA g = TA (Te G) ’ Te (TA G)

’

=

is the Lie algebra of TA G, and the Lie bracket is just TA ([ , ]). Since the bracket

is bilinear, the description of (31.5) implies that [X — a, Y — b]TA g = [X, Y ]g —

ab. Also TA expG = expTA G . If expG is a di¬eomorphism near 0, (TA )0 (expG ) :

(TA )0 g ’ (TA )e G is also a di¬eomorphism near 0, since TA is local. The natural

transformation 0G : G ’ TA G is a homomorphism which splits the bottom row

of the diagram, so TA G is the semidirect product (TA )0 g G via the mapping

TA ρ : (u, g) ’ TA (ρg )(u). So from (38.9) we may conclude that TA G is also a

31.12

31.14 31. Product preserving functors on manifolds 313

regular Lie group, if G is regular. If ω G : T G ’ Te G is the Maurer Cartan form of

G (i.e., the left logarithmic derivative of IdG ) then

κ0 —¦ 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 Maurer Cartan form of TA G.

Product preserving functors from ¬nite

dimensional manifolds to in¬nite dimensional ones

31.13. Product preserving functors. Let Mf¬n denote the category of all

¬nite dimensional separable Hausdor¬ smooth manifolds, with smooth mappings

as morphisms. Let F : Mf¬n ’ Mf be a functor which preserves products in the

following sense: The diagram

F (pr ) F (pr )

F (M1 ) ← ’1’ F (M1 — M2 ) ’ ’ 2 F (M2 )

’’ ’’ ’

is always a product diagram.

Then F (point) = point, by the following argument:

u (pr ) F (point u— point) F (pr )T (point)

∼ wF

F

£

∼

1

=R

2

F (point)

RRR

f

=

RRR f

f 1 2

point

Each of f1 , f , and f2 determines each other uniquely, thus there is only one mapping

f1 : point ’ F (point), so the space F (point) is a single point.

We also require that F has the following two properties:

(1) The map on morphisms F : C ∞ (Rn , R) ’ C ∞ (F (Rn ), F (R)) is smooth,

where we regard C ∞ (F (Rn ), F (R)) as Fr¨licher space, see section (23).

o

Equivalently, the associated map C ∞ (Rn , R) — F (Rn ) ’ F (R) is smooth.

(2) There is a natural transformation π : F ’ Id such that for each M the

mapping πM : F (M ) ’ M is a ¬ber bundle, and for an open submanifold

U ‚ M the mapping F (incl) : F (U ) ’ F (M ) is a pullback.

31.14. C ∞ -algebras. An R-algebra is a commutative ring A with unit together

with a ring homomorphism R ’ A. Then every map p : Rn ’ Rm which is given

by an m-tuple of real polynomials (p1 , . . . , pm ) can be interpreted as a mapping

A(p) : An ’ Am in such a way that projections, composition, and identity are

preserved, by just evaluating each polynomial pi on an n-tuple (a1 , . . . , an ) ∈ An .

Compare with (17.1).

A C ∞ -algebra A is a real algebra in which we can moreover interpret all smooth

mappings f : Rn ’ Rm . There is a corresponding map A(f ) : An ’ Am , and

again projections, composition, and the identity mapping are preserved.

31.14

314 Chapter VI. In¬nite dimensional manifolds 31.15

More precisely, a C ∞ -algebra A is a product preserving functor from the category

C ∞ to the category of sets, where C ∞ has as objects all spaces Rn , n ≥ 0, and all

smooth mappings between them as arrows. Morphisms between C ∞ -algebras are

then natural transformations: they correspond to those algebra homomorphisms

which preserve the interpretation of smooth mappings.

Let us explain how one gets the algebra structure from this interpretation. Since A

is product preserving, we have A(point) = point. All the laws for a commutative

ring with unit can be formulated by commutative diagrams of mappings between

products of the ring and the point. We do this for the ring R and apply the product

preserving functor A to all these diagrams, so we get the laws for the commutative

ring A(R) with unit A(1) with the exception of A(0) = A(1) which we will check

later for the case A(R) = point. Addition is given by A(+) and multiplication by

A(m). For » ∈ R the mapping A(m» ) : A(R) ’ A(R) equals multiplication with

the element A(») ∈ A(R), since the following diagram commutes:

eeeeee

eeeeg )

A(R)

e

A(m

u

»

∼

=

w A(R) — A(R) Aw A(R)

Id —A(»)