_{1}

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The principal objective of this paper is to study the relationship between the old kingdom of differential geometry (the category of smooth manifolds) and its new kingdom (the category of functors on the category of Weil algebras to some smooth category). It is shown that the canonical embedding of the old kingdom into the new kingdom preserves Weil functors.

Roughly speaking, the path to axiomatic differential geometry is composed of five acts. Act One was Weil’s algebraic treatment of nilpotent infinitesimals in [

In our previous two papers [

The category of topological spaces and continuous mappins is by no means cartesian closed. In 1967 Steenrod [

At about the same time, an attempt to give a convenient category of smooth spaces began, and we have a few candidates at present. For a thorough study upon the relationship among these already proposed candidates, the reader is referred to [

Weil algebras were introduced by Weil himself [

Notation 1 We denote by W e i l R the category of Weil algebras over R .

Let us endow the category S m o o t h with Weil functors.

Proposition 2 Let W be an object in the category W e i l R with its finite presentation

W = C ∞ ( R n ) / I

as a smooth algebra in the sense of III.5 of [

f ~ W g ,

then

h ∘ f ~ W h ∘ g

Proof. Given ς ∈ S m o o t h ( Y , R ) , we have

ς ∘ ( h ∘ f ) − ς ∘ ( h ∘ g ) = ( ς ∘ h ) ∘ f − ( ς ∘ h ) ∘ g ∈ I

so that we have the desired result. ■

Corollary 3 We can naturally make T _ S m o o t h W a functor

T _ S m o o t h W : S m o o t h → S m o o t h

Proposition 4 Let W 1 and W 2 be objects in the category W e i l R with their finite presentations

W 1 = C ∞ ( R n ) / I

W 2 = C ∞ ( R m ) / J

as smooth algebras. Let

φ : W 1 → W 2

be a morphism in the category W e i l R , so that there exists a morphism

φ ← : R m → R n

in the category S m o o t h such that the composition with φ ← renders a mapping

C ∞ ( R n ) → C ∞ ( R m )

inducing φ . Let X ∈ S m o o t h and f , g ∈ S m o o t h ( R n , X ) . If

f ~ W 1 g

then

f ∘ φ ← ~ W 2 g ∘ φ ←

Proof. Given any ς ∈ S m o o t h ( Y , R ) , we have

ς ∘ ( f ∘ φ ← ) − ς ∘ ( g ∘ φ ← ) = ( ς ∘ f ) ∘ φ ← − ( ς ∘ g ) ∘ φ ← = ( ς ∘ f − ς ∘ g ) ∘ φ ← ∈ J

since ς ∘ f − ς ∘ g ∈ I , and the composition with φ ← : R n → R m maps I into J . ■

Corollary 5 The above procedure automatically induces a natural transfor- mation

α _ φ S m o o t h : T _ S m o o t h W 1 ⇒ T _ S m o o t h W 2

Notation 6 Given an object W in the category W e i l R , the restriction of the functor T _ S m o o t h W to the category M f is denoted by T _ M f W . Given a morphism φ : W 1 → W 2 in the category W e i l R , the corresponding restriction of α _ φ S m o o t h is denoted by α _ φ M f .

Remark 7 Weil functors

T _ M f W : M f → M f

are given distinct (but equivalent) definitions and studied thoroughly in Chapter VIII of [

It is well known that

Proposition 8 We have the following:

1) Given an object W in the category W e i l R , the functor

T _ M f W : M f → M f

abides by the following conditions:

・ T _ M f W preserves finite products.

・ The functor

T _ M f R : M f → M f

is the identity functor.

・ We have

T _ M f W 2 ∘ T _ M f W 1 = T _ M f W 1 ⊗ R W 2

for any objects W 1 and W 2 in the category W e i l R .

2) Given a morphism φ : W 1 → W 2 in the category W e i l R , α _ φ M f : T _ M f W 1 ⇒ T _ M f W 2 is a natural transformation subject to the following conditions:

・ We have

α _ id W M f = id T _ M f W

for any identity morphism id W : W → W in the category W e i l R .

・ We have

α _ ψ M f ⋅ α _ φ M f = α _ ψ ∘ φ M f

for any morphisms φ : W 1 → W 2 and ψ : W 2 → W 3 in the category W e i l R .

・ Given an object W and a morphism φ : W 1 → W 2 in the category W e i l R , the diagrams

T _ M f W ∘ T _ M f W 1 T _ M f W ∘ α _ φ M f ⇒ T _ M f W ∘ T _ M f W 2 ∥ ∥ T _ M f W 1 ⊗ R W ⇒ α _ φ ⊗ R id W M f T _ M f W 2 ⊗ R W

and

T _ M f W ⊗ R W 1 α _ id W ⊗ R φ M f ⇒ T _ M f W ⊗ R W 2 ∥ ∥ T _ M f W 1 ∘ T _ M f W ⇒ α _ φ M f ∘ T _ M f W T _ M f W 2 ∘ T _ M f W

are commutative.

3) Given an object W in the category W e i l R , we have

T _ W ( R ) = W

4) Given a morphism φ : W 1 → W 2 in the category W e i l R , we have

α _ φ ( R ) = φ

Notation 9 We introduce the following notation:

1) We denote by K S m o o t h the category whose objects are functors from the category W e i l R to the category S m o o t h and whose morphisms are their natural transformations.

2) Given an object W in the category W e i l R , we denote by

T S m o o t h W : K S m o o t h → K S m o o t h

the functor obtained as the composition with the functor

W ⊗ R ⋅ : W e i l R → W e i l R

so that for any object M in the category � S m o o t h , we have

T S m o o t h W ( M ) = M ( W ⊗ R ⋅ )

3) Given a morphism φ : W 1 → W 2 in the category W e i l R , we denote by

α φ S m o o t h : T S m o o t h W 1 ⇒ T S m o o t h W 2

the natural transformation such that, given an object W in the category W e i l R , the morphism

α φ S m o o t h ( M ) : T S m o o t h W 1 ( M ) → T S m o o t h W 2 ( M )

is

M ( φ ⊗ R id W ) : M ( W 1 ⊗ R W ) → M ( W 2 ⊗ R W )

4) We denote by ℝ S m o o t h the functor

R ⊗ R ⋅ : W e i l R → S m o o t h

We have established the following proposition in [

Proposition 10 We have the following:

1) K S m o o t h is a category which is complete and cartesian closed.

2) Given an object W in the category W e i l R , the functor

T S m o o t h W : K S m o o t h → K S m o o t h

abides by the following conditions:

・ T S m o o t h W preserves limits.

・ The functor

T S m o o t h R : K S m o o t h → K S m o o t h

is the identity functor.

・ We have

T S m o o t h W 1 ∘ T S m o o t h W 2 = T S m o o t h W 1 ⊗ R W 2

for any objects W 1 and W 2 in the category W e i l R .

・ We have

T S m o o t h W ( M N ) = T S m o o t h W ( M ) T S m o o t h W ( N )

for any objects M and N in the category K S m o o t h .

3) Given a morphism φ : W 1 → W 2 in the category W e i l R ,

α φ : T S m o o t h W 1 ⇒ T S m o o t h W 2

is a natural transformation subject to the following conditions:

・ We have

α id W S m o o t h = id T W

for any identity morphism id W : W → W in the category W e i l R .

・ We have

α ψ S m o o t h ∘ α φ S m o o t h = α ψ ∘ φ S m o o t h

for any morphisms φ : W 1 → W 2 and ψ : W 2 → W 3 in the category W e i l R .

・ Given objects M and N in the category K S m o o t h , the diagram

T S m o o t h W 1 ( M ) T S m o o t h W 1 ( N ) ∥ T S m o o t h W 1 ( M N ) α φ S m o o t h ( M ) T S m o o t h W 1 ( N ) → T S m o o t h W 2 ( M ) T S m o o t h W 1 ( N ) α φ S m o o t h ( M N ) ↓ ↗ T S m o o t h W 2 ( M ) α φ S m o o t h ( N ) T S m o o t h W 2 ( M N ) ∥ T S m o o t h W 2 ( M ) T S m o o t h W 2 ( N )

is commutative.

・ Given an object W and a morphism φ : W 1 → W 2 in the category W e i l R , the diagrams

T S m o o t h W ∘ T S m o o t h W 1 T S m o o t h W ∘ α φ S m o o t h ⇒ T S m o o t h W ∘ T S m o o t h W 2 ∥ ∥ T S m o o t h W ⊗ R W 1 ⇒ α id W ⊗ R φ S m o o t h T S m o o t h W ⊗ R W 2

and

T S m o o t h W 1 ⊗ R W α φ ⊗ R id W S m o o t h ⇒ T S m o o t h W 2 ⊗ R W ∥ ∥ T S m o o t h W 1 ∘ T S m o o t h W ⇒ α φ S m o o t h ∘ T S m o o t h W T S m o o t h W 2 ∘ T S m o o t h W

are commutative.

4) Given an object W in the category W e i l R , we have

T S m o o t h W ( ℝ S m o o t h ) = ℝ S m o o t h ⊗ R W

5) Given a morphism φ : W 1 → W 2 in the category W e i l R , we have

α φ S m o o t h ( ℝ S m o o t h ) = ℝ S m o o t h ⊗ R φ

Notation 11 We write

i S m o o t h : S m o o t h → K S m o o t h

for the functor

i S m o o t h ( M _ ) : W ∈ Obj W e i l R ↦ T _ S m o o t h W M _ ∈ Obj K S m o o t h

i S m o o t h ( M _ ) : φ ∈ Mor W e i l R ↦ α _ φ S m o o t h ( M _ ) ∈ Mor K S m o o t h

provided with an object object M _ in the category S m o o t h , and

i S m o o t h ( f ) ( W ) = T _ S m o o t h W f : T _ S m o o t h W M _ 1 → T _ S m o o t h W M _ 2

provided with a morphism f : M _ 1 → M _ 2 in the category S m o o t h and an object W in the category W e i l R . The restriction of i S m o o t h to the subcategory M f is denoted by

i M f : M f → K S m o o t h

Theorem 12 Given an object W in the category W e i l R , the diagram

M f i M f → K S m o o t h T _ M f W ↓ ↓ T S m o o t h W M f i M f → K S m o o t h

is commutative.

Proof. Given an object M _ in the category M f , we have

( T S m o o t h W ∘ i M f ) ( M _ ) = i M f ( M _ ) ∘ ( W ⊗ R ⋅ ) = T _ M f W ⊗ R ⋅ M _ = T _ M f ⋅ ( T _ M f W M _ ) = i M f ( T _ M f W M _ ) = ( i M f ∘ T _ M f W ) ( M _ )

Given a morphism

f _ : M _ 1 → M _ 2

in the category M f , we have

( T S m o o t h W ∘ i M f ) ( f _ ) = i M f ( f _ ) ∘ ( W ⊗ R ⋅ ) = T _ M f W ⊗ R ⋅ f _ = T _ M f ⋅ ( T _ M f W f _ ) = i M f ( T _ M f W f _ ) = ( i M f ∘ T _ M f W ) ( f _ )

■

Theorem 13 Given a morphism φ : W 1 → W 2 in the category W e i l R , the diagram

i M f ∘ T _ M f W 1 i M f ∘ α _ φ M f ⇒ i M f ∘ T _ M f W 2 ∥ ∥ T S m o o t h W 1 ∘ i M f ⇒ α φ S m o o t h ∘ i M f T S m o o t h W 2 ∘ i M f

is commutative.

Proof. Given an object M _ in the category M f , we have

( i M f ∘ α _ φ M f ) ( M _ ) = i M f ( α _ ϕ M f ( M _ ) ) = T _ M f ⋅ ( α _ φ M f ( M _ ) ) = α _ φ M f ( T _ M f ⋅ ( M _ ) ) = α φ S m o o t h ( i M f ( M _ ) ) = ( α φ S m o o t h ∘ i M f ) ( M _ )

■

Definition 14 Given a category K endowed with a functor T W : K → K for each object W in the category W e i l R and a natural transformation α φ : T W 1 ⇒ T W 2 for each morphism φ : W 1 → W 2 in the category W e i l R , an object M in the category K is called microlinear if any limit diagram D in the category W e i l R makes the diagram T D M a limit diagram in the category K , where the diagram T D M consists of objects

T W M

for any object W in the diagram D and morphisms

α φ ( M ) : T W 1 M → T W 2 M

for any morphism φ : W 1 → W 2 in the diagram D .

Proposition 15 Every manifold as an object in the category S m o o t h is microlinear.

Proof. This can be established in three steps.

1) The first step is to show that R n is micorlinear for any natural number n , which follows easily from

T _ M f W R n = T _ S m o o t h W R n = W n

and

α _ φ M f ( R n ) = α _ φ S m o o t h ( R n ) = φ n

for any morphism φ : W 1 → W 2 in the category W e i l R .

2) The second step is to show that any open subset of R n is microlinear in homage to the result in the first step.

3) The third step is to establish the desired result by remarking that a smooth manifold is no other than an overlapping family of open subsets of R n .

The details can safely be left to the reader. ■

Theorem 16 The embedding

i S m o o t h : S m o o t h → K S m o o t h

maps smooth manifolds to microlinear objects in the category K S m o o t h .

Proof. Let D be a limit diagram in the category W e i l R . Let M _ be a smooth manifold in the category S m o o t h . Given an object W in the category W e i l R , the diagram ( T S m o o t h D ( i S m o o t h ( M _ ) ) ) ( W ) , which consists of objects

T _ M f W ′ ⊗ R W M _ = T _ M f W ⊗ R W ′ M _ = T _ M f W ′ ( T _ M f W M _ )

for any object W ′ in the category W e i l R and morphisms

T _ M f W 1 ⊗ R W M _ α _ φ ⊗ R id W M f ( M _ ) → T _ M f W 2 ⊗ R W M _ ∥ ∥ T _ M f W ⊗ R W 1 M _ T _ M f W ⊗ R W 2 M _ ∥ ∥ T _ M f W 1 ( T _ M f W M _ ) α _ φ M f ( T _ M f W M _ ) ← T _ M f W 2 ( T _ M f W M _ )

for any morphism φ : W 1 → W 2 in the category W e i l R , is a limit diagram in the category S m o o t h , because T _ M f W M _ is a microlinear object in the category S m o o t h in homage to Proposition 15. Therefore the diagram T S m o o t h D ( i S m o o t h ( M _ ) ) is a limit diagram in the category K S m o o t h thanks to The- orem 7.5.2 and Remarks 7.5.3 in [

Definition 17 A cone D in the category S m o o t h is called a transversal limit diagram if the diagram T S m o o t h W D is a limit diagram for any object W in the category W e i l R . In this case, the vertex of the cone is called a transversal limit.

It is easy to see that

Proposition 18 A transversal limit diagram is a limit diagram, so that a transversal limit is a limit.

Proof. Since

T S m o o t h R D = D

for any cone D in the category S m o o t h , the desired conclusion follows immediately. ■

What makes the notion of a transversal limit significant is the following theorem.

Theorem 19 The embedding

i S m o o t h : S m o o t h → K S m o o t h

maps transversal limit diagrams in the category S m o o t h to limit diagrams in the category K S m o o t h .

Proof. This follows directly in homage to Theorem 7.5.2 and Remarks 7.5.3 in [

Now we are going to show that the above embedding preserves vertical Weil functors, as far as fibered manifolds are concerned. Let us recall the definition of vertical Weil functor given in [

Definition 20 Let us suppose that we are given a left exact category K en- dowed with a functor T W : K → K for each object W in the category W e i l R and a natural transformation α φ : T W 1 ⇒ T W 2 for each morphism φ : W 1 → W 2 in the category W e i l R . Given a morphism π : E → M in the category K , its vertical Weil functor T → W ( π ) is defined to be the equalizer of the parallel morphisms

T W ( E ) T W ( π ) → T W ( π ) → T W ( M ) α W → R ( M ) → T R ( M ) α R → W ( M ) → T W ( M )

Lemma 21 The equalizer of the above diagram in the category S m o o t h is transversal, as far as π : E → M is a fibered manifold in the sense of 2.4 in [

Proof. The proof is similar to that in Proposition 15.

1) In case that E = R m + n , M = R m , and π is the canonical projection, the equalizer is the canonical injection

R m × W n → W m + n = T W ( E )

and it is easy to see that it is transversal.

2) Then we prove the statement in case that E = U × V , M = U , and π is the canonical projection, where U is an open subset of R m , and V is an open subset of R n .

3) The desired statement in full generality follows from the above case by remarking that the fiber bundle π : E → M is no other than an overlapping family of such special cases.

The details can safely be left to the reader. ■

Theorem 22 Given an object W in the category W e i l R and a fibered manifold π : E → M in the category S m o o t h , we have

i S m o o t h ( T _ → S m o o t h W ( π ) ) = T → S m o o t h W ( i S m o o t h ( π ) )

Proof. In homage to Theorems 12 and 13, the functor i S m o o t h maps the diagram

T _ S m o o t h W ( E ) T _ S m o o t h W ( π ) ← T _ S m o o t h W ( π ) → T _ S m o o t h W ( M ) α _ W → R S m o o t h ( M ) → T _ S m o o t h R ( M ) α _ R → W S m o o t h ( M ) → T _ S m o o t h W ( M )

in the category S m o o t h into the diagram

T S m o o t h W ( i S m o o t h ( E ) ) T S m o o t h W ( i S m o o t h ( π ) ) ← T S m o o t h W ( i S m o o t h ( π ) ) → T S m o o t h W ( i S m o o t h ( M ) ) α W → R S m o o t h ( i S m o o t h ( M ) ) → T S m o o t h R ( i S m o o t h ( M ) ) α R → W S m o o t h ( i S m o o t h ( M ) ) → T S m o o t h W ( i S m o o t h ( M ) )

in the category K S m o o t h . Since the equalizer of the former diagram is transversal by Lemma 21, it is preserved by the functor i S m o o t h by Theorem 19, so that the desired result follows. ■

Corollary 23 Given a morphism φ : W 1 → W 2 in the category W e i l R and a fibered manifold π : E → M in the category S m o o t h , we have

i S m o o t h ( α → _ φ S m o o t h ( π ) ) = α φ S m o o t h ( ( i S m o o t h ( π ) ) ) .

Nishimura, H. (2017) Axiomatic Differential Geometry III-3-Its Landscape-Chapter 3: The Old Kingdom of Differential Geometers. Applied Mathematics, 8, 835-845. https://doi.org/10.4236/am.2017.86065