Axiomatic Differential Geometry III-3-Its Landscape-Chapter 3: The Old Kingdom of Differential Geometers

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.


Introduction
Roughly speaking, the path to axiomatic differential geometry is composed of five acts.Act One was Weil's algebraic treatment of nilpotent infinitesimals in [1], namely, the introduction of so-called Weil algebras.It showed that nilpotent infinitesimals could be grasped algebraically.While nilpotent infinitesimals are imaginary entities, Weil algebras are real ones.Act Two began almost at the same time with Steenrod's introduction of convenient categories of topological spaces (cf.[2]), consisting of a string of proposals of convenient categories of smooth spaces.Its principal slogan was that the category of differential geometry should be (locally) cartesian closed.The string was panoramized by [3] as well as [4].Act Three was so-called synthetic differential geometry, in which synthetic methods as well as nilpotent infinitesimals play a predominant role.It demonstrated amply that differential geometry could be made axiomatic in the same sense that Euclidean geometry is so, though it should resort to reincarnation of nilpotent infinitesimals.In any case, synthetic differential geo-meters were forced to fabricate their own world and call well-adapted models, where they could indulge in their favorite nilpotent infinitesimals incessantly.Their unblushing use of moribund nilpotent infinitesimals alienated most of orthodox mathematicians, because nilpotent infinitesimals were almost eradicated as genuine hassle and replaced by so-called ε δ − arguments in the 19th century.The reader is referred to [5] and [6] for good treatises on synthetic differential geometry.Act Four was the introduction of Weil functors and their thorough study by what was called the Czech school of differential geometers in the 1980's, for which the reader is referred to Chapter VIII of [7] and §31 of [8].Weil functors, which are a direct generalization of the tangent bundle functor, open a truly realistic path of axiomatizing differential geometry without nilpotent infinitesimals.Then Act Five is our axiomatic differential geometry, which is tremendously indebted to all previous four acts.For axiomatic differential geometry, the reader is referred to [9]- [15].
In our previous two papers [14] and [15], we have developed model theory for axiomatic differential geometry, in which the category  Smooth of functors on the category R Weil of Weil algebras to the smooth category Smooth (by which we mean any proposed or possible convenient category of smooth spaces) and their natural transformations play a crucial role.We will study the relationship between the category Mf of smooth manifolds and smooth mappings and our new kingdom  Smooth as well as that between Smooth and  Smooth in this paper.

Convenient Categories of Smooth Spaces
The category of topological spaces and continuous mappins is by no means cartesian closed.In 1967 Steenrod [2] popularized the idea of convenient category by announcing that the category of compactly generated spaces and continuous mappings renders a good setting for algebraic topology.The proposed category is cartesian closed, complete and cocomplete, and contains all CW complexes.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 [3], in which he or she will find, by way of example, that the category of Frölicher spaces is a full subcategory of that of Souriau spaces, and the category of Souriau spaces is in turn a full subcategory of that of Chen spaces.
We have no intention to discuss which is the best convenient category of smooth spaces here, but we note in passing that both the category of Souriau spaces and that of Chen spaces are locally cartesian closed, while that of Frölicher spaces is not.At present we content ourselves with denoting some of such convenient categories of smooth spaces by Smooth , which is required to be complete and cartesian closed at least, containing the category Mf of smooth manifolds as a full subcategory.Obviously the category Mf contains the set R of real num- bers.

Weil Functors
Weil algebras were introduced by Weil himself [1].For a thorough treatment of Weil algebras as smooth algebras, the reader is referred to III.5 in [5].
as a smooth algebra in the sense of III.5 of [5].Let , :W W ϕ → be a morphism in the category R Weil , so that there exists a morphism in the category Smooth such that the composition with ϕ  renders a mapping , and the composition with : The above procedure automatically induces a natural transformation are given distinct (but equivalent) definitions and studied thoroughly in Chapter VIII of [7] in the finite-dimensional case and §31 of [8] in the infinitedimensional case.
It is well known that Proposition 8 We have the following: 1) Given an object W in the category R Weil , the functor abides by the following conditions:

T Mf Mf
is the identity functor.
• We have : T T is a natural transformation subject to the following conditions: for any identity morphism id : for any morphisms 3) Given an object W in the category R Weil , we have ( )

A New Kingdom for Differential Geometers
Notation 9 We introduce the following notation: 1) We denote by  Smooth the category whose objects are functors from the category R Weil to the category Smooth and whose morphisms are their natural transformations.
2) Given an object W in the category R Weil , we denote by T the functor obtained as the composition with the functor

Weil Weil
so that for any object M in the category Smooth  , we have 3) Given a morphism : id : We denote by  Smooth the functor

R Weil Smooth
We have established the following proposition in [14] and [15].
Proposition 10 We have the following: 1)  Smooth is a category which is complete and cartesian closed.
2) Given an object W in the category R Weil , the functor

T
abides by the following conditions:

T
is the identity functor.
• We have for any objects M and N in the category  Smooth .
3) Given a morphism : T is a natural transformation subject to the following conditions: for any identity morphism id : for any morphisms      )

5.
From the Old Kingdom to the New One Notation 11 We write

Smooth
for the functor

Weil
provided with an object object M in the category Smooth , and :

T T T
provided with a morphism Proof.Given an object M in the category Mf , we have

T T T T T T
Given a morphism

T T T T T T ■
Theorem 13 Given a morphism Proof.Given an object M in the Mf , we have

Microlinearity
Definition 14 Given a category  endowed with a functor :  Proof.This can be established in three steps.1) The first step is to show that n R is micorlinear for any natural number n , which follows easily from for any morphism 2) The second step is to show that any open subset of n R 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 n R .
The details can safely be left to the reader.■ Theorem The embedding

Smooth
maps smooth manifolds to microlinear objects in the category  Smooth .
Proof.Let  be a limit diagram in the category R Weil .Let M be a smooth manifold in the category Smooth .Given an object W in the for any object W ′ in the category R Weil and morphisms ( )

T T T T T T T T T
for any morphism

Transversal Limits
Definition 17 A cone  in the category Smooth is called a transversal limit diagram if the diagram W  Smooth T is a limit diagram for any object W in the category R Weil .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.This follows directly in homage to Theorem 7.5.2 and Remarks 7.5.3 in [16].■ 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 [9].
Definition 20 Let us suppose that we are given a left exact category  endowed with a functor : for each object W in the category R Weil and a natural transformation Lemma 21 The equalizer of the above diagram in the category Smooth is transversal, as far as : E M π → is a fibered manifold in the sense of 2.4 in [7].
Proof.The proof is similar to that in Proposition 15. 1) In case that , and π is the canonical projection, the equalizer is the canonical injection ( ) 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 m R , and V is an open subset of n R .
3) The desired statement in full generality follows from the above case by remarking that the fiber bundle : E M π → 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

Notation 1
We denote by R Weil the category of Weil algebras over R .Let us endow the category Smooth with Weil functors.Proposition 2 Let W be an object in the category R Weil with its finite presentation we have the desired result.■ Corollary 3 We can naturally make W

•
Given an object W and a morphism

1
natural transformation such that, given an object W in the category R Weil , the morphism

•
Given objects M and N in the category  Smooth , the diagram

•
Given an object W and a morphism

4 )
Given an object W in the category R Weil , we have

(
Smooth and an object W in the category R Weil .The restriction of i Smooth to the subcategory Mf is denoted by : the category Mf , we have

R
Weil and a natural transformation object M in the category  is called microlinear if any limit diagram  in the category R Weil makes the diagram M  T a limit diagram in the category  , where the diagram M  T consists of objects W M T for any object W in the diagram  and morphisms  .Proposition 15 Every manifold as an object in the category Smooth is microlinear.
object in the category Smooth in homage to Proposition 15.Therefore the diagram diagram in the category  Smooth thanks to The- orem 7.5.2 and Remarks 7.5.3in[16].■ cone  in the category Smooth , the desired conclusion follows immediately.■ What makes the notion of a transversal limit significant is the following theorem.Theorem 19 The embedding : i →  Smooth Smooth Smooth maps transversal limit diagrams in the category Smooth to limit diagrams in the category  Smooth .
be the equalizer of the parallel morphisms

(
homage to Theorems 12 and 13, the functor i Smooth maps the diagram  Smooth .Since the equalizer of the former diagram is transversal by Lemma 21, it is preserved by the functor i Smooth by Theorem 19, so that the desired result follows.■