Axiomatic Differential Geometry III-3-Its Landscape-Chapter 3: The Old Kingdom of Differential Geometers ()
1. 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 nil- potent 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 re- incarnation 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 era- dicated as genuine hassle and replaced by so-called
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
of functors on the category
of Weil algebras to the smooth category
(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
of smooth manifolds and smooth mappings and our new kingdom
as well as that between
and
in this paper.
2. 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 pro- posed 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
, which is required to be complete and cartesian closed at least, containing the category
of smooth manifolds as a full subcategory. Obviously the category
contains the set
of real num- bers.
3. 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] .
Notation 1 We denote by
the category of Weil algebras over
.
Let us endow the category
with Weil functors.
Proposition 2 Let W be an object in the category
with its finite presentation
as a smooth algebra in the sense of III.5 of [5] . Let
,
, and
. If
then
Proof. Given
, we have
so that we have the desired result. ■
Corollary 3 We can naturally make
a functor
Proposition 4 Let
and
be objects in the category
with their finite presentations
as smooth algebras. Let
be a morphism in the category
, so that there exists a morphism
in the category
such that the composition with
renders a mapping
inducing
. Let
and
. If
then
Proof. Given any
, we have
since
, and the composition with
maps
into
. ■
Corollary 5 The above procedure automatically induces a natural transfor- mation
Notation 6 Given an object W in the category
, the restriction of the functor
to the category
is denoted by
. Given a morphism
in the category
, the corresponding restriction of
is denoted by
.
Remark 7 Weil functors
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 infinite- dimensional case.
It is well known that
Proposition 8 We have the following:
1) Given an object
in the category
, the functor
abides by the following conditions:
・
preserves finite products.
・ The functor
is the identity functor.
・ We have
for any objects
and
in the category
.
2) Given a morphism
in the category
,
is a natural transformation subject to the following conditions:
・ We have
for any identity morphism
in the category
.
・ We have
for any morphisms
and
in the category
.
・ Given an object
and a morphism
in the category
, the diagrams
and
are commutative.
3) Given an object
in the category
, we have
4) Given a morphism
in the category
, we have
4. A New Kingdom for Differential Geometers
Notation 9 We introduce the following notation:
1) We denote by
the category whose objects are functors from the category
to the category
and whose morphisms are their natural transformations.
2) Given an object
in the category
, we denote by
the functor obtained as the composition with the functor
so that for any object
in the category
, we have
3) Given a morphism
in the category
, we denote by
the natural transformation such that, given an object
in the category
, the morphism
is
4) We denote by
the functor
We have established the following proposition in [14] and [15] .
Proposition 10 We have the following:
1)
is a category which is complete and cartesian closed.
2) Given an object
in the category
, the functor
abides by the following conditions:
・
preserves limits.
・ The functor
is the identity functor.
・ We have
for any objects
and
in the category
.
・ We have
for any objects
and
in the category
.
3) Given a morphism
in the category
,
is a natural transformation subject to the following conditions:
・ We have
for any identity morphism
in the category
.
・ We have
for any morphisms
and
in the category
.
・ Given objects
and
in the category
, the diagram
is commutative.
・ Given an object
and a morphism
in the category
, the diagrams
and
are commutative.
4) Given an object
in the category
, we have
5) Given a morphism
in the category
, we have
5. From the Old Kingdom to the New One
Notation 11 We write
for the functor
provided with an object object
in the category
, and
provided with a morphism
in the category
and an object
in the category
. The restriction of
to the subcategory
is denoted by
Theorem 12 Given an object
in the category
, the diagram
is commutative.
Proof. Given an object
in the category
, we have
Given a morphism
in the category
, we have
■
Theorem 13 Given a morphism
in the category
, the diagram
is commutative.
Proof. Given an object
in the category
, we have
■
6. Microlinearity
Definition 14 Given a category
endowed with a functor
for each object
in the category
and a natural transformation
for each morphism
in the category
, an object
in the category
is called microlinear if any limit diagram
in the category
makes the diagram
a limit diagram in the category
, where the diagram
consists of objects
for any object
in the diagram
and morphisms
for any morphism
in the diagram
.
Proposition 15 Every manifold as an object in the category
is microlinear.
Proof. This can be established in three steps.
1) The first step is to show that
is micorlinear for any natural number
, which follows easily from
and
for any morphism
in the category
.
2) The second step is to show that any open subset of
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
.
The details can safely be left to the reader. ■
Theorem 16 The embedding
maps smooth manifolds to microlinear objects in the category
.
Proof. Let
be a limit diagram in the category
. Let
be a smooth manifold in the category
. Given an object
in the category
, the diagram
, which consists of objects
for any object
in the category
and morphisms
for any morphism
in the category
, is a limit diagram in the category
, because
is a microlinear object in the category
in homage to Proposition 15. Therefore the diagram
is a limit diagram in the category
thanks to The- orem 7.5.2 and Remarks 7.5.3 in [16] . ■
7. Transversal Limits
Definition 17 A cone
in the category
is called a transversal limit diagram if the diagram
is a limit diagram for any object
in the category
. 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
for any cone
in the category
, the desired conclusion follows immediately. ■
What makes the notion of a transversal limit significant is the following theorem.
Theorem 19 The embedding
maps transversal limit diagrams in the category
to limit diagrams in the category
.
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
en- dowed with a functor
for each object
in the category
and a natural transformation
for each morphism
in the category
. Given a morphism
in the category
, its vertical Weil functor
is defined to be the equalizer of the parallel morphisms
Lemma 21 The equalizer of the above diagram in the category
is transversal, as far as
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
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
,
, and
is the canonical projection, where
is an open subset of
, and
is an open subset of
.
3) The desired statement in full generality follows from the above case by remarking that the fiber bundle
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
in the category
and a fibered manifold
in the category
, we have
Proof. In homage to Theorems 12 and 13, the functor
maps the diagram
in the category
into the diagram
in the category
. Since the equalizer of the former diagram is transversal by Lemma 21, it is preserved by the functor
by Theorem 19, so that the desired result follows. ■
Corollary 23 Given a morphism
in the category
and a fibered manifold
in the category
, we have