Weil Diffeology I: Classical Differential Geometry

TL;DR

This paper develops a category-theoretic framework for classical differential geometry using the Weil topos and Dubuc functor, aiming to formalize tangent spaces and modules in an axiomatic setting.

Contribution

It introduces a category-theoretical axiomatization of the Weil topos with the Dubuc functor for classical differential geometry.

Findings

Axiomatization of the Weil topos with Dubuc functor

Formalization of tangent spaces as modules over the canonical ring

Proof of tangent space properties in the axiomatic framework

Abstract

Topos theory is a category-theoretic axiomatization of set theory. Model categories are a category-theoretical framework for abstract homotopy theory. They are complete and cocomplete categories endowed with three classes of morphisms (called fibrations, cofibrations and equivalences) satisfying certain axioms. Functors from the category of Weil algebras to the category of sets are called Weil spaces by Wolfgang Bertram and form the Weil topos} after Eduardo J. Dubuc. The Weil topos is endowed intrinsically with the Dubuc functor, a functor from a larger category containing Weil algebras to the Weil topos standing for the incarnation of each algebraic entity of the category in the Weil topos. The Weil functor and the canonical ring object are to be defined in terms of the Dubuc functor. The principal object in this paper is to present a category-theoretical axiomatization of the Weil topos with the Dubuc functor intended to be an adequate framework for axiomatic classical differential geometry and hopefully comparable with model categories. We will give an appropriate formulation and a rather complete proof of a generalization of the familiar and desired fact that the tangent space of a microlinear Weil space is a module over the canonical ring object.

Equations

$$F\left(\left(\_\right)\otimes W\right):\mathfrak{W}\to \mathbf{Sets}$$

$${\left(F^{W_1}\right)}^{W_2}=F^{W_1\otimes W_2}$$

$$\mathbf{y}:{\mathfrak{W}}^{\mathrm{op}}\to \mathbf{Weil}$$

$$F\left(\_\right)\cong {\mathrm{Hom}}_{\mathbf{Weil}}\left(\mathbf{y}\left(\_\right),F\right)$$

$$\tilde{\mathbf{y}}:{\tilde{\mathfrak{W}}}^{\mathrm{op}}\to \mathbf{Weil}$$

$$\tilde{\mathbf{y}}\left(A\right)={\mathrm{Hom}}_{\mathbf{R}-\mathbf{Alg}}\left(A,\_\right)$$

$$\mathbf{y}{W}_1\times \mathbf{y}{W}_2\cong \mathbf{y}\left(W_1\otimes W_2\right)$$

$${\mathrm{Hom}}_{\mathbf{Weil}}\left(\mathbf{y}\_\times G,F\right)$$

$$F^W\cong F^{\mathbf{y}W}$$

$$F^{\mathbf{y}W}$$

$$=\mathrm{Hom}\left(\mathbf{y}\_\times \mathbf{y}W,F\right)$$

$$\cong \mathrm{Hom}\left(\mathbf{y}\left(\_\otimes W\right),F\right)$$

$$\cong F\left(\_\otimes W\right)$$

$$=F^W$$

$${\left(F^G\right)}^W\cong {\left(F^W\right)}^G$$

$${\left(F^G\right)}^W$$

$$\cong {\left(F^G\right)}^{\mathbf{y}W}$$

$$\cong {\left(F^{\mathbf{y}W}\right)}^G$$

$$\cong {\left(F^W\right)}^G$$

$$\mathbb{R}=\tilde{\mathbf{y}}\left(\mathbf{R}\left[X\right]\right)$$

$${\mathbb{R}}^{\mathbf{y}\left(\_\right)},\mathbb{R}\otimes \left(\_\right):\mathfrak{W}\to \mathbf{R}-\mathbf{Alg}\left(\mathbf{Weil}\right)$$

$${\mathbb{R}}^{\mathbf{y}W}\left(W^{\prime }\right)$$

$$\cong {\mathbb{R}}^W\left(W^{\prime }\right)$$

$$=W^{\prime }\otimes W$$

$$F\left(W\otimes \mathcal{D}\right)$$

$$\mathbf{D}\left(\mathbf{R}\right)=1$$

$$\mathcal{K}$$

$$\mathbf{D}$$

$$i_{\tilde{\mathfrak{W}},C^{\infty }-\mathbf{Alg}}\left(\mathbf{R}\left[X_1,...,X_n\right]\otimes W\right)=C^{\infty }\left({\mathbf{R}}^n\right)\otimes W$$

$$\mathbf{D}=\mathbf{y}\circ i_{\tilde{\mathfrak{W}},C^{\infty }-\mathbf{Alg}}$$

$$\mathbf{T}\left(\left(\_\right),\left(\cdot \right)\right)\cong {\left(\cdot \right)}^{\mathbf{D}\left(\_\right)}$$

$$\mathbf{T}\left(\mathbf{R},\left(\_\right)\right)\cong \left(\_\right)$$

$$\mathbf{T}\left(\left({\cdot }_2\right),\mathbf{T}\left(\left({\cdot }_1\right),\left(\_\right)\right)\right)\cong \mathbf{T}\left(\left({\cdot }_1\right)\otimes \left({\cdot }_2\right),\left(\_\right)\right)$$

$$\mathbf{T}\left(\left(\_\right),{\left({\cdot }_1\right)}^{\left({\cdot }_2\right)}\right)\cong \mathbf{T}{\left(\left(\_\right),\left({\cdot }_1\right)\right)}^{\left({\cdot }_2\right)}$$

$$\mathbf{T}\left(\left(\_\right),{\left({\cdot }_1\right)}^{\left({\cdot }_2\right)}\right)$$

Full text

1 Introduction

Differential geometry usually exploits not only the techniques of differentiation but also those of integration. In this paper we would like to use the term ”differential geometry” in its literal sense, that is, genuinely differential geometry, which is vast enough as to encompass a large portion of the theory of connections and the core of the theory of Lie groups. Now we know well that there is a horribly deep and overwhelmingly gigantic valley between differential calculus of the 17th and 18th centuries (that is to say, that of the good old days of Newton, Leibniz, Lagrange, Laplace, Euler and so on) and that of our modern age since the 19th century when Angustin Louis Cauchy was active. The former exquisitely resorts to nilpotent infinitesimals, while the latter grasps differentiation in terms of limits by using so-called $\varepsilon-\delta$ arguments formally. Differential geometry based on the latter style of differentiation is generally called smootheology, while we propose that differential geometry based on the former style of differentiation might be called Weilology.

As is well known, the category of topological spaces and continuous mappins is not cartesian closed. The classical example of a convenient category of topological spaces for working topologisits was suggested by Norman Steenrod [29] in the middle of the 1960s, namely, the category of compactly generated spaces. Now the category of finite-dimensional smooth manifolds and smooth mappings is not cartesian closed, either. Convenient categories for smootheology have been proposed by several authors in several corresponding forms. Among them Souriau’s [27] approach based upon the category $\mathfrak{O}$ of open subsets $O$’s of $\mathbf{R}^{n}$’s and smooth mappings between them has developed into a galactic volume of diffeology, for which the reader is referred to [9]. A diffeological space is a set $X$ endowed with a subset $\mathcal{D}\left(O\right)\subseteq X^{O}$ for each $O\in\mathfrak{O}$ such that, for any morphism $f:O\rightarrow O^{\prime}$ in $\mathfrak{O}$ and any $\gamma\in\mathcal{D}\left(O^{\prime}\right)$, we have $\gamma\circ f\in\mathcal{D}\left(O\right)$. A diffeological map between diffeological spaces $\left(X,\mathcal{D}\right)$ and $\left(X,\mathcal{D}^{\prime}\right)$ is a mapping $f:X\rightarrow X^{\prime}$ such that, for any $O\in\mathfrak{O}$ and any $\gamma\in\mathcal{D}\left(O\right)$, we have $f\circ\gamma\in\mathcal{D}^{\prime}\left(O\right)$.

Roughly speaking, there are two approaches to geometry in representing spaces, namely, contravariant (functional) and covariant (parameterized) ones, for which the reader is referred, e.g., to Chapter 3 of [26] as well as [24] and [25]. Diffeology finds itself in the covariant realm. The contravariant approach boils down spaces to their function algebras. We are now accustomed to admitting all algebras to stand for abstract spaces in some way or other, whatever they may be. This is a long tradition of algebraic geometry since as early as Alexander Grothendieck. Now we are ready to acknowledge any functor $\mathfrak{O}^{\mathrm{op}}\rightarrow\mathbf{Sets}$ as an *abstract *diffeological space. Then it is pleasant to enjoy

Theorem 1

The category of abstract diffeological spaces and natural transformations between them is a topos.

Turning to Weilology, a space should be represented as a functor $\mathbf{Inf}^{\mathrm{op}}\rightarrow\mathbf{Sets}$, where $\mathbf{Inf}$ stands for the category of nilpotent infinitesimal spaces. Since our creed tells us that the category $\mathbf{Inf}^{\mathrm{op}}$ is equivalent to $\mathfrak{W}$, a space should be no other than a functor $\mathfrak{W}\rightarrow\mathbf{Sets}$, for which Wolfgang Bertram [6] has coined the term ”Weil space”. To be sure, we have

Theorem 2

The category of Weil spaces and natural transformations between them is a topos.

2 Cahiers Algebras

Unless stated to the contrary, our base field is assumed to be $\mathbf{R}$ (real numbers) throughout the paper, so that we will often say ”Weil algebra” simply in place of Weil $\mathbf{R}$-algebra”. For the exact definition of a Weil algebra, the reader is referred to §I.16 of [10].

Notation 3

We denote by $\mathfrak{W}$ the category of Weil algebras.

Remark 4

$\mathbf{R}$* is itself a Weil algebra, and it is an initial object in the category $\mathfrak{W}$.*

Definition 5

An $\mathbf{R}$-algebra isomorphic to an $\mathbf{R}$-algebra of the form $\mathbf{R}\left[X_{1},...,X_{n}\right]\otimes W$ with \mathbf{R}\left[X_{1},...,X_{n}\right]\being the polynomial algebra over \mathbf{R}\in indeterminates $X_{1},...,X_{n}$ (possibly $n=0$, when the definition degenerates to Weil algebras) and $W$ being a Weil algebra is called a cahiers algebra.

Remark 6

This definition of a cahiers algebra is reminiscent of that in the definition of Cahiers topos, where we consider a product of a Cartesian space $\mathbf{R}^{n}$ and a formal dual of a Weil algebra.

Notation 7

We denote by $\widetilde{\mathfrak{W}}$ the category of cahiers algebras.

Remark 8

The category $\mathfrak{W}$ is a full subcategory of the category $\widetilde{\mathfrak{W}}$. Both are closed under the tensor product  $\otimes$.

Notation 9

We will use such a self-explanatory notation as $Z\rightarrow X/\left(X^{2}\right)$ or $X/\left(X^{2}\right)\leftarrow Z$ for the morphism $\mathbf{R}\left[Z\right]\rightarrow\mathbf{R}\left[X\right]/\left(X^{2}\right)$ assigning $X$ modulo $\left(X^{2}\right)$ to $Z$.

3 Weil Spaces

Definition 10

A Weil space is simply a functor $F$ from the category $\mathfrak{W}$ of Weil algebras to the category $\mathbf{Sets}$ of sets. A Weil morphism from a Weil space $F$ to another Weil space $G$ is simply a natural transformation from the functor $F$ to the functor $G$.

Remark 11

The term ”Weil space” has been coined in [6].

Example 12

The Weil prolongation of a ”manifold” in its broadest sense (cf. [4]) by a Weil algebra was fully discussed by Bertram and Souvay, for which the reader is cordially referred to [5]. We are happy to know that any manifold naturally gives rise to its associated Weil space, which can be regarded as a functor from the category of manifolds to the category $\mathbf{Weil}$. It should be stressed without exaggeration that the functor is not full in general, for which the reader is referred to exuberantly readable §1.6 (discussion) of [6].

Example 13

The Weil prolongation $A\otimes W$ of a $C^{\infty}$-algebra $A$ by a Weil algebra $W$ was discussed in Theorem III.5.3 of [10]. We are happy to know that any $C^{\infty}$-algebra naturally gives rise to its associated Weil space.

Notation 14

We denote by $\mathbf{Weil}$ the category of Weil spaces and Weil morphisms.

Remark 15

Dubuc [7] has indeed proposed the topos $\mathbf{Weil}$ as the first step towards the well adapted model theory of synthetic differential geometry, but we would like to contend somewhat radically that the topos $\mathbf{Weil}$ is verbatim the central object of study in classical differential geometry

It is well known (cf. Chapter 1 of [14]) that

Theorem 16

The category $\mathbf{Weil}$ is a topos. In particular, it is locally cartesian closed.

Remark 17

Dubuc [7] has called the category $\mathbf{Weil}$ the Weil topos.

Remark 18

The category of Frölicher spaces is indeed cartesian closed, but it is not locally cartesian closed. On the other hand, the category of diffeological spaces is locally cartesian closed. For these matters, the reader is referred to [st1]. It was shown by Baez and Hoffnung [2] that diffeological spaces as well as Chen spaces are no other than concrete sheaves on concrete sites.

Definition 19

The Weil prolongation $F^{W}$ of a Weil space $F$ by a Weil algebra $W$ is simply the composition of the functor $\left(\_\right)\otimes W:\mathfrak{W}\rightarrow\mathfrak{W}$ and the functor $F:\mathfrak{W}\rightarrow\mathbf{Sets}$, namely

[TABLE]

which is surely a Weil space.

Remark 20

$\left(\_\right)^{\left(\cdot\right)}$* assigning $F^{W}$ to each $\left(W,F\right)\in\mathfrak{W}\times\mathbf{Weil}$ can naturally be regarded as a bifunctor $\mathfrak{W}\times\mathbf{Weil}\rightarrow\mathbf{Weil}$.*

Trivially we have

Proposition 21

For any Weil space $F$ and any Weil algebras $W_{1}$ and $W_{2}$, we have

[TABLE]

Remark 22

The so-called Yoneda embedding

[TABLE]

is full and faithful. The famous Yoneda lemma claims that

[TABLE]

for any Weil space $F$. The Yoneda embedding can be extended to

[TABLE]

by

[TABLE]

for any $A\in\widetilde{\mathfrak{W}}$, where $\mathbf{R-Alg}$ denotes the category of $\mathbf{R}$-algebras.

Remark 23

Given Weil algebras $W_{1}$ and $W_{2}$, we have

[TABLE]

Remark 24

As is well known (cf. §8.7 of [1]), given Weil spaces $F$ and $G$, their exponential $F^{G}$ in $\mathbf{Weil}$ is provided by

[TABLE]

Proposition 25

For any Weil space $F$ and any Weil algebra $W$, $F^{W}$ and $F^{\mathbf{y}W}$ are naturally isomorphic, namely,

[TABLE]

where the left-hand side stands for the Weil prolongation $F^{W}$ of $F$ by $W$, while the right-hand side stands for the exponential $F^{\mathbf{y}W}$ in the topos $\mathbf{Weil}$.

Proof. The proof is so simple as follows:

[TABLE]

Corollary 26

Given a Weil algebra $W$ together with Weil spaces $F$ and $G$, $\left(F^{G}\right)^{W}$ and $\left(F^{W}\right)^{G}$ are naturally isomorphic, namely,

[TABLE]

Proof. We have

[TABLE]

Corollary 27

For any Weil algebra $W$, the functor $\left(\_\right)^{W}:\mathbf{Weil}\rightarrow\mathbf{Weil}$ preserves limits, particularly, products.

Proof. Since the functor $\left(\_\right)^{W}$ is of its left adjoint $\left(\_\right)\times\mathbf{y}W$ (cf. Proposition 8.13 of [1]), the desired result follows readily from the well known theorem claiming that a functor being of its left adjoint preserves limits (cf. Proposition 9.14 of [1]).

Notation 28

We denote by $\mathbb{R}$ the forgetful functor $\mathfrak{W}\rightarrow\mathbf{Sets}$, which is surely a Weil space. It can be defined also as

[TABLE]

Remark 29

The Weil space $\mathbb{R}$ is canonically regarded as an $\mathbf{R}$-algebra object in the category $\mathbf{Weil}$.

Remark 30

Since $\mathbb{R}$ is an $\mathbf{R}$-algebra object in the category $\mathbf{Weil}$, we can define, after §I.16 of [10], another $\mathbf{R}$-algebra object $\mathbb{R}\otimes W$ in the category $\mathbf{Weil}$ for any Weil algebra $W$.

Notation 31

We denote by $\mathbf{R-Alg}\left(\mathbf{Weil}\right)$ the category of $\mathbf{R}$-algebra objects in the category $\mathbf{Weil}$.

Proposition 32

The functors

[TABLE]

are naturally isomorphic.

Proof. We have

[TABLE]

4 Microlinearity

Not all Weil spaces are susceptible to the techniques of classical differential geometry, so that there should be a criterion by which we can select decent ones.

Definition 33

A Weil space $F$ is called microlinear provided that a finite limit diagram $\mathcal{D}$ in $\mathfrak{W}$ always yields a limit diagram $F^{\mathcal{D}}$ in $\mathbf{Weil}$.

Proposition 34

We have the following:

The Weil space $\mathbb{R}$ is microlinear. 2. 2.

The limit of a diagram of microlinear Weil spaces is microlinear. 3. 3.

Given Weil spaces $F$ and $G$, if $F$ is microlinear, then the exponential $F^{G}$ is also microlinear.

Proof. The first statement follows from Proposition 32. The second statement follows from the well-known fact that double limits commute. The third statement follows from Corollary 26.

It is easy to see that

Proposition 35

A Weil space $F$ is microlinear iff the diagram

[TABLE]

is a limit diagram for any Weil algebra $W$ and any finite limit diagram $\mathcal{D}$ of Weil algebras.

Proof. By Proposition 8.7 of .[1]

5 Weil Categories

Definition 36

A Weil category is a couple $\left(\mathcal{K},\mathbf{D}\right)$, where

$\mathcal{K}$* is a topos.* 2. 2.

$\mathbf{D}:\widetilde{\mathfrak{W}}^{\mathrm{op}}\rightarrow\mathcal{K}$* is a product-preserving functor. In particular, we have*

[TABLE]

where $1$ denotes the terminal object in $\mathcal{K}$.

Remark 37

The entity $\mathbf{D}$ is called a Dubuc functor with enthroning his pioneering work in [7].

Now some examples are in order.

Example 38

The first example of a Weil category has already been discussed in §3, namely,

[TABLE]

Indeed, this is the paradigm of our new concept of a Weil category, just as the category $\mathbf{Sets}$ is the paradigm of the prevailing concept of a topos.

Notation 39

We denote by $C^{\infty}\mathbf{-Alg}$ the category of $C^{\infty}$-algebras.

Example 40

Let $\mathbf{L}$ be a class of $C^{\infty}$-algebras encompassing all $C^{\infty}$-algebras of the form $C^{\infty}\left(\mathbf{R}^{n}\right)\otimes W$ with $W$ being a Weil algebra (cf. Theorem III.5.3 of [10].). We define a functor $i_{\widetilde{\mathfrak{W}},C^{\infty}\mathbf{-Alg}}:\widetilde{\mathfrak{W}}\rightarrow C^{\infty}\mathbf{-Alg}$ as

[TABLE]

Putting down $\mathbf{L}$ as a full subcategory of the category $C^{\infty}\mathbf{-Alg}$, consider a subcanonical Grothendieck topology $J$ on the category $\mathbf{L}^{\mathrm{op}}$. We let $\mathcal{K}$ be the category of all sheaves on the site $\left(\mathbf{L}^{\mathrm{op}},J\right)$. The Dubuc functor $\mathbf{D}$ is defined as

[TABLE]

where $\mathbf{y}$ stands for the Yoneda embedding.

Remark 41

Such examples have been discussed amply in the context of well-adapted models of synthetic differential geometry without being conscious of Weil categories at all. The reader is referred to [10] and [15] for them.

Now we fix a Weil category $\left(\mathcal{K},\mathbf{D}\right)$ throughout the rest of this section. Weil functors are to be defined within our framework of a Weil category.

Definition 42

The bifunctor $\mathbf{T}:\mathfrak{W}\times\mathcal{K}\rightarrow\mathcal{K}$ is defined to be

[TABLE]

We give some elementary properties with respect to $\mathbf{T}$.

Proposition 43

We have the following:

• •

The functor $\mathbf{T}\left(\mathbf{R},\left(\_\right)\right)$ and the identity functor of $\mathcal{K}$, both of which are $\mathcal{K}\rightarrow\mathcal{K}$, are naturally isomorphic, namely,

[TABLE]

• •

The trifunctors $\mathbf{T}\left(\left(\cdot_{2}\right),\mathbf{T}\left(\left(\cdot_{1}\right),\left(\_\right)\right)\right)$ and $\mathbf{T}\left(\left(\cdot_{1}\right)\otimes\left(\cdot_{2}\right),\left(\_\right)\right)$, both of which are $\mathfrak{W}\times\mathfrak{W}\times\mathcal{K}\rightarrow\mathcal{K}$, are naturally isomorphic, namely,

[TABLE]

for any Weil space $F$ and any Weil algebras $W_{1}$ and $W_{2}$.

Proposition 44

Given a Weil algebra $W$, the functor $\mathbf{T}\left(W,\cdot\right):\mathcal{K}\rightarrow\mathcal{K}$ preserves limits.

Proof. Since the functor $\mathbf{T}\left(W,\cdot\right):\mathcal{K}\rightarrow\mathcal{K}$ is of its left adjoint $\left(\cdot\right)\times\mathbf{D}W:\mathcal{K}\rightarrow\mathcal{K}$, the desired result follows readily from the well known theorem claiming that a functor being of its left adjoint preserves limits (cf. Proposition 9.14 of [1]).

Proposition 45

The trifunctors $\mathbf{T}\left(\left(\_\right),\left(\cdot_{1}\right)^{\left(\cdot_{2}\right)}\right),\mathbf{T}\left(\left(\_\right),\left(\cdot_{1}\right)\right)^{\left(\cdot_{2}\right)}:\mathfrak{W}\times\mathcal{K}\times\mathcal{K}\rightarrow\mathcal{K}$ are naturally isomorphic, namely,

[TABLE]

Proof. We have

[TABLE]

An $\mathbf{R}$-algebra object is to be introduced within our framework of a Weil category.

Notation 46

The entity $\mathbf{D}\left(\mathbf{R}\left[X\right]\right)$ is denoted by $\mathbb{R}$.

It is in nearly every mathematician’s palm to see that

Proposition 47

The entity $\mathbb{R}$ is a commutative $\mathbf{R}$-algebra object in $\mathcal{K}$ with respect to the following addition, multiplication, scalar multiplication by $\alpha\in\mathbf{R}$ and unity:

[TABLE]

Notation 48

The above four morphisms are denoted by

[TABLE]

in order.

Notation 49

The entity $\mathbf{D}\left(\mathbf{R}\left[X\right]/\left(X^{2}\right)\right)$ is denoted by $D$.

Proposition 50

The $\mathbf{R}$-algebra object $\mathbb{R}$ operates canonically on $D$ in $\mathcal{K}$. To be specific, we have the following morphism:

[TABLE]

Notation 51

The above morphism is denoted by $\cdot_{\mathbb{R},D}$.

Proposition 52

It makes the following diagrams commutative:

[TABLE]

:where the horizontal arrow is $+_{\mathbb{R}}\times D:\mathbb{R}\times\mathbb{R}\times D\rightarrow\mathbb{R}\times D$, the vertical arrow is $\cdot_{\mathbb{R},D}:\mathbb{R}\times D\rightarrow D$, and the slant arrow is

[TABLE] 2. 2.

[TABLE]

where the upper horizontal arrow is $\cdot_{\mathbb{R}}\times D:\mathbb{R}\times\mathbb{R}\times D\rightarrow\mathbb{R}\times D$, the lower horizontal arrow is $\cdot_{\mathbb{R},D}:$,$\mathbb{R}\times D\rightarrow D$* the left vertical arrow is $\mathbb{R}\times\cdot_{\mathbb{R},D}:\mathbb{R}\times\mathbb{R}\times D\rightarrow\mathbb{R}\times D$, and the right vertical arrow is $\cdot_{\mathbb{R},D}:\mathbb{R}\times D\rightarrow D$.* 3. 3.

[TABLE]

where the horizontal arrow is $\cdot_{\mathbb{R},D}:\mathbb{R}\times D\rightarrow D$, the vertical arrow is $1_{\mathbb{R}}\times D:1\times D\rightarrow\mathbb{R}\times D$, and the slant arrow is $\mathrm{id}_{D}:D\rightarrow D$.

Remark 53

We have no canonical addition in $D$. In other words, we could not define addition in $D$ in such a way as

[TABLE]

This would simply be meaningless, because

[TABLE]

is not well-defined.

Remark 54

We have the canonical morphism $D\rightarrow\mathbb{R}$. Specifically speaking, it is to be

[TABLE]

Many significant concepts and theorems of topos theory can quite easily be transferred into the theory of Weil categories surely with due modifications. In particular, we have

Theorem 55

(The Fundamental Theorem for Weil Categories, cf. Theorem 4.19 in [3] and Theorem 1 in §IV.7 of [14]) Let $\left(\mathcal{K},\mathbf{D}\right)$ be a Weil category with $M\in\mathcal{K}$. Then the slice category $\mathcal{K}/M$ endowed with a Dubuc functor $\mathbf{D}_{M}:\rightarrow\mathcal{K}/M$ is a Weil category, where

• •

$\mathbf{D}_{M}\left(A\right)$* is the canonical projection $\mathbf{D}\left(A\right)\times M\rightarrow M$ for any $A\in\widetilde{\mathfrak{W}}$, and*

• •

$\mathbf{D}_{M}\left(f\right)$* is $f\times M$ for any morphism $f$ in $\widetilde{\mathfrak{W}}$.*

Remark 56

This theorem corresponds to so-called fiberwise differential geometry. In other words, the theorem claims that we can do differential geometry fiberwise.

6 Axiomatic Differential Geometry

We fix a Weil category $\left(\mathcal{K},\mathbf{D}\right)$ throughout this section.

Notation 57

We introduce the following aliases:

• •

The entity $\mathbf{D}\left(\mathbf{R}\left[X,Y\right]/\left(X^{2},Y^{2},XY\right)\right)$ is denoted by $D\left(2\right)$.

• •

The entity $\mathbf{D}\left(\mathbf{R}\left[X,Y,Z\right]/\left(X^{2},Y^{2},Z^{2},XY,XZ,YZ\right)\right)$ is denoted by $D\left(3\right)$.

As a corollary of Proposition 47 and Theorem 55, we have

Proposition 58

The canonical projection $\mathbb{R}\times M\rightarrow M$ is a commutative $\mathbf{R}$-algebra object in the slice category $\mathcal{K}/M$.

Definition 59

An object $M$ in $\mathcal{K}$ is called microlinear provided that a finite limit diagram $\mathcal{D}$ in $\mathfrak{W}$ always yields a limit diagram $\mathbf{T}\left(\mathcal{D},M\right)$ in $\mathcal{K}$.

As in Proposition 34, we have

Proposition 60

We have the following:

The limit of a diagram of microlinear objects in $\mathcal{K}$ is microlinear. 2. 2.

Given objects $M$ and $N$ in $\mathcal{K}$, if $M$ is microlinear, then the exponential $M^{N}$ is also microlinear.

Theorem 61

Let $M$ be a microlinear object in $\mathcal{K}$. The entity $M^{\mathbf{D}\left(\mathbf{R}\rightarrow\mathbf{R}\left[X\right]/(X^{2})\right)}:M^{D}=M^{\mathbf{D}\left(\mathbf{R}\left[X\right]/(X^{2})\right)}\rightarrow M^{\mathbf{D}\left(\mathbf{R})\right)}=M$ is a $\left(\mathbb{R}\times M\rightarrow M\right)$-module object in the slice category $\mathcal{K}/M$ with respect to the following addition and scalar multiplication:

• •

The following diagram

[TABLE]

is a pullback, where the upper horizontal arrow is

[TABLE]

the lower horizontal arrow is

[TABLE]

the left vertical arrow is

[TABLE]

and the right vertical arrow is

[TABLE]

Since $M$ is microlinear, the diagram

[TABLE]

is a pullback, where the upper horizontal arrow is

[TABLE]

the lower horizontal arrow is

[TABLE]

the left vertical arrow is

[TABLE]

and the right vertical arrow is

[TABLE]

Therefore we have

[TABLE]

The morphism

[TABLE]

stands for addition and is denoted by $\varphi$.

• •

The composition of the morphism

[TABLE]

and the evaluation morphism

[TABLE]

is denoted by $\widehat{\psi}_{1}:D\times\mathbb{R}\times M^{D}\rightarrow M$. Its transpose $\psi_{1}:\mathbb{R}\times M^{D}\rightarrow M^{D}$ stands for scalar multiplication.

Proof. Here we deal only with the associativity of addition and the distibutivity of scalar multiplication over addition, leaving verification of the other rquisites of $M^{\mathbf{D}\left(\mathbf{R}\rightarrow\mathbf{R}\left[X\right]/(X^{2})\right)}:M^{D}=M^{\mathbf{D}\left(\mathbf{R}\left[X\right]/(X^{2})\right)}\rightarrow M^{\mathbf{D}\left(\mathbf{R})\right)}=M$ being a $\left(\mathbb{R}\times M\rightarrow M\right)$-module object in the category $\mathcal{K}/M$ to the reader.

• •

The diagram

[TABLE]

is a limit diagram, where the upper three arrows are

[TABLE]

from left to right, and the lower three arrows are the same

[TABLE]

Since $M$ is microlinear, the diagram

[TABLE]

is a limit diagram, where the upper three arrows are

[TABLE]

from left to right, and the lower three arrows are the same

[TABLE]

Therefore we have

[TABLE]

It is now easy to see that the diagram

[TABLE]

is commutative, where the upper horizontal arrow is

[TABLE]

the lower horizontal arrow is

[TABLE]

the left vertical arrow is

[TABLE]

and the right vertical arrow is

[TABLE]

We have just established the associativity of addition.

• •

The proof of the distibutivity of scalar multiplication over addition is divided into three steps:

The composition of the morphism

[TABLE]

and the evaluation morphism

[TABLE]

is denoted by $\widehat{\psi}_{2}:D\left(2\right)\times\mathbb{R}\times M^{D\left(2\right)}\rightarrow M$. Its transpose is denoted by $\psi_{2}:\mathbb{R}\times M^{D\left(2\right)}\rightarrow M^{D\left(2\right)}$. And the composition of the morphism

[TABLE]

and the evaluation morphism

[TABLE]

is denoted by $\widehat{\chi}:D\left(2\right)\times\mathbb{R}\times\mathbb{R}\times M^{D\left(2\right)}\rightarrow M$. Its transpose is denoted by $\chi:\mathbb{R}\times\mathbb{R}\times M^{D\left(2\right)}\rightarrow M^{D\left(2\right)}$. It is easy to see that the diagram

[TABLE]

commutes, where the vertical arrow is

[TABLE]

the horizontal arrow is

[TABLE]

and the slant arrow is

[TABLE]

It is also easy to see that the morphism $\chi:\mathbb{R}\times\mathbb{R}\times M^{D\left(2\right)}\rightarrow M^{D\left(2\right)}$ can be defined to be

[TABLE] 2. 2.

Let us consider the following diagram:

[TABLE]

where the upper two horizontal arrows are

[TABLE]

from left to right, the lower two horizontal arrow are

[TABLE]

from left to right, the three vertical arrows are

[TABLE]

from left to right, and the two slant arrows are the evaluation morphisms $D\times M^{D}\rightarrow M$ and $D\left(2\right)\times M^{D\left(2\right)}\rightarrow M$. In order to establish the commutativity of the diagram (4), we will be engaged in the commutativity of the three subdiagrams 1, 2 and 3 in order. It is easy to see that both the diagram 1 and the digaram 2 commute. The commutativity of the diagram 1 is a simple consequence of the fact that $\left(\_\right)\times\left(\_\right)$ is a bifunctor, while the commutativity of the diagram 2 follows directly from that of the following diagram

[TABLE]

where the two horizontal arrows are

[TABLE]

from top to bottom, and the two vertical arrows are

[TABLE]

from left to right. The commutativity of the diagram 3 follows from the following commutative diagram of so-called parametrized adjunction (cf. Theorem 3 in §IV.7 of [13]):

[TABLE]

where the left two vertical arrows are

[TABLE]

from top to bottom, while the right vertical arrows are

[TABLE]

from top to bottom. Choose

[TABLE]

on the right of the diagram.(5). Then both yield the same morphism in $\mathrm{Hom}_{\mathcal{K}}\left(M^{D\left(2\right)},M^{D}\right)$ by application of their adjacent vertical arrows. The corresponding morphism of $\mathrm{id}_{M^{D\left(2\right)}}$ in $\mathrm{Hom}_{\mathcal{K}}\left(D\left(2\right)\times M^{D\left(2\right)},M\right)$ is no other than the evaluation morphism $D\left(2\right)\times M^{D\left(2\right)}\rightarrow M$, and the corresponding morphism of $\mathrm{id}_{M^{D}}$ in $\mathrm{Hom}_{\mathcal{K}}\left(D\times M^{D},M\right)$ is no other than the evaluation morphism $D\times M^{D}\rightarrow M$, Therefore both the evaluation morphisms $D\left(2\right)\times M^{D\left(2\right)}\rightarrow M$ and $D\times M^{D}\rightarrow M$ yield the same morphism in $\mathrm{Hom}_{\mathcal{K}}\left(D\times M^{D\left(2\right)},M\right)$ by application of their adjacent vertical arrows, which is tantamount to the commutativity of the diagram 3. We have just established the commutativity of the whole diagram (4). In particular, the outer hexagon of the diagram (4) is commutative, which means that the diagram

[TABLE]

is commutative, where the two horizontal arrows are

[TABLE]

from top to bottom, and the two vertical arrows are

[TABLE]

from left to right. 3. 3.

The following is a commutative diagram of parametrized adjunction (cf. Theorem 3 in §IV.7 of [13]):

[TABLE]

where the left two vertical arrows are

[TABLE]

from top to bottom, while the right vertical arrows are

[TABLE]

from top to bottom. Choose

[TABLE]

on the left of the diagram.(7). Then both yield the same morphism in $\mathrm{Hom}_{\mathcal{K}}\left(D\times\mathbb{R}\times M^{D\left(2\right)},M\right)$ by application of their adjacent vertical arrows by dint of the commutativity of the diagram (6). The corresponding morphism of $\widehat{\psi}_{2}$ in $\mathrm{Hom}_{\mathcal{K}}\left(\mathbb{R}\times M^{D\left(2\right)},M^{D\left(2\right)}\right)$ is $\psi_{2}$, and the corresponding morphism of $\widehat{\psi}_{1}$ in $\mathrm{Hom}_{\mathcal{K}}\left(\mathbb{R}\times M^{D},M^{D}\right)$ is $\psi_{1}$, Therefore both $\psi_{2}$ and $\psi_{1}$ yield the same morphism in $\mathrm{Hom}_{\mathcal{K}}\left(\mathbb{R}\times M^{D\left(2\right)},M^{D}\right)$ by application of their adjacent vertical arrows, which is tantamount to the commutativity of the following diagram:

[TABLE]

where the two horizontal arrows are

[TABLE]

from top to bottom, and the two vertical arrows are

[TABLE]

from left to right. We have just established the distibutivity of scalar multiplication over addition.

7 Concluding Remarks

Weilology began with André Weil’s algebraic treatment of nilpotent infinitesimals [30]. Its second step is synthetic differential geometry [10]and the study of Weil functors of Czech geometers [11]. Its third step is the author’s axiomatic differential geometry ([16]-[23]). Now we have its final form in this paper.

A subsequent paper is devoted to fixing the syntax of Weil categories after the manner of [3], under which we can develop axiomatic differential geometry naively (i.e., without tears), just as René Lavendhomme did for synthetic differential geometry [12].

Another important point is that we can investigate Weilology for supergeometry, braided geometry, noncommutative geometry, homotopical differential geometry, arithmetical differential geometry and so on in the same vein, which is the topic of subsequent papers.