# nLab Dubuc topos

Contents

### Context

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

The Dubuc topos is a model in which one may apply synthetic differential geometry to attain results in classical differential geometry. It is constructed as the topos of sheaves for the open covering topology on the dual of the category of C-infinity rings presented by a germ determined ideal.

## Definition

###### Definition

(germ-determined C-infinity ring)

A finitely generated C-infinity ring of the form $C^\infty(M)/I$ for some smooth manifold $M$ and ideal $I$ is germ-determined if the ideal $I$ is a germ-determined ideal, which means that for any $f\in I$, if the germ of $f$ at any point $x$ in the zero locus of $I$ in $M$ coincides with the germ of some element of $I$ at $x$, then $f$ itself belongs to $I$.

Equivalently, germ-determined ideals are precisely those ideals that are closed under (possibly infinite) sums of locally finite families. One can also take sums with coefficients in a partition of unity.

###### Definition

(Dubuc topos)

The Dubuc topos is the sheaf topos on the site $G$, whose underlying category is the opposite category of the category of germ-determined finitely generated C-infinity rings (Def. ).

The resulting opposite category is equipped with a Grothendieck topology as follows. A collection of morphisms $\{A_i\to B\}_{i\in I}$ in this category is a covering family if for any $i\in I$, the corresponding homomorphism of C-infinity rings $B \to A_i$ is isomorphic to a localization map at some element $b_i\in B$.

Here localization is defined using a universal property formulated in the category of C-infinity rings and need not coincide with the usual localization of commutative rings. Furthermore, we require that applying the functor $\gamma$ to the family $\{A_i\to B\}_{i\in I}$ produces a jointly surjective family of continuous maps of topological spaces.

Here the functor $\gamma$ is the right adjoint functor of the opposite functor of the inclusion of finitely generated point-determined C-infinity rings into finitely generated germ-determined C-infinity rings. Concretely, $\gamma(C^\infty(M)/I)$ for a germ-determined ideal $I$ can be described as the zero locus of $I$ in $M$, converted into a finitely generated point-determined C-infinity ring using its C-infinity ring of smooth functions.

Here a finitely generated C-infinity ring of the form $C^\infty(M)/I$ for some smooth manifold $M$ and ideal $I$ is point-determined if the ideal $I$ is point-determined, which means that for any $f\in I$, if $f(x)=0$ for any point $x$ in the zero locus of $I$ in $M$, then $f$ itself belongs to $I$.

## Some history

This section is written by Eduardo Dubuc.

In papers on SDG one can find phrases of the type “We build our model after the standard manner of Moerdijk & Reyes” ([MR] “Models for Smooth Infinitesimal Analysis”, Moerdijk and Reyes, Springer Verlag 1991). Now, this “standard manner” of building models was developed by Dubuc, and not by Moerdijk-Reyes. Of course, I understand that when a monograph is available, the proper reference is that, and not the original papers. But, phrases as the one quoted above are ambiguous, and the fault is not of the authors.

I resume the history of the subject: In a series of papers and many lectures given specially in Montreal, Sydney, Oberwolfach and elsewhere, I created and started the development of the subject of models of SDG adapted for the applications to classical differential geometry.

1. Sur les Modeles de la Geometrie Differentielle Sinthetyque, Cahiers de Topologie et Geometrie Differentielle Vol. XX-3 (1979).

2. Schemas C-inf (amplified version of , with detailed proofs and many examples), “Prepublications de la Universite de Montreal” 80-81 edited by G. Reyes (1980).

3. C-inf Schemes, American Journal of Mathematics, John Hopkins University, Vol. 103-4 (1981).

4. Open Covers and Infinitary Operations in C-inf-rings, Cahiers de Topologie et Geometrie Differentielle Vol. XXII-3 (1981).

5. Integracion de campos vectoriales y Geometria Diferencial Sintetica, Proceedings of the VII Sem. Nac. de Mat., FAMAF, Univ. Nac. de Cordoba (1984).

6. Germ representability and Local integration of vector fields in a well adapted model of SDG, Aarhus Univ. Math. Inst. preprint series (1985/1986), (Journal of Pure and Applied Algebra Vol. 64, 1990).

7. Archimedian Local C-inf-rings and Models of SDG (with Marta Bunge), Cahiers de Topologie et Geometrie Differentielle Vol. XXVII-3 (1986).

8. Local concepts in SDG and germ representability (with Marta Bunge), in D. Kueker et al. (ads), Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics 106, Marcel Dekker (1987)

A) I introduced the concept, coined the name “well-adapted model?”, and constructed the first ones. I considered the two essential properties:

i) preservation of all open covers.

ii) preservation of transversal pullbacks.

B) I started a systematic study of C-infinity rings as such. Of course, they were already there, but nothing had been done with them. I even had to state and prove such simple facts as that the algebraic quotient of a C-infinity ring by an R-algebra ideal had a canonical structure of C-infinity ring, and was then the quotient in this category.

I introduced the key notion of Germ Determined Ideal (or ideal of local nature), which was, as such, nowhere to be found in the literature, and stated and proved their basic properties. This, I think, is the most important concept in the subject. It is the basic definition to start to build upon. It is just the right concept needed. Among other things, I first proved it contains all finitely generated ideals, and defines the largest possible class of C-infinity rings consistent with the Nullstellensatz. This means that the ring can be seen as the ring of global sections of a C-infinity scheme. The notion of germ determined ideal also determines the right notion of C-infinity local ring (notice that I do not say local C-infinity ring), that I then developed. I also developed the relative C-infinity version of inverting elements universally, and proved the essential fact that the ring of C-infinity functions defined in an open set $U$ inverts universally a function which is non-zero exactly on $U$ (every Euclidean open is C-infinity Zariski).

When incorporating this work in their monograph Moerdijk and Reyes say (I quote): “Although this general notion of C-infinity ring does not occur as such in classical analysis and differential geometry, the main examples do … Given the role of these examples of C-infinity rings in the classical literature, it is not surprising that although the statements of several of the results in this chapter seem new, most of their proofs are either known or easily derivable from known ones”. This is, at the least, misleading, and I see a clear intention to disqualify my work. Of course, when the new concepts are introduced, the examples are already there, and the proof of the basic properties is easy. The important thing is to identify explicitly the concept, and to identify the right statements and properties, and this does not come easily. And I repeat, even if C-infinity rings may have been there, the concept of germ determined ideal was not, neither the concept of $C^\infty$-local ring, and several derived concepts and the statements of their basic essential properties neither).

C) With this in hand, I introduced the topos $G$ of sheaves for the open covering topology on the dual of the category of C-infinity rings presented by a germ determined ideal, and proved all the basic important properties, which many times are the correct relative C-infinity versions of corresponding properties in algebraic geometry. This is the analogue in SDG of the Zariski topos of algebraic geometry (here we should notice that the Zariski topos is defined by the topology of all open covers, this is essential. Moerdijk-Reyes, misunderstanding Zariski, call Smooth-Zariski, the topos determined by the finite covers).

The topos $G$ is the best known model in order to do applications of SDG to classical differential geometry, and as such, it is the most utilized in practice. Many early workers in the subject (J. Penon, O. Bruno, M. Bunge, F. Gago, Yetter, among others) called this topos “The Dubuc Topos”. Even Moerdijk and Reyes did so in some preprints, although they changed this in the published versions.

In their monograph they say (I quote): “As far as terminology is concerned, we have tried to avoid descriptions of the type ”the Moerdijk envelop of the Reyes topos“, in favor of more informative ones”. But the true fact was that the only name that was involved was “Dubuc”, since no other new name was being utilized at the time (of course, things as Kock-Lawvere axiom or Weil algebras were not aimed by this philosophy, and “Moerdijk envelop of the Reyes topos” was an invention to reduce to the ridiculous, the fact of naming things after “Dubuc”.

D) I should mention here that I do not ignore the fact that my work is acknowledged and referenced in the monograph, but the matter is much more subtle, and nobody can deny the evidence of the following consequence of their maneuvering:

My name, as time passes, and as young people appear, is less and less credited with a subject that I created and developed, namely, models of SDG adapted to classical differential geometry. People talk as if the well adapted models were always there, or start referring to them in a way that lead inexperienced (in the subject) readers to believe that these constructions are “M-R way of doing models”, as I quoted at the beginning of this note.

E) This does not do justice to my work, and does not corresponds to the true history of the subject. As an starting point to remedy it, I request all workers that need to use the topos $G$, to refer to it as “Dubuc Topos”. After all, it is a long tradition in mathematics to associate proper names to important concepts or constructions when it is justified, as I believe it is the case here.

REMARK. Proposition 18 and Theorem 22 of 6. cited above establish that postulate WA2 (II, 3.2, 3.3) of 8. holds in the Dubuc Topos. However, there is a mistake (discovered by M. Makkai) in Lemma 21 of 6. which invalidates the proof. I believe that the result is true but I am not working to fix the proof now. The external statement holds, it is easy and it proved in 5. cited above.

This preprint is an amplified version of the following article, with detailed proofs and many examples:

Penon’s thesis has an appendix on the Dubuc topos: