De l’infinitésimal au local. (Thèse de Doctorat d’Etat, Université Paris VII). (French) Zbl 0558.18003

Penon, Jacques

Diagrammes 13, 191 p. (1985).

This work comprises a summary of various approaches to the notion of infinitesimal, and a new ”intrinsic” approach of the author’s.
The very readable introduction traces the antecedent ideas: a quick nod in the direction of Abraham Robinson, Weil’s ”Théorie des points proches sur les variétés différentiables”, the use of nilpotents in algebraic geometry, the étale topology, the rise of topos theory, and synthetic differential geometry.
Chapter 0 contains a quick survey of the interpretation of intuitionistic logic in a topos.
Chapter 1 is entitled ”Use of Nilpotents”. It covers some of Kock’s work on linear algebra in an intuitionistic setting, the Kock-Lawvere axiom for a ring of line type, models of the axiom in algebraic and differential geometry, infinitesimal linearity and the tangent bundle, 1- étale maps, and Reyes’ notion of a Fermat ring.
Chapter 2 is entitled ”Use of Infinitesimals”. The basic idea of this work is to exploit the observation that in the particular toposes discussed above as models for synthetic differential geometry it so happens that nilpotent elements in the ring of line type are also the elements x satisfying \(NOT(NOT(x=0))\); recall that we are working in an intuitionistic setting. The author defines in general the relation of being infinitesimally close to be the double negation of equality. Of course, it is the fact that this notion is not geometric, i.e. not preserved by inverse image functors, which makes it difficult to work with, and also rather deeper. Infinitesimal invertibility and formal manifolds are examined from this point of view, and comparison is made with the nilpotent approach. Section 2 is particularly interesting; in it a class of toposes is introduced for which a ”Nullstellensatz” holds, making the interpretation of infinitesimals work more smoothly.
Chapter 3 is entitled ”Use of Intrinsic Neighbourhoods”. The idea here is that open neighbourhoods in a topology are interpreted as containing all points ”sufficiently close”. It follows that in an intuitionistic setting we have an intrinsic notion of ”open set”, namely a set closed under the relation of being infinitesimally close. This intrinsic topology is explored in this chapter.
Chapter 4 investigates the relationship between local invertibility in the different senses now available, and what they mean in the different models under consideration.
There is an appendix on the Dubuc topos.

Reviewer: G.C.Wraith

MSC:

18B25	Topoi
51K10	Synthetic differential geometry
18F15	Abstract manifolds and fiber bundles (category-theoretic aspects)
03F55	Intuitionistic mathematics
14A99	Foundations of algebraic geometry

Keywords:

infinitesimal; nilpotents; synthetic differential geometry; intuitionistic logic; tangent bundle; ring of line type; formal manifolds; Nullstellensatz; intrinsic topology; local invertibility; Dubuc topos