This chapter gives the basic configurations in the infinitesimal geometry: it describes and draws pictures of some of the figures which can be made out of the first-order neighbour relation ∼1: infinitesimal simplices, infinitesimal parallelepipeda, geometric distributions, parallelograms given by affine connections. (Some further pictures, which derive from a symmetric affine connection, are found in Chapter 8.)

The section on jets deals with the kth-order neighbour relation ∼k for general k.

Manifolds

A manifold M of dimension n is a space such that there exists a family {Ui| i ∈ I} of spaces equipped with open inclusions Ui → M and Ui → Rn; the family Ui → M is supposed to be jointly surjective.

The meaning of this “definition” depends on the meaning of the term “open inclusion”(= “open subspace”), and on the meaning of “family”.

For “open inclusion”, we take the viewpoint that this is a primitive notion: we assume that among all the arrows in ℰ, there is singled out a subclass ℛ of “open inclusions”, with suitable stability properties, e.g. stability under pullback, as spelled out in the Appendix, Section A.6. Also, we require that the inclusion Inv(R) ⊆ R of the set of invertible elements in the ring R should be open.

It will follow that all maps that are inclusions, which are open according to ℛ, are also formally open; this is the openness notion considered in Kock (1981/2006). For V a finite-dimensional vector space, U ⊆ V is formally open if x ∈ U and y ∼kx implies y ∈ U.