The tangent bundle T(M) → M of a manifold M is traditionally the main vehicle for encoding the geometry of infinitesimals; a substantial part of existing literature on SDG deals with aspects of this, see e.g. Kock (1981/2006) and the references therein, notably the references for the second edition. The main tool for comparing the tangent bundle approach to the approach based on the (first-order) neighbour relation is what we call the log-exp bijection, which we introduce in Section 4.3 below.

Tangent vectors and vector fields

It is a classical conception in algebraic geometry (schemes) that the notion of tangent vectors may be represented by a scheme D, namely D = the spectrum of the ring k[∈] = k[Z]/(Z2) of dual numbers, cf. e.g. Mumford (1965/1988, p. 338/238), who calls this D (in his notation I) “a sort of disembodied tangent vector”, so that “the set of all morphisms from D to M” is a “sort of settheoretic tangent bundle to M”.

In a seminal lecture in 1967, Lawvere proposed to axiomatize the object D, together with the category ℰ of spaces in which it lives, and to exploit the assumed cartesian closedness of ℰ (existence of function space objects) to comprehend the tangent vectors of a space M into a space MD, which thus is not just a set, but a space (an object of ℰ). This was the seed that was to grow into SDG.

In the present text, D is, as in Section 1.2, taken to be the subspace of the ring R consisting of elements of square 0.