One does not get far in differential geometry without calculations. This also applies for that synthetic approach which we present. We develop in this chapter the basic calculus and algebra needed. The fundamental differentiation process (formation of directional derivatives) here actually becomes part of the algebra, since the classical use of limit processes is eliminated in favour of the use of infinitesimal subspaces of the number line R and of the coordinate vector spaces Rn. These infinitesimal spaces are defined in an algebraic, and ultimately coordinate-free, way, so that they may be defined as subspaces of arbitrary finite-dimensional vector spaces V. The combinatorial notion of “pairs of points in V which are k-neighbours” (k = 0, 1, 2, …), written x ∼k y, is introduced as an aspect of these infinitesimal spaces. The neighbour relations ∼k are invariant under all, even locally defined, maps. This opens up consideration of the neighbour relations in general manifolds in Chapter 2.

The content of this chapter has some overlap with the existing textbooks on SDG (notably with Part I of Kock, 1981/2006) and is, as these, based on the KL axiom scheme.

The number line R

The axiomatics and the theory to be presented involve a sufficiently nice category ℰ, equipped with a commutative ring object R, the “number line” or “affine line”; the symbol R is chosen because of its similarity with ℝ, the standard symbol for the ring of real numbers. The category ℰ is typically a topos (although for most of the theory, less will do).