This book deals with a certain aspect of the theory of smooth manifolds, namely (for each k) the kth neigbourhood of the diagonal. A part of the theory presented here also applies in algebraic geometry (smooth schemes).

The neighbourhoods of the diagonal are classical mathematical objects. In the context of algebraic geometry, they were introduced by the Grothendieck school in the early 1960s; the Grothendieck ideas were imported into the context of smooth manifolds by Malgrange, Kumpera and Spencer, and others. Kumpera and Spencer call them “prolongation spaces of order k”.

The study of these spaces has previously been forced to be rather technical, because the prolongation spaces are not themselves manifolds, but live in a wider category of “spaces”, which has to be described. For the case of algebraic geometry, one passes from the category of varieties to the wider category of schemes; for the smooth case, Malgrange, Kumpera and Spencer, and others, described a category of “generalized differentiable manifolds with nilpotent elements” (Kumpera and Spencer, 1973, p. 54).

With the advent of topos theory, and of synthetic differential geometry, it has become possible to circumvent the construction of these various categories of generalized spaces, and instead to deal axiomatically with the notions. This is the approach we take; in my opinion, it makes the neighbourhood notion quite elementary and expressive, and in fact, provides a non-technical and geometric gateway to many aspects of differential geometry; I hope the book can be used as such a gateway, even with very little prior knowledge of differential geometry.