This Appendix is a mixed bag; it contains things which are either foundational or technical. Of foundational nature is in particular Sections A.2 and A.3 – for those who need to lift the hood of the car to get an idea how the engine works, before getting into the car and driving it. Section A.4 deals with microlinearity, a technical issue specific to SDG; and the remaining sections are completely standard mathematical technique, with due care taken of the constructive character of the reasoning, but otherwise probably found in many standard texts.

Category theory

Basic to the development of SDG is category theory, with emphasis on what the maps between the mathematical objects are.

In particular, SDG depends on the notion of cartesian closed category ℰ; recall that in such ℰ, one not only has a set of maps X → Y, but an object (or “space”) YX ∈ ℰ of maps, with a well-known universal property, cf. e.g. Section A.3 below for the relevant diagram.

The category of sets is cartesian closed, in fact, any topos is so.

The category Mf of smooth manifolds is not cartesian closed, and this failure has historically caused difficulties for subjects like calculus of variations, path integrals, continuum mechanics, …, and has led to the invention of more comprehensive “smooth categories” by Chen, Frölicher, Kriegl, Michor, and many others (see e.g. Frölicher and Kriegl, 1988; Kriegl and Michor, 1997 and the references therein), and also to the invention of toposes containing Mf like Grothendieck's “smooth topos” (terminology of Kock, 1981/2006) and the “well-adapted toposes” for SDG, as sketched below.