Kock on synthetic differential geometry
Tuesday, July 25th, 2006Synthetic differential geometry is an attempt to reformulate differential geometry to allow infinitesimals. Unlike nonstandard analysis, these infinitesimals are nilpotent, and the operation of taking the derivative of a function at a point becomes just evaluating the function at a nilpotent infinitesimal near that point. The idea was used heuristically in the nineteenth century, but the inspiration from the modern reformulation comes from commutative algebra, where the idea is unproblematic.
Anders Kock has made his book on the subject, Synthetic Differential Geometry, available for download on his website. The book is being reprinted, so he asks readers not to circulate printed copies.