Kock A. and G.E. Reyes (2003). Some calculus with extensive quantities: wave equation. arXiv: math.CT/0303297 v1 24 Mar 2003
This paper is a contribution to a synthetic theory of distributions. The sense in which we understand “synthetic” in this context is that we place ourselves in a setting (category) with a ring object having suitable properties, “the smooth reals” and where everything is smooth (differentiable). A main assumption about the category in which we work is that it is cartesian closed, meaning that function “spaces”, and hence some of the methods of functional analysis, are available. The blunt assumption of smoothness allows a simplification of the theory in the sense that distributions are functionals which are linear, continuity comes for free in this smooth universe. We provide topos models which contain both classical smooth manifolds and infinitesimal structures that relate the theory to the classical one. As a pilot project, we show how to construct the fundamental solution of the wave equation: the description of the evolution of a point (Dirac) distribution over time.
article Gonzalo E. Reyes Mathematics synthetic differential geometry