Kock.A. and G.E. Reyes (2004) Categorical distribution theory; heat equation.arXiv:math.CT/0407242 13 Dec 2004
The theory of distributions with compact support has been developed “synthetically”, i.e., in the context of a cartesian closed category with a ring object having suitable properties, “the smooth reals” and in which everything is smooth. Topos models which contain both classical smooth manifolds and infinitesimal structures have been provided to relate it to the classical theory and the wave equation has been shown to have solutions in these models (see “Some calculus with extensive quantities: wave equation” by the authors). In this paper, we provide a similar theory for distributions which are not necessarily of compact support. As a test case, we show that the Cahiers topos is a model for this theory and admits a fundamental (distributional) solution of the heat equation on the unlimited line.