Categorical distribution theory; heat equation

The aim of this paper is to develop the theory of distributions, not necessarily of compact support, in a topos model of Synthetic Differential Geometry, the so-called "Cahiers Topos". As an application, we study the evolution through time of a heat distribution. In the first part of the paper we study distributions in the sense of Schwartz in the context of convenient vector spaces. In particular, we study smoothness (with respect to time) of the fundamental distribution solution of the heat equation. In the second part, we show how the results of the first part may be used to internalize these notions to prove the existence of the fundamental distribution solution of the heat equation in the Cahiers topos.}