Abstract: Function spaces appearing in calculus usually have extra structure besides being a topological vector space. In this talk, we start by examining Borel's lemma from the viewpoint of diffeological vector spaces, where diffeology is a sheaf theoretical way to unify smooth manifolds, infinite-dimensional spaces and singular spaces. This leads to homological algebra in the smooth setting. We will see how fibre bundle theory plays a role there. At the end, we will see some interesting phenomena, together with several conjectures and open questions. This talk includes some joint work with Dan Christensen.