Couplings and primitives on topological spaces

For an open covering $\mathcal U$ of a topological space and a mapping $d\colon\mathcal I\to\mathbb{K}$, where $\mathcal I:=\bigl\{(U,V)\in\mathcal U\times\mathcal U;\ U\cap V\ne\varnothing\bigr\}$, we present a context for the existence of a mapping $C\colon\mathcal U\to\mathbb{K}$ satisfying $C_V-C_U=d_{UV}$ for all $(U,V)\in\mathcal I$. The result is applied to a Poincar\'e type theorem concerning distributional potentials. We also put the result into the context of algebraic topology.