An integral affine structure on a topological manifold $M$ is equivalent to the datum of a subsheaf $\mathrm{Aff}_{M}$ of the sheaf of continuous functions on $M$ such that $(M,\mathrm{Aff}_{M})$ is locally isomorphic to $(\mathbb{R}^{n},\mathrm{Aff}_{\mathbb{R}^{n}})$.