In topological data analysis we study a data set given as a finitepoint cloud by embedding it insome parameter-dependent topological spaces, and computing their homology. This can be formalised as thecomposition of two functors: first from a poset to the category of topological spaces, and then on to the category of vector spaces over a field. This is known as a filtration. Filtrations can be seen as modules over the path algebra of a quiver given by the Hasse diagram of the initial poset, or equivalently as quiver representations of this quiver. We obtain the ranks of morphisms in this quiver representation via relative projective resolutions for a suitably chosen exact structure. In order to show that this structure is exact we follow the proof of a theorem by Botnan, Opperman and Oudot, which we extend to work over all abelian categories rather than only categories of modules over Artin algebras. We conclude with a discussion of Auslander-Reiten theory in the context of exact categories.