One can alternatively use a set-theoretic definition of a persistence module that is equivalent to the categorical viewpoint: A persistence module is a pair (V,\pi) where V is a collection \{V_z\}_{z\in P} of K -vector spaces and \pi is a collection \{\pi_{y,z}\}_{y\leq z\in P} of linear maps where \pi_{y,z} : V_y \to V_z for each y\leq z \in P , such that \pi_{y,z} \circ \pi_{x,y} = \pi_{x,z} for any x \leq y \leq z \in P (i.e., all the maps commute).
