We are fundamentally interested in the algebraic foundations of applied and computational algebraic topology, in particular in a unifying theory for the various flavors of persistent homology that have emerged so far: standard persistence, multidimensional persistence and zigzag persistence. In this talk, we explore a topos-based approach, pursuing a formulation that can make the topos setting clear and amenable for generalization. This permits us to consider substituting the category of sets underlying semi-simplicial sets, by the topos of sheaves over H. We will also discuss the theory of persistent homology as the internal homology of simplicial complexes over a set theory in which elements have life-times; where elements of sets, simplices of simplicial complexes, and base vectors of vector spaces are born, live, and die, much like the language has evolved for discussing persistent homology. In the end of the talk we will propose applications of this theory to the topological analysis of medical data.