Filtered Topology and Persistence in Stable Homotopy

We define a category of filtered topological spaces and explore some of its homotopy theoretic properties, including a filtered analogue of CW approximation. With this, we define and study a filtered (weighted) variant of the Euler characteristic and show this is a `filtered homotopy invariant'. We then go on to use the recent work of Biran, Cornea and Zhang by considering a persistence Spanier-Whitehead category of filtered CW complexes and show this is a triangulated persistence category and discuss the fragmentation metrics induced by this structure. We go on to show that the K-group of this persistence category is isomorphic to the ring of Novikov polynomials and this isomorphism is induced by the weighted Euler characteristic. Finally, we discuss how these constructions extend to a filtered stable homotopy category and its relation to filtered/persistence homologies.