Rips Complexes as Nerves and a Functorial Dowker-Nerve Diagram

Using ideas of the Dowker duality we prove that Rips complex at scale $r$ is homotopy equivalent to nerve a cover consisting sets prescribed diameter. We then develop functorial version Nerve theorem coupled with duality, which presented as Functorial Dowker-Nerve Diagram. These results are incorporated into systematic theory filtrations arising from covers. As result provide general framework for reconstruction spaces by complexes, short proof Hausmann, and completely classify scales metric graphs. Furthermore introduce new extraction method homology space based on nested complexes single scale, requires no conditions neighboring nor Euclidean structure ambient space.