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