# Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

## TL;DR

This paper develops a functorial framework connecting Rips complexes, nerves, and Dowker duality, enabling space reconstruction, classification of scales for metric graphs, and a novel homology extraction method without Euclidean assumptions.

## Contribution

It introduces a functorial Dowker-Nerve diagram, unifies Rips complexes with nerve theorems, and presents a new homology extraction method applicable in general metric spaces.

## Key findings

- Rips complex at scale r is homotopy equivalent to a nerve of a cover with prescribed diameter
- Provides a systematic theory for filtrations from covers and space reconstruction
- Introduces a scale-independent homology extraction method

## Abstract

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.