Categorified Reeb Graphs

@article{Silva2016CategorifiedRG,
  title={Categorified Reeb Graphs},
  author={Vin de Silva and E. Munch and Amit K. Patel},
  journal={Discrete \& Computational Geometry},
  year={2016},
  volume={55},
  pages={854-906}
}
The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs… 
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This paper shows that the two metrics are strongly equivalent on the space of Reeb graphs, and gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
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