2019/10/07 by Simon Cho, Cho, Simon · 1 citation
Computer Science · Mathematics · Medicine · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Advanced Neuroimaging Techniques and Applications
paper · pdf · doi:10.48550/arxiv.1910.02905
We construct a nerve functor parametrized by a choice of quantale, exhibiting both the Vietoris-Rips complex and the magnitude nerve as instances of this nerve for different choices of monoidal structure on ℝ. Furthermore, the difference between how persistent homology processes the Vietoris-Rips complex and how magnitude homology processes the magnitude nerve is cast as a choice of whether or not to "localize" the corresponding nerves along ℝ in a precise sense. Lastly, we mention some application-oriented observations naturally suggested by the perspective mentioned above.