Ephemeral persistence modules and distance comparison

Abstract

We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of $\gamma$-sheaves. In the case of one-dimensional persistence, our definition agrees with the usual one showing that the observable category and the category of $\gamma$-sheaves are equivalent. We also establish isometry theorems between the category of persistent modules and $\gamma$-sheaves both endowed with their interleaving distance. Finally, we compare the interleaving and convolution distances.

27 pages, minor changes, several typos corrected

References in corpus (7)

• The Theory of the Interleaving Distance on Multidimensional Persistence Modules
• Sheaves, Cosheaves and Applications
• Interactive Visualization of 2-D Persistence Modules
• Multidimensional Interleavings and Applications to Topological Inference
• Persistence-like distance on Tamarkin's category and symplectic displacement energy
• Theory of interleavings on categories with a flow
• Stable resolutions of multi-parameter persistence modules

Cited by in corpus (13)

• Functors on Posets Left Kan Extend to Cosheaves: an Erratum
• A derived isometry theorem for constructible sheaves on $\mathbb{R}$
• Hybrid transforms of constructible functions
• Essential graded algebra over polynomial rings with real exponents
• Global dimension of real-exponent polynomial rings
• Modules over posets: commutative and homological algebra
• Completeness of derived interleaving distances and sheaf quantization of non-smooth objects
• Multiplicative persistent distances
• Multi-Parameter Persistence Modules are Generically Indecomposable
• Projected distances for multi-parameter persistence modules
• A property of the interleaving distance for sheaves
• Convolution of Persistence Modules
• Discrete Microlocal Morse Theory