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Categorification of Persistent Homology
TLDR
This work redevelops persistent homology (topological persistence) from a categorical point of view and gives a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category and shows that if the target category is abelian, so is this category of interleavments.
Metrics for Generalized Persistence Modules
TLDR
This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems.
An algebraic Wasserstein distance for generalized persistence modules
TLDR
The algebraic Wasserstein distances do not require computing a persistence diagram, and they apply to persistence modules that are not interval decomposable and also to generalized persistence modules, such as multi-parameter persistence modules.
Co-rings over operads characterize morphisms
Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let
CoHohschild and cocyclic homology of chain coalgebras
Generalizing work of Doi and of Farinati and Solotar, we define coHochschild and cocyclic homology theories for chain coalgebras over any commutative ring and prove their naturality with respect to
Interleaving and Gromov-Hausdorff distance
One of the central notions to emerge from the study of persistent homology is that of interleaving distance. It has found recent applications in symplectic and contact geometry, sheaf theory,
Categorification of Gromov-Hausdorff Distance and Interleaving of Functors
One of the central notions to emerge from the study of persistent homology is that of interleaving distance. Here we present a general study of this topic. We define interleaving of categories and of
Wasserstein distance for generalized persistence modules and abelian categories
In persistence theory and practice, measuring distances between modules is central. The Wasserstein distances are the standard family of L^p distances for persistence modules. They are defined in a
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