nLab diamond principle

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Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Representation theory

Contents

Idea

In zig-zag persistent homology-theory, the diamond principle (Carlsson & de Silva 2010, Sec. 5.2) states that persistence diagrams of pairs of zigzag persistence modules are bijective if the modules coincide away from two consecutive edges where their difference forms an “exact diamond”.

Applied specifically to persistence modules given by the homology groups of zig-zags of topological spaces, the diamond principle is an incarnation of the Mayer-Vietoris sequence in algebraic topology.

In this case it says that the two canonical zigzag persistence modules associated with a filtered topological space (one built from union-zigzags, the other from intersection-zigzags) in fact yield the same persistence diagrams.

Notice how the diamond principle is a result genuinely about persistent invariants, in that the two persistence modules forming an exact diamond do not have the same shape as diagrams and hence are non-comparable when regarded, say, as quiver representations (since they are objects of distinct categories).

References

The diamond principle originates in

Review:

Last revised on May 23, 2022 at 07:09:48. See the history of this page for a list of all contributions to it.