Table of Contents

Introduction

Two Setups in TDA

Sensing continuous Functions

Point Clouds of Shapes

Distances on Functions

Absolute Distance of Functions

Relative Distance of Functions

Lower Bounds

Further References

Reeb Graphs

Join Trees

Interlude on Precosheaves

Monoidal Posets for 1D-Interleavings

Interleaving Reeb Graphs

Interleaving Precosheaves on D

Interleavings in D-Categories

Properties of Interleavings

Homomorphisms of D-Categories

Positive Persistence-Enhancements

Complete Persistence-Enhancements

Some Equivalences

Equivalence to descending Precosheaf

A second type of Interleavings

A negative Enhancement for Join Trees

Equality of Interleaving Distances

Relation to the Join Precosheaf

Appendix

Constructible Spaces over the Reals

Graphs over the Reals

A Skeleton for the Epigraph

References