Abstract
The magnitude homology has a variant called the blurred magnitude homology. It is constructed from a filtration by a distance function on a simplex spanned by the points on a metric space.
From this point of view, it is thought of as a sibling of the persistent homology. The goal of this talk is to introduce a common foundation for this type of parameterized homologies. More precisely, I will explain how they are understood as semantics of ordinary simplicial homology in the intuitionistic logic. As an application, one can make a provably correct implementation of these homologies.
Overview
Persistence Homology is used to detecs the structure of point clouds homologically. Intuitively we’re studying the points with different resolutions. Using this we can study how much a point clout looks like a circle. (Edelsbrunner, Letscher and Zomorodian, 2002)
Blurred magnitude homology
The Blurred magnitude homology is calculated similarly to the normal magnitude but instead of a we use a . The magnitude homology can be reconstructed. Isn’t this just an alternative reformulation of morse theory?
Today we will be studying persistent homology as homology sheaves. We want t reformulate parameterized homology theory in terms of the language of categorical logic.
Why logic? We can use logic to describe complex combinations of open subsets (interpreted as truth variables).
Slogan: The category oof sheaves comes equipped with a functional programming language to define objects and morphisms.