# Convergence of Leray Cosheaves for Decorated Mapper Graphs

## TL;DR

This paper introduces decorated mapper graphs as a way to capture richer topological information from data sets, establishing their convergence to Leray cosheaves as covers become finer.

## Contribution

It provides a theoretical foundation showing that decorated mapper graphs approximate Leray cosheaves, linking discrete structures to continuous topological invariants.

## Key findings

- Decorated mapper graphs generalize standard mapper graphs.
- The cellular Leray cosheaf converges to the actual Leray cosheaf with finer covers.
- Theoretical proof of convergence as cover resolution approaches zero.

## Abstract

We introduce decorated mapper graphs as a generalization of mapper graphs capable of capturing more topological information of a data set. A decorated mapper graph can be viewed as a discrete approximation of the cellular Leray cosheaf over the Reeb graph. We establish a theoretical foundation for this construction by showing that the cellular Leray cosheaf with respect to a sequence of covers converges to the actual Leray cosheaf as the resolution of the covers goes to zero.