Combinatorial presentation of multidimensional persistent homology

TL;DR

This paper provides a combinatorial description of multidimensional persistent homology by characterizing the module structure of multifiltrations of simplicial complexes, revealing they are sums of monomial ideals.

Contribution

It explicitly describes the natural graded module structure of multifiltration homology and characterizes modules arising from multifiltrations of sets as sums of monomial ideals.

Findings

Modules from multifiltrations of sets are sums of monomial ideals.

Provides an explicit combinatorial description of multidimensional persistent homology.

Characterizes the algebraic structure of homology modules in this context.

Abstract

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.