Combinatorial presentation of multidimensional persistent homology

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 $R[x_1,\dots ,x_r]$-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 $R[x_1,\dots ,x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.

arXiv comment21 pages, 3 figures