# Combinatorial presentation of multidimensional persistent homology

arXiv: https://arxiv.org/abs/1409.7936 · HTML: https://mlsanity.com/paper/1409.7936/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2014.

## Built on by (most cited first, 2 of 2)

- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

A multifiltration is a functor indexed by mathbbNr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural mathbbNr-graded R[x1,ldots, xr]-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 mathbbNr-graded R[x1,ldots, xr]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.