Combinatorial presentation of multidimensional persistent homology

A multifiltration is a functor indexed by Nr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural Nr-graded R[x1,…,xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and R-modules. We prove in particular that the Nr-graded R[x1,…,xr]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.