Combinatorial presentation of multidimensional persistent homology

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.