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[x(1),...x(r)]-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[x(1),...,x(r)]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals. (C) 2016 Elsevier B.V. All rights reserved.