Differential calculus for free algebra and constrained homology

Abstract

In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via differential calculus on discrete sets. In this paper, we study the differential calculus for free algebra and subsequently study the homomorphisms of constrained (co)homology induced by inclusions of simplicial complexes and independence hypergraphs. We apply the differential calculus to hypergraphs. We realize simplicial complexes and independence hypergraphs as certain invariant traces of hypergraphs. As an application, we give the constrained persistent (co)homology for filtrations of simplicial complexes and filtrations of independence hypergraphs.