On preservation of relative resolutions for poset representations

TL;DR

This paper investigates Galois connections in poset representations, focusing on interior systems and their impact on persistence modules, especially regarding interval decomposability and resolutions in multiparameter persistent homology.

Contribution

It introduces aligned interior systems and proves that induction and contraction functors preserve interval-decomposability, advancing understanding of persistence module structures.

Findings

Induction and contraction functors preserve interval-decomposability.

Global dimensions of interval resolutions are computed for specific finite posets.

A new framework for analyzing persistence modules via Galois connections is established.

Abstract

The concept of Galois connections (i.e., adjoint pairs between posets) is ubiquitous in mathematics. In representation theory, it is interesting because it naturally induces the adjoint quadruple between the categories of persistence modules (representations) of the posets via Kan extensions. One of central subjects in multiparameter persistent homology analysis is to understand structures of persistence modules. In this paper, we mainly study a class of Galois connections whose left adjoint is the canonical inclusion of a full subposet. We refer to such a subposet as an interior system, with its corresponding right adjoint given by the floor function. In the induced adjoint quadruple, we call the left Kan extension along its floor function the contraction functor. From its construction, it is left adjoint to the induction functor. Under this setting, we firstly prove that this adjoint pair gives an adjoint pair between finitely presentable persistence modules. Moreover, we introduce a special class of interior systems called aligned interior systems, and prove that both induction and contraction functors over them preserve interval-decomposability of modules. Then, we use them to analyze interval covers and resolutions. We also compute interval resolution global dimensions for certain classes of finite posets.