Realisations of posets and tameness

TL;DR

This paper introduces the concept of realisations of posets, demonstrating their similarities to upper semilattices and their usefulness in studying homological properties and tameness of functors.

Contribution

It defines realisations of posets, explores their properties, and applies them to analyze homological and homotopical aspects of tame functors, extending the theory of upper semilattices.

Findings

Realisations of posets share key features with upper semilattices.

Discrete approximations of realisations effectively capture homological properties.

Koszul complexes are useful for computing Betti diagrams of tame functors.

Abstract

We introduce a construction called realisation which transforms posets into posets. We show that realisations share several key features with upper semilattices. For example, we define local dimensions of points in a poset and show that these numbers for realisations behave in a similar way as they do for upper semilattices. Furthermore, similarly to upper semilattices, realisations have well-behaved discrete approximations which are suitable for capturing homological properties of functors indexed by them. These discretisations are convenient and effective for describing tameness of functors. Homotopical and homological properties of tame functors, particularly those indexed by realisations, are discussed, with emphasis on the use of Koszul complexes to compute Betti diagrams of minimal free resolutions of tame functors indexed by upper semilattices and realisations.