The cylinder of a relation and generalized versions of the Nerve Theorem

TL;DR

This paper introduces the concept of a cylinder of a relation in posets, generalizes Quillen's Theorem A, and extends the Nerve Theorem to covers with non-contractible intersections, broadening topological tools.

Contribution

It defines the cylinder of a relation in posets and generalizes key theorems, providing new formulations applicable to more complex covers.

Findings

Established a local-to-global result for relations.

Generalized Quillen's Theorem A for order-preserving maps.

Derived new formulations of the Nerve Theorem for posets and simplicial complexes.

Abstract

We introduce the notion of cylinder of a relation in the context of posets, extending the construction of the mapping cylinder. We establish a local-to-global result for relations, generalizing Quillen's Theorem A for order preserving maps, and derive novel formulations of the classical Nerve Theorem for posets and simplicial complexes, suitable for covers with not necessarily contractible intersections.