Discrete homotopy and homology theories for finite posets

TL;DR

This paper develops discrete homotopy and homology theories for finite posets, establishing their relationship with classical theories and introducing a discrete Hurewicz map to connect homology and homotopy.

Contribution

It introduces a novel discrete homotopy and homology framework for finite posets, showing their isomorphism with classical theories and defining a discrete Hurewicz map.

Findings

Discrete and classical homotopy groups of finite posets are always isomorphic.

The paper establishes a discrete analogue of the Hurewicz map linking homology and homotopy.

The new theories provide a combinatorial approach to topological properties of finite posets.

Abstract

This paper presents a discrete homotopy theory and a discrete homology theory for finite posets. In particular, the discrete and classical homotopy groups of finite posets are always isomorphic. Moreover, this discrete homology theory is related to the discrete homotopy theory through a discrete analogue of the Hurewicz map.