On the inverse limits of finite posets

TL;DR

This paper extends Clader's result by proving that any finite simplicial complex is not only weakly homotopy equivalent but actually homeomorphic to an inverse limit of a sequence of finite posets. The construction relies on barycentric subdivisions to build an inverse system of posets whose limit reproduces the geometric realization of the original complex, with a detailed map $F$ establishing the homeomorphism. It further shows how continuous maps between polyhedra correspond to morphisms of inverse systems, delivering an approximation theorem (Theorem M3) that translates topological questions into finite-poset morphisms. Overall, the work provides a robust finite-poset framework for representing and approximating finite simplicial complexes and their maps, with potential for simplified computations in finite topology.

Abstract

In this paper, we show that any finite simplicial complex is homeomorphic to the inverse limit of a sequence of finite posets, which is an extension of Claders result.

Theorems & Definitions (18)

• Theorem 2.1
• Proposition 2.2
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4
• Theorem 3.5
• Theorem 3.6