Algebraic Topology 2.0

TL;DR

This paper proposes a new algebraic framework for topology by introducing a (-1)-dimensional space as a join unit, aligning topological spaces with simplicial complexes and algebraic categories.

Contribution

It introduces a novel algebraic structure with a (-1)-dimensional space as a join unit, rectifying the role of the empty space in topology and algebraizing Hausdorff's definition.

Findings

Aligns topological spaces with simplicial complexes

Provides an algebraic reformulation of topology

Implications for algebraic topological methods

Abstract

The efficiency of contemporary algebraic topology is not optimal since the category of topological spaces can be made more algebraic by introducing a profoundly new (-1)-dimensional topological space as a topological join unit. Thereby synchronizing the category of topological spaces with the structures within the contemporary category of simplicial complexes as well as with the structures within the algebraic categories. In the category of topological spaces, the empty space {\O} has since long been given the role as a join unit - ad-hoc though. Since it is {{\O}}, not {\O}, that is the join unit within the category of simplicial complexes, the role of {\O} within general topology has to be rectified. This article presents an algebraization of Hausdorff's century old definition of the category of topological spaces as well as some useful algebraic topological consequences thereof.