On topological motives

TL;DR

This paper develops a universal homology theory framework using Eilenberg-Steenrod axioms, representing theories in abelian categories, and describes a universal abelian category for CW-complexes as hieratic modules.

Contribution

It introduces a universal construction for ordinary homology theories applicable to various categories, including a concrete description for CW-complexes.

Findings

Constructed a universal homology theory for any homological structure.

Represented ordinary theories with values in abelian categories.

Described the universal abelian category for CW-complexes as hieratic modules.

Abstract

Following Eilenberg-Steenrod axiomatic approach we construct the universal ordinary homology theory for any homological structure on a given category by representing ordinary theories with values in abelian categories. For a convenient category of spaces we then obtain a universal abelian category which can be actually described for CW-complexes as the category of hieratic modules.