Grothendieck groups and a categorification of additive invariants

TL;DR

This paper develops a categorical framework to transform additive homology classes into natural transformations, providing a new perspective on characteristic classes of singular varieties through Grothendieck groups.

Contribution

It introduces a generalized relative Grothendieck group and a categorification method for additive invariants, advancing the understanding of characteristic classes in algebraic topology.

Findings

Established a categorical approach to homology class categorification

Developed a theory of characteristic homology classes for singular varieties

Provided a generalized framework connecting Grothendieck groups and additive invariants

Abstract

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e., a "categorification" of it. In a general categorical set-up we introduce a generalized relative Grothendieck group from a cospan of functors of categories and also consider a categorification of additive invariants on objects. As an example, we obtain a general theory of characteristic homology classes of singular varieties.