Simplicial Waldhausen categories and topological $K$-theory

TL;DR

This paper proves that the geometric realization of a certain simplicial Waldhausen category models connective topological K-theory for complex and real numbers, using a generalized Waldhausen comparison theorem.

Contribution

It introduces a novel application of simplicial Waldhausen categories to model topological K-theory as an infinite loop space.

Findings

The geometric realization of the category of bounded chain complexes over complex numbers models connective complex K-theory.

Similarly, over real numbers, it models connective real K-theory.

A generalized Waldhausen comparison theorem is key to the proof.

Abstract

Utilizing simplicial Waldhausen theory, we prove that the geometric realization of the topologized category of bounded chain complexes over complex numbers (resp. real numbers) is an infinite loop space that represents connective complex (resp. real) topological K-theory. The key ingredient in our proof is a generalized Waldhausen comparison theorem.