Algebraic $K_0$ for unpointed homotopy Categories

TL;DR

This paper extends algebraic K-theory, specifically K_0, to unpointed homotopy categories using Grothendieck heaps, broadening the scope of K-theoretic studies in unpointed topological spaces.

Contribution

It introduces Grothendieck heaps for unpointed Waldhausen and stable ∞-categories, enabling the extension of K_0 to unpointed homotopy categories.

Findings

Defined Grothendieck heaps for unpointed categories

Extended K_0 to unpointed topological spaces

Provided new tools for unpointed homotopy theory

Abstract

We introduce the notion of Grothendieck heaps for unpointed Waldhausen categories and unpointed stable $\infty$-categories. This allows an extension of the studies of $\mathrm{K}_0$ to the homotopy category of unpointed topological spaces.