On the Mac Lane $Q$-Construction for Exact $\infty$-Categories

TL;DR

This paper generalizes Mac Lane's $Q$-construction to exact $ abla$-categories within the framework of $ abla$-categories, providing a new method to compute stable homology in a broad categorical context.

Contribution

It extends McCarthy's stabilization construction to exact $ abla$-categories, introducing a coherent chain complex framework for stable homology calculations.

Findings

Constructed a coherent chain complex in stable $ abla$-categories.

Generalized Mac Lane's cubical $Q$-complex to exact $ abla$-categories.

Provided a new tool for computing stable homology in higher categorical settings.

Abstract

We extend McCarthy's stabilization construction to exact $\infty$-categories. This is achieved by constructing, for any functor from exact $\infty$-categories to a fixed stable $\infty$-category $\mathcal{A}$, a coherent chain complex in $\mathcal{A}$ that is an immediate generalization of Mac Lane's cubical $Q$-complex computing the stable homology of abelian groups.