On the Mac Lane $Q$-Construction for Exact $\infty$-Categories
Ettore Aldrovandi, Arash Karimi
TL;DR
This work extends McCarthy's cubical Q-construction to exact ∞-categories by developing an ∞-categorical $S_n$-construction and proving its exactness. It then defines a coherent, connective Q-type complex in a fixed stable ∞-category $\mathcal{A}$ via a generalized $Q'$-construction and a cofiber assembly to obtain $Q(F;\mathcal{C})$, for any functor $F:\mathrm{Exact}_{\infty}\to\mathcal{A}$. The authors provide concrete realizations in the stable category of spectra, using the suspension spectrum, the Map functor, and the Eilenberg–Mac Lane functor to produce explicit connective chain complexes in $\mathrm{Sp}$. The constructions yield a robust algebraic model for the stable homology of the K-theory spectrum of exact ∞-categories and demonstrate how higher categorical exactness interacts with stable homotopy theory, via explicit chain maps, cofibrations, and cofibers in a coherent framework.
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.