A theorem of Retakh for exact $\infty$-categories and higher extension functors

TL;DR

This paper generalizes Retakh's theorem to exact $ abla$-categories by defining extension $ abla$-categories as bifibrations, establishing their invariance, and relating their homotopy groups to higher extension groups.

Contribution

It introduces extension $ abla$-categories for exact $ abla$-categories via bifibrations, proving they form an $oldsymbol{ ext{ extOmega}}$-spectrum and connect to higher extension groups.

Findings

Extension $ abla$-categories form an $oldsymbol{ ext{ extOmega}}$-spectrum.

Homotopy groups of extension $ abla$-categories are isomorphic to higher extension groups.

Extension $ abla$-categories are invariant under passing to the stable hull.

Abstract

We define extension $\infty$-categories for exact $\infty$-categories in terms of bifibrations. Extension $\infty$-categories are invariant when passing to the stable hull, and consequently we show that they form an $\Omega$-spectrum, generalizing a theorem of Retakh. Finally, we show that the homotopy groups of extension $\infty$-categories are naturally isomorphic to the higher extension groups of the extriangulated category given by the homotopy category.