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.
Categories: math.CT math.AT