$$\infty $$-Dold-Kan Correspondence via Representation Theory

Abstract We give a purely derivator-theoretical reformulation and proof of a classic result of Happel and Ladkani, showing that it occurs uniformly across stable derivators and it is then independent of coefficients. The resulting equivalence provides a bridge between homotopy theory and representation theory: indeed, our result is a derivator-theoretic version of the $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -Dold-Kan correspondence for bounded chain complexes. Moreover, our equivalence can also be realized as an action of a spectral bimodule in the setting of universal tilting theory developed by Groth and Šťovíček.