$\infty$-Dold-Kan correspondence via representation theory

TL;DR

This paper presents a derivator-theoretic reformulation of the $ abla$-Dold-Kan correspondence, establishing a uniform equivalence between homotopy and representation theory that is coefficient-independent and applicable in stable derivators.

Contribution

It introduces a derivator-based proof of the $ abla$-Dold-Kan correspondence, connecting homotopy and representation theory through a spectral bimodule action.

Findings

Provides a coefficient-independent equivalence in stable derivators.

Reformulates the $ abla$-Dold-Kan correspondence derivator-theoretically.

Connects homotopy theory with representation theory via spectral bimodules.

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 $\infty$-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 \v{S}\v{t}ov\'i\v{c}ek.