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

Both Happel and Ladkani proved that, for commutative rings, the quiver $A_n$ is derived equivalent to the diagram generated by $A_n$ where any composition of two consecutive arrows vanishes. We give a purely derivator-theoretic reformulation and proof of this result 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, not only can it be realised as an action of a spectral bimodule, but it also factors through an equivalence arising in the setting of abstract representation theory developed by Groth and \v{S}\v{t}ov\'i\v{c}ek. Moreover, we explain how our result is a derivator-theoretic version of the $\infty$-Dold-Kan correspondence for bounded cochain complexes.Comment: 27 pages, comments are welcome. arXiv admin note: substantial text overlap with arXiv:1409.5003 by other author