Homotopy coherent theorems of Dold-Kan type

TL;DR

This paper generalizes classical homotopy equivalences like the Dold-Kan correspondence to a broad, homotopy coherent setting within $at$-categories, establishing new Morita-equivalences and categorical correspondences.

Contribution

It introduces a large class of homotopy coherent Morita-equivalences of Dold-Kan type in $at$-categories, extending known categorical equivalences to a homotopy coherent framework.

Findings

Establishes homotopy coherent Morita-equivalences in $at$-categories.

Generalizes classical Dold-Kan and related theorems.

Provides an $at$-categorical Dold-Kan correspondence.

Abstract

We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $\Gamma$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.