nLab
categorified Dold-Kan correspondence

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

A categorification of the Dold-Kan correspondence appears in (Dyckerhoff17), where Ab, the category of abelian groups, is replaced by the (∞,2)-category of stable (∞, 1)-categories.

The equivalence between simplicial abelian groups and connective chain complexes from the ordinary Dold-Kan correspondence becomes an equivalence between 2-simplicial stable (∞, 1)-categories and connective chain complexes of stable (∞, 1)-categories.

This should lead to a categorified version of homological algebra and of cohomology, for example, through a categorified version of an Eilenberg-Mac Lane spectrum.

Categorified nerve functor

The nerve in the ordinary Dold-Kan correspondence sends a connective chain complex in an abelian category 𝒜\mathcal{A} to a simplicial object in 𝒜\mathcal{A}. The categorified nerve, 𝒩\mathcal{N}, is playing an analogous role on connective chain complexes of stable (∞, 1)-categories. This nerve unifies various known constructions from algebraic K-theory, in particular, for , 0, 1\mathcal{B}, \mathcal{B}_0, \mathcal{B}_1 all stable (∞, 1)-categories:

References