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categorified Dold-Kan correspondence

Contents

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:

  • For [1]\mathcal{B}[1], the chain complex concentrated in degree 1, 𝒩([1])\mathcal{N}(\mathcal{B}[1]) is a (,1)(\infty,1)-version of the Waldhausen S-construction.

  • For f: 1 0f: \mathcal{B}_1 \to \mathcal{B}_0, then applying 𝒩\mathcal{N} to the corresponding chain complex concentrated in degrees 00 and 11 is an (,1)(\infty,1)-version of the relative Waldhausen S-construction.

  • For the chain complex concentrated in degree 22, 𝒩([2])\mathcal{N}(\mathcal{B}[2]) is an (,1)(\infty,1)-version of Hesselholt-Madsen’s S 2,1S^{2,1}_{\bullet}-construction. (See the reference at real algebraic K-theory.)

  • For the chain complex concentrated in degree kk, 𝒩([k])\mathcal{N}(\mathcal{B}[k]) can be seen as a categorification of the Eilenberg-Mac Lane space N(B[k])N(B[k]).

References

Last revised on October 25, 2017 at 09:51:44. See the history of this page for a list of all contributions to it.