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infinity-Dold-Kan correspondence

Contents

Context

Higher algebra

Stable Homotopy theory

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Statement

Proposition

Let 𝒞\mathcal{C} be a stable (∞,1)-category. Then the (∞,1)-categories of non-negatively graded sequences in CC is equivalent to the (∞,1)-category of simplicial objects in an (∞,1)-category in 𝒞\mathcal{C}

Fun(N( 0),C)Fun(N(Δ) op,C). Fun(N(\mathbb{Z}_{\geq 0}), C) \simeq Fun(N(\Delta)^{op}, C) \,.

Under this equivalence, a simplicial object X X_\bullet is sent to the sequence of geometric realizations ((∞,1)-colimits) of its simplicial skeleta

|sk 0X ||sk 1X ||sk 2X |. {\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots \,.

This constitutes a filtering on the geometric realization of X X_\bullet itself

|X |lim n|sk nX |. {\vert X_\bullet \vert} \simeq \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert} \,.

(Higher Algebra, theorem 1.2.4.1)

Remark

Given a simplicial object X X_\bullet in a stable (∞,1)-category 𝒞\mathcal{C}, its image in the triangulated homotopy category Ho(𝒞)Ho(\mathcal{C}) is identified by the ordinary Dold-Kan correspondence with a chain complex. On the other hand, by the discussion at spectral sequence of a filtered stable homotopy type in the section Filtered objects and their chain complexes, the skelton sequence sk X sk_\bullet X_\bullet also induces a chain complex. These are naturally isomorphic.

In particular therefore first page of the spectral sequence of a filtered stable homotopy type associated with the simplicial skeleton filtration consists of the Moore complexes of the simplicial objects π q(X )𝒜 Δ op\pi_q(X_\bullet) \in \mathcal{A}^{\Delta^{op}}

E 1 ,qN(π qX ). E_1^{\bullet,q} \simeq N(\pi_q X_\bullet) \,.

(Higher Algebra, remark 1.2.4.3, 1.2.4.4)

Applications

References

This infinity-Dold-Kan correspondence is theorem 12.8, p. 50 of

later absorbed in

Last revised on April 28, 2014 at 09:25:05. See the history of this page for a list of all contributions to it.