nLab infinity-Dold-Kan correspondence



Higher algebra

Stable Homotopy theory

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories





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


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,



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.