nLab infinity-Dold-Kan correspondence

Statement

Proposition

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

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

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

$${\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots \,.$$

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

$${\vert X_\bullet \vert} \simeq \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert} \,.$$

(Higher Algebra, theorem 1.2.4.1)

Remark

Given a simplicial object $X_\bullet$ in a stable (∞,1)-category $\mathcal{C}$, its image in the triangulated homotopy category $Ho(\mathcal{C})$ 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 $sk_\bullet X_\bullet$ 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 $\pi_q(X_\bullet) \in \mathcal{A}^{\Delta^{op}}$

$$E_1^{\bullet,q} \simeq N(\pi_q X_\bullet) \,.$$

(Higher Algebra, remark 1.2.4.3, 1.2.4.4)

Applications

• J-homomorphism and chromatic homotopy

Related concepts

• spectral sequence of a simplicial stable homotopy type

• stable Dold-Kan correspondence

References

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

• Jacob Lurie, Stable ∞-Categories

later absorbed in

• Jacob Lurie, section 1.2.4 of Higher Algebra