nLab categorified Dold-Kan correspondence

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 $\mathcal{A}$ to a simplicial object in $\mathcal{A}$. The categorified nerve, $\mathcal{N}$, 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 $\mathcal{B}, \mathcal{B}_0, \mathcal{B}_1$ all stable (∞, 1)-categories:

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

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

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

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

Related concepts

• Waldhausen S-construction
• real algebraic K-theory

References

• Tobias Dyckerhoff, A categorified Dold-Kan correspondence (arXiv:1710.08356)