The past few posts have been focused on a discussion of Lurie’s version of the Dold-Kan correspondence in stable -categories. I’ve made these posts more detailed than usual: while I’ve been trying to treat such category theory as a black box on this blog, it should be interesting (at least for me) to see how the machines work beneath the surface, in some specific examples. In previous posts, I stated the result, and described an important lemma on the structure of simplicial objects in a stable -category, which depended on the combinatorics of cubes.

The goal of this post is to (finally) prove the result, an equivalence of -categories

valid for any stable -category . As before, the intuition behind this version of the Dold-Kan correspondence is that a simplicial object determines a filtered object by taking successive geometric realizations of the -truncations. The fact that one can go in reverse, and reconstruct the simplicial object from the geometric realizations of the truncations, is specific to the stable case. (more…)