The higher-categorical Dold-Kan correspondence II

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.

 

The cube lemma in a stable infinity-category

Let be a stable -category. In the previous post, we needed to consider cubical diagrams

These diagrams come with an initial object and a terminal object: in fact, they are the cones on smaller diagrams. For instance, is the nerve of all subsets of , which is the cone on the nerve of all nonempty subsets of , and also the cone on the nerve of all proper subsets of . So it makes sense to talk about whether is a limit diagram, or whether is a colimit diagram.

The main result is:

Proposition 11 (Cube lemma) If is stable, then is a limit diagram if and only if it is a colimit diagram.

 

When , this is automatic: any diagram is a limit diagram if and only if it is an equivalence, and ditto for colimit diagrams. When , this is particular to the stable case: a square is a push-out if and only if it is a pull-back. We took this as more or less axiomatic, though it can be deduced from much weaker axioms, as in “Higher Algebra.”

The purpose of this post is to work through the proof of the “cube lemma.” This is more or less a piece of an attempt to work through Lurie’s version of the Dold-Kan correspondence. I’ve been doing it in a fair bit of detail for my own benefit — this means that the posts are a little more detailed than usual. In any event, the present post should stand alone from the others.

 

The higher-categorical Dold-Kan correspondence

Let be a stable -category. For us, this means that we have three important properties:

1. admits finite limits and colimits.
2. has a zero object: that is, the initial object is also final.
3. A square in is a pull-back if and only if it is a push-out.

This is equivalent to the stability of . Actually, stability is usually defined using slightly weaker conditions, and then it takes a little work to show that these stronger ones are implied. We’ll just work with these. Stability can be thought of as a higher-categorical version of being triangulated; a general stable -category has many of the properties (in a higher categorical sense) of the homotopy category of spectra, or the (classical) derived category.

Our goal is to show that in this case, we have an equivalence of -categories

between simplicial objects in and filtered (nonnegatively) objects in . The idea here is that the geometric realization of a simplicial object comes with a canonical filtration, given by geometric realizing the -truncations for each . This is going to give the associated filtered object. (We don’t know that the geometric realization exists, but the realizations of the truncations will.)

We will actually prove something stronger: for each , there is an equivalence

where is the subcategory of elements . In other words, -truncated simplicial objects are the same as -filtered objects of . (Note that, as a simplicial set, the nerve of is .) These equivalences will be compatible, and taking inverse limits will give the Dold-Kan correspondence.