Let ${\mathcal{C}}$ be a stable ${\infty}$-category. In the previous post, we needed to consider cubical diagrams

$\displaystyle f: (\Delta^1)^{n+1} \rightarrow \mathcal{C}.$

These diagrams come with an initial object and a terminal object: in fact, they are the cones on smaller diagrams. For instance, ${(\Delta^1)^{n+1}}$ is the nerve of all subsets of ${[n]}$, which is the cone on the nerve of all nonempty subsets of ${[n]}$, and also the cone on the nerve of all proper subsets of ${[n]}$. So it makes sense to talk about whether ${f}$ is a limit diagram, or whether ${f}$ is a colimit diagram.

The main result is:

Proposition 11 (Cube lemma) If ${\mathcal{C}}$ is stable, then ${f: (\Delta^1)^{n+1} \rightarrow \mathcal{C}}$ is a limit diagram if and only if it is a colimit diagram.

When ${n = 0}$, this is automatic: any diagram ${\Delta^1 \rightarrow \mathcal{C}}$ is a limit diagram if and only if it is an equivalence, and ditto for colimit diagrams. When ${n = 1}$, 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. (more…)