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.
In general, we will prove this by induction on . Start with an observation which is particular to the stable case: a diagram
is a push-out (or pull-back) if the map on cofibers is an equivalence. This can be verified by contemplating the diagram
Each vertical line is a cofiber sequence, and if is an equivalence, a version of the “five-lemma” shows that must be an equivalence, too. (Namely: use the stability hypothesis to realize as cofibers of , etc.)
Let’s now prove the cube lemma, by induction on . For , as we said, it is a fact we take for granted about stable -categories (or as part of the definition). Suppose given . Then we have a decomposition of (for the terminal object) as
Here is the top face, and is the top face minus the endpoint.
This means that we have an analogous decomposition
(Ideally, I would draw this.)
Now the colimit over of is evaluated at the terminal vertex. Let’s call that : so sits right above the terminal vertex of . The colimit over of is the same as the colimit over the bottom copy of of , by cofinality. (How do we prove cofinality? Any map is cofinal: more generally, any right anodyne morphism is.)
So we get
Here refers to the top copy , and refers to the bottom copy .
Let be the terminal vertex of , lying right below . Then to say that is a colimit diagram is to say that one has a pushout diagram:
However, we are working in the stable setting, so this is equivalent to the condition that we have an equivalence on cofibers:
We can identify the objects on both sides of the equation. If we consider as a natural transformation between two maps and , then the above equivalence is saying precisely that the diagram
is a colimit diagram.
In a completely analogous manner, we find that to say that is a limit diagram is equivalent to saying that
is a limit diagram.
Now, by induction on , to say that is a colimit diagram is the same as saying that it is a limit diagram (since this is a cubical diagram of a smaller size). This in turn is the same as saying that is a limit diagram since is a shift of the cofiber. But we have just seen that this is equivalent to saying that is a limit diagram.