3 Basics of Goodwillie calculus
The calculation of the derivative will use the explicit description of in the proof of its existence, which is the focus of this section.
We keep the assumptions of the previous section. To construct the universal approximation of by a -excisive functor, we force to send certain strongly coCartesian diagrams to Cartesian diagrams, and it will turn out that this is enough.
More specifically, we find some strongly coCarteisan diagrams , where and . We then replace by (which would equal if were -excisive). We repeat this procedure and hope we eventually end up with something -excisive.
To define such an , we need to specify , since we know and everything is the left Kan extension of this. Since is the only thing we know, the only thing we can do is to set (placing there does no good).
We define to be the unique strongly coCartesian diagram such that and . Concretely, is the colimit of the diagram
where the vertices are indexed by .
There is then a natural transformation .
If is any functor, define as the sequential colimit
Then is -excisive and the natural map is the universal natural transformation to such a functor.
In the -categorical case, to prove the universal property, we first need to know that is an equivalence if is already -excisive, which is clear in our case. This means if we have a map where is -excisive, then we have a diagram
Since is an equivalence, lifts to a map to that makes the diagram commute.
To show that the extension to is unique, suppose we have two extensions
Applying to the whole diagram, we know that . If were an equivalence, then this implies . Since essentially acts as the identity on and , this implies .
In the -categorical case, HTT 18.104.22.168 implies these two conditions are also sufficient.
As we said, the first condition is immediate from construction, and to prove that is an equivalence, it suffices to show that is an equivalence. But this is clear, since we are just shifting the sequential colimit.
So all we have to do is to show that is in fact -excisive.
Let be a strongly coCartesian -cube. Then the canonical map factors through a Cartesian -cube in .
The Cartesian -cube we seek admits a very simple description. Indeed, we simply take and replace the vertex with the pullback of the rest of the diagram. This is then by construction a Cartesian cube, and there is a canonical map by the universal property.
To show that factors through this map, we need to use a funny description of , and this description uses the fact that is strongly coCartesian.
Fix a . We define by setting to be the pushout of the diagram
We make the following observations:
If is strongly coCartesian, then essentially by definition, .
If we replace the bottom vertices with , then this gives . So there is a canonical map .
The last fact gives us a map
natural in and . These assemble to give maps
It remains to show that the middle object is equal to . But it is not difficult to use the second fact to see that