3 Basics of Goodwillie calculus

The calculation of the derivative will use the explicit description of $P_k F$ 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 $F$ by a $k$ -excisive functor, we force $F$ 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 $\mathcal{X}: \mathbb {P}(S) \to \mathcal{C}$ , where $|S| = k + 1$ and $\mathcal{X}(\emptyset ) = X$ . We then replace $F(X)$ by $\lim _{\emptyset \neq T \subseteq S}F(\mathcal{X}(T))$ (which would equal $F(X)$ if $F$ were $k$ -excisive). We repeat this procedure and hope we eventually end up with something $k$ -excisive.

To define such an $\mathcal{X}$ , we need to specify $\mathcal{X}|_{\mathbb {P}_{=1}(S)}$ , since we know $\mathcal{X}(\emptyset ) = X$ and everything is the left Kan extension of this. Since $X$ is the only thing we know, the only thing we can do is to set $\mathcal{X}(\{ s\} ) = *$ (placing $X$ there does no good).

We define $C_{(-)}(X): \mathbb {P}(S) \to \mathcal{C}$ to be the unique strongly coCartesian diagram such that $C_{\emptyset }(X) = X$ and $C_{\{ s\} } = *$ . Concretely, $C_T(X)$ is the colimit of the diagram

$\begin{useimager} \[ \begin{tikzcd} & & X \ar[dll] \ar[dl] \ar[d] \ar[dr] \ar[drr]\\ * & * & \cdots & *& * \end{tikzcd} \] \end{useimager}$

where the vertices are indexed by $T$ .

We define

T_k F(X) = \lim _{\emptyset \neq T \subseteq S}F(C_T(X)).

There is then a natural transformation $t_F\colon F \to T_k F$ .

Theorem 19

If $F\colon \mathcal{C} \to \mathcal{D}$ is any functor, define $P_k F\colon \mathcal{C} \to \mathcal{D}$ as the sequential colimit

$\begin{useimager} \[ \begin{tikzcd} F \ar[r, "t_F"] & T_k F \ar[r, "t_{T_k F}"] & T_k T_k F \ar[r] & \cdots \end{tikzcd} \] \end{useimager}$

Then $P_k F$ is $k$ -excisive and the natural map $\theta _F: F \to P_k F$ is the universal natural transformation to such a functor.

Proof

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We first figure out what we have to prove. Of course, we have to prove that

P_k F

k

-excisive. It turns out this is the only non-trivial thing to prove.

In the $1$ -categorical case, to prove the universal property, we first need to know that $\theta _F$ is an equivalence if $F$ is already $k$ -excisive, which is clear in our case. This means if we have a map $\alpha : F \to G$ where $G$ is $k$ -excisive, then we have a diagram

$\begin{useimager} \[ \begin{tikzcd} F \ar[r, "\alpha"] \ar[d, "\theta_F"] & G \ar[d, "\theta_G"]\\ P_k F \ar[r, "P_k \alpha"] & P_k G \end{tikzcd} \] \end{useimager}$

Since $\theta _G$ is an equivalence, $P_k\alpha$ lifts to a map to $G$ that makes the diagram commute.

To show that the extension to $P_k F$ is unique, suppose we have two extensions $\tilde{\alpha }, \tilde{\alpha }'$

$\begin{useimager} \[ \begin{tikzcd} F \ar[r, "\alpha"] \ar[d, "\theta_F"] & G\\ P_k F \ar[ur, dashed] \end{tikzcd} \] \end{useimager}$

Applying $P_k$ to the whole diagram, we know that $P_k(\tilde{\alpha }) \circ P_k(\theta _F) = P_k(\tilde{\alpha }') \circ P_k(\theta _F)$ . If $P_k(\theta _F)$ were an equivalence, then this implies $P_k(\tilde{\alpha }) = P_k(\tilde{\alpha }')$ . Since $P_k$ essentially acts as the identity on $P_k F$ and $G$ , this implies $\tilde{\alpha } = \tilde{\alpha }'$ .

In the $\infty$ -categorical case, HTT 5.2.7.4 implies these two conditions are also sufficient.

As we said, the first condition is immediate from construction, and to prove that $P_k(\theta _F)$ is an equivalence, it suffices to show that $P_k(t_F: F \to T_k F)$ 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 $P_k F$ is in fact $k$ -excisive.

Claim

Let $\mathcal{X}\colon \mathbb {P}(S) \to \mathcal{C}$ be a strongly coCartesian $k$ -cube. Then the canonical map $\theta _F\colon F(\mathcal{X}) \to (T_kF)(\mathcal{X})$ factors through a Cartesian $k$ -cube in $\mathcal{D}$ .

If this were true, then

P_k F(\mathcal{X})

would be the sequential colimit of Cartesian cubes, hence Cartesian (since we assumed finite limits commute with sequential colimits).

The Cartesian $k$ -cube $Y: \mathbb {P}(S) \to \mathcal{D}$ we seek admits a very simple description. Indeed, we simply take $F(\mathcal{X})$ and replace the $\emptyset$ vertex with the pullback of the rest of the diagram. This is then by construction a Cartesian cube, and there is a canonical map $F(\mathcal{X}) \to Y$ by the universal property.

To show that $\theta _F$ factors through this map, we need to use a funny description of $Y$ , and this description uses the fact that $\mathcal{X}$ is strongly coCartesian.

Fix a $T \subseteq S$ . We define $\mathcal{X}_T: \mathbb {P}(S) \to \mathcal{C}$ by setting $\mathcal{X}_T(I)$ to be the pushout of the diagram

$\begin{useimager} \[ \begin{tikzcd}[column sep=small] & & \mathcal{X}(I) \ar[dll] \ar[dl] \ar[d] \ar[dr] \ar[drr]\\ \mathcal{X}(I \cup \{s_1\}) & \mathcal{X}(I \cup \{s_2\}) & \cdots & \mathcal{X}(I \cup \{s_{k - 1}\}) & \mathcal{X}(I \cup \{s_k\}) \end{tikzcd} \] \end{useimager}$

where $T = \{ s_1, \ldots , s_k\}$ .

We make the following observations:

$\mathcal{X}_{\emptyset }(I) = \mathcal{X}(I)$ .
If $\mathcal{X}$ is strongly coCartesian, then essentially by definition, $\mathcal{X}_T(I) = \mathcal{X}(I \cup T)$ .
If we replace the bottom vertices with $*$ , then this gives $C_T(\mathcal{X}(I))$ . So there is a canonical map $\mathcal{X}_T(I) \to C_T(\mathcal{X}(I))$ .

The last fact gives us a map

F(\mathcal{X}_U(T)) \to F(C_U(\mathcal{X}(T)))

natural in $U$ and $T$ . These assemble to give maps

F(\mathcal{X}(I)) \cong F(\mathcal{X}_{\emptyset }(I)) \to \lim _{\emptyset \not= T \subseteq S} F(\mathcal{X}_T(I)) \to \lim _{\emptyset \not= T \subseteq S} F(C_T(\mathcal{X}(I))) \cong T_kF(\mathcal{X})(I)

It remains to show that the middle object is equal to $Y$ . But it is not difficult to use the second fact to see that

\lim _{\emptyset \neq T \subseteq S}F(\mathcal{X}_T(I)) = \begin{cases} \lim _{\emptyset \neq T \subseteq S}F(\mathcal{X}(T)) & I = \emptyset \\ F(\mathcal{X}(I)) & I \neq \emptyset \end{cases}.

Proof

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