2 The Goodwillie filtration

Most of the material here about Goodwillie calculus are due to Goodwillie, and a “modern” account can be found in Chapter 6 of Higher Algebra. We will provide references to Higher Algebra when we omit proofs of theorems.

Goodwillie calculus is a method of approximating functors $F\colon \mathcal{C} \to \mathcal{V}$ by a sequence of “polynomial” functors. For the theory to work out, we have to make the following assumptions:

$\mathcal{C}$ and $\mathcal{D}$ are pointed.
$\mathcal{C}$ has finite colimits.
$\mathcal{D}$ has finite limits and sequential colimits which commute past each other.

The first hypothesis is not necessary, but our examples are all of this form. Moreover, the theorems we seek are usually first proven for the pointed case and then transferred to unpointed case, so we might as well focus on the pointed case only.

Before we go into the definition of a “polynomial functor”, we give the special case of a linear functor. We should think of linear functors as functors like homology, which satisfy Mayer–Vietoris.

Example 3

A linear functor is a functor that sends pushout squares to pullback squares.

Example 4

If $\mathcal{D}$ is stable, then pushouts are the same as pullbacks. So any colimit preserving functor is in particular linear.

A $k$ -excisive functor, which should be thought of as a polynomial functor of degree $\leq k$ , satisfies a higher-dimensional analogue of this axiom involving higher-dimensional cubes.

Definition 5

Let $S$ be a finite set with $|S| = k$ . A $k$ -cube in $\mathcal{C}$ is a functor $\mathbb {P}(S) \to \mathcal{C}$ , where $\mathbb {P}(S)$ is the power set of $S$ , considered as a poset category.

Example 6

$\mathbb {P}(\{ 0, 1\} )$ and $\mathbb {P}(\{ 0, 1, 2\} )$ can be depicted as follows

$\begin{useimager} \[ \begin{tikzcd} \emptyset \ar[r] \ar[d] & \{0\} \ar[d]\\ \{1\} \ar[r] & \{0, 1\} \end{tikzcd},\quad \begin{tikzcd}[row sep=small, column sep=small] & \emptyset \ar[rr] \ar[ld] \ar[dd] & & \{0\} \ar[ld] \ar[dd]\\ \{2\} \ar[dd] \ar[rr, crossing over] & & \{0, 2\}\\ & \{1\} \ar[ld] \ar[rr]& & \{0, 1\} \ar[ld]\\ \{1, 2\} \ar[rr] & & \{1, 2, 3\} \ar[from=uu, crossing over] \end{tikzcd} \] \end{useimager}$

Definition 7

A $k$ -cube is (co)Cartesian if it is a (co)limit diagram.

It is strongly coCaretsian if it is the left Kan extension of the restriction to $\mathbb {P}_{\leq 1}(S)$ , the sub-poset of subsets of $S$ of cardinality at most $1$ (strongly Cartesian is defined similarly).

Definition 8

A functor is $k$ -excisive if it sends strongly coCartesian $(k + 1)$ -cubes to Cartesian $(k + 1)$ -cubes.

It is not too hard to see that

Lemma 9 ([HA 6.1.1.14])

If $k' \geq k$ , then every $k$ -excisive functor is also $k'$ -excisive.

Theorem 10 (Goodwillie)

For each $k$ , there is a universal approximation $P_k F$ that is polynomial of degree $\leq k$ and a natural transformation $F \to P_k F$ universal amongst natural transformations to polynomial functors of degree $\leq k$ . These assemble to give a “Taylor tower”

$\begin{useimager} \[ \begin{tikzcd} & \vdots \ar[d]\\ & P_2 F \ar[d]\\ & P_1 F \ar[d]\\ F \ar[r] \ar[ur] \ar[uur] & P_0 F \end{tikzcd} \] \end{useimager}$

We will prove this theorem in the next section. Note that the theorem does not claim whether the tower actually converges to

F

or not. This is an issue that has to be addressed separately.

The goal of this talk is to understand the polynomial approximations $P_k \int _{M_*}(-)$ .

Example 11

A $1$ -cube is just a morphism. It is always strongly coCartesian and is Cartesian iff it is an equivalence. So $0$ -excisive functors are constant functors, and $P_0 F(X) = F(*)$ .

Example 12

A $2$ -cube is a square, and being $1$ -excisive means sending pushouts to pullbacks, as promised.

We will prove the theorem by providing an explicit model for $P_k F$ . We can give an indication of what this explicit model looks like in the case where $F$ is reduced, i.e. $F(*) = *$ .

Example 13

If $F$ is reduced, then

P_1 F(X) = \operatorname*{colim}_{n \to \infty } \Omega ^n F(\Sigma ^n X).

The fiber of the map $P_k F \to P_{k - 1} F$ is called the $k$ th derivative of $F$ (at $*$ ). It is $k$ -homogeneous:

Definition 14

A functor $F$ is $k$ -homogeneous if $F = P_k F$ and $P_{k - 1}F = *$ .

In general, the polynomial approximations $P_k F$ are rather difficult to understand, but often times, the derivatives admit rather explicit descriptions. This is the case, for example, when $F$ is the identity map from the category of spaces to itself. Our actual goal is to understand the derivatives of the functor $\int _{M_*} (-)\colon \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})\to \mathcal{V}$ for a fixed $M_*$ .

While polynomial functors of degree $k$ can be pretty complicated, $k$ -homogeneous functors are not.

Theorem 15 ([HA 6.1.4.14, HA 6.1.2.9])

An $k$ -homogeneous functor $F\colon \mathcal{C} \to \mathcal{D}$ is uniquely of the form

F(X) = G(\Sigma ^\infty X, \cdots , \Sigma ^\infty X)_{\Sigma _k},

where $\Sigma ^\infty \colon \mathcal{C} \to \operatorname{Sp}(\mathcal{C})$ is the stabilization functor and $G\colon \operatorname{Sp}(\mathcal{C})^k \to \mathcal{D}$ is symmetric in the $k$ variables and $1$ -homogeneous in each variable.

Conversely, every functor of this form is $k$ -homogeneous.

This is an actual, non-trivial theorem to prove, and is a crucial ingredient in our identification of the derivatives. We will, however, not prove it.

Example 16

$F$ is reduced iff $P_0 F$ is trivial. So if $F$ is reduced, then $P_1 F$ is also the first derivative. Our explicit formula above shows that it is indeed of the form we claimed.

In the case of interest, the map $\Sigma ^\infty \colon \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})\to \operatorname{Sp}(\mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V}))$ is exactly the functor $L$ we had previously. So the $k$ th derivative of $\int _{M_*}(-)$ is determined by its values on free algebras. The main theorem we want to prove is

Theorem 17

The $k$ th derivative of $\int _{M_*}(-)$ is

A \mapsto \operatorname{Conf}_k^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _k \wr \mathrm{O}(n)} L(A)^{\otimes k}.

Note that by the theorem above, this functor is indeed

k

-homogeneous, with

G(X_1, \ldots , X_k) = \operatorname{Conf}_k^{{\mathrm{fr}}}(M_*) \bigotimes _{\mathrm{O}(n)^k} (X_1 \otimes \cdots \otimes X_k),

which is cocontinuous and in particular linear in each variable. We will prove this theorem by explicitly calculating the polynomial approximations evaluated on free algebras:

Example 18

In the case where $A = \mathbb {F}^{\mathrm{aug}}(V)$ , we have

P_k\int _{M_*} \mathbb {F}^{\mathrm{aug}}(V) = \bigoplus _{0 \leq i \leq k} \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i}.