Chromotopy

Homogeneous functors

In a previous post, Eric discussed a bit about the Goodwillie calculus, and mentioned some adorable-looking formulas that one gets out for the “layers” in the Taylor approximation of a functor. I recently gave a talk at the MIT Talbot Workshop (which was about the calculus of functors) on the classification of homogeneous functors, and so I thought that it would be worthwhile to distill that talk into a post on this not-updated-enough mathblog.

Homogeneity

Recall from Eric’s post that we have the construction of the Taylor tower of a functor, which mimics the construction of the polynomial Taylor approximation of a smooth, real-valued function. Letting $f: \mathbb{R} \rightarrow \mathbb{R}$ be such a function, we have that the $n$-th approximation looks like (if we base everything at $0$)

$$(P_n f) (x) = f(0) + f'(0) x + \frac{f''(0)}{2} x^2 + \dots + \frac{f^{(n)} (0)}{n!} x^n.$$

We can think of this approximation as being built up out of “layers”, and letting $D_n f:= P_n f- P_{n-1}f$, we have that

$$(D_n f ) (x) = \frac{f^{(n)}(0)}{n!}x^n.$$

These layers have some nice properties:

1. They are degree $n$;
2. they are of exactly degree $n$, or they are homogeneous of degree $n$;
3. they are determined by their coefficient, $\frac{f^{(n)}(0)}{n!}$.

We’re going to keep trying to push our “functors $\leftrightarrow$ functions” analogy, and hope that the layers of the Taylor tower of a functor behave nicely as well. So, let’s just copy our definitions over to the land of functors – this amounts to replacing differences with taking homotopy fibers, etc. Recall that we have the natural map $P_{n} F \rightarrow P_{n-1} F$, so it makes sense to make the following

Definition. Let $F: Top_\ast \rightarrow D$ be a decently nice functor (where $D$ is either based spaces or spectra). The $n^{\mathrm{th}}$ layer of the Taylor tower of* $F$ is defined to be

$$D_n F := \text{hofib} (P_n F \rightarrow P_{n-1} F).$$

Our hope is that these guys behave like the layers of the Taylor approximation, namely that they be homogeneous of degree $n$. What should that mean?

1. Degree $n$: We already have a notion of what it means for a functor to be of degree $n$: it is $n$-excisive. As Eric discussed, this roughly means that it is “determined by its value on $(n+1)$ points”.
2. Homogeneous: A characterization of homogeneous polynomials of degree $n$ is that they have no lower degree terms, which is to say that their $(n-1)$-st Taylor approximation is zero. For a functor, this should mean that $P_{n-1} F$ is trivial, or that $F$ is $n$-reduced.

Based on these translations, we make the following

Definition. A functor $F$ is homogeneous of degree $n$ if it is $n$-reduced and $n$-excisive.

How well does this definition work? Well, let’s go through some examples!

Examples

Example: Layers of the Taylor Tower.

The motivating examples for this construction were the layers of the Taylor tower, so we would hope that these guys satisfy our definition of homogeneity. Lucky for us, they do, and we have the following

Proposition. The $n$-th layers of the Taylor tower of a functor $F$ are $n$-homogeneous.

This follows from

1. $P_n$ preserves homotopy fiber sequences;
2. $P_{n-1} P_n \rightarrow P_{n-1} P_{n-1}$ is an equivalence;
3. if we have a fiber sequence of functors (objectwise fiber sequences) $F \rightarrow G \rightarrow H$ and both $G$ and $H$ are $n$-excisive, so is $F$. Roughly, “the difference of two degree $n$ things is at most degree $n$”.

That’s a good start! There are two more examples, and it will be an important fact that there are in fact all of them.

Example: Smashin’.

Consider the functor from spaces to spectra given by $F(X) = \Sigma^\infty(X^{\wedge n})$, where $X^{\wedge n}$ is the $n$-fnew smash product of spaces and $\Sigma^\infty$ is taking the suspension spectrum. Similarly, we can take  the functor from spectra to spectra given by $G(X)=X^{\wedge n}$.

Proposition. Both of these functors are homogeneous of degree $n$.

This proposition follows from the following two lemmas, which are sort of interesting in their own right.

Lemma. Let $L: C^n \rightarrow D$ be a functor of several variables. If $L$ is $n_i$-excisive in each slot, then the composite functor

$$C \xrightarrow{\Delta} C^n \xrightarrow{L} D$$

is $\sum n_i$-excisive, where $\Delta$ is the diagonal map $C \rightarrow C^n$.

This process of using the diagonal map to produce a single-variable functor from a multi-variable functor will show up a bunch, so this is important to know. We also have

Lemma. If $L: C^n \rightarrow D$ is 1-reduced in each slot, then the composite functor $(L \circ \Delta)$ is $n$-reduced.

To obtain the proposition from those lemmas, we apply them to (for example) the functor $L: \text{Top}^n_\ast \rightarrow \text{Spectra}$ given by

$$L(X_1, \dots, X_n) = X_1 \wedge \dots \wedge X_n.$$

Composing with the diagonal gives us $X^{\wedge n}$, and it satisfies the conditions of the lemmas.

Note. There are some variants of this which also produce $n$-homogeneous functors. For example, if we smash with a fixed spectrum $C$, the functor $F(X) = C \wedge X^{\wedge n}$ is $n$-homogeneous. If there is an action of the symmetric group $\Sigma_n$ floating around, then taking the homotopy quotient by this action preserves $n$-homogeneous functors (by virtue of the fact that the quotient can be defined as a homotopy colimit). For example, there is an obvious $\Sigma_n$ action on $X^{\wedge n}$ obtained by permuting the factors, so if $C$ has a $\Sigma_n$ action, the functor

$$F(X) = (C \wedge X^{\wedge n})_{h \Sigma_n}$$

is $n$-homogeneous. This is the functorial version of $\frac{c x^n}{n!}$!

A key property is the following, which says that space-valued homogeneous functors are actually nicer than we could have imagined.

Theorem. Let $F: \text{Top}_\ast \rightarrow \text{Top}_\ast$ be a homogeneous, degree $n$ functor. Then for all $X \in \text{Top}_\ast$, $F(X)$ is an infinite loop space.

This tells us that these functors are really spectra-valued, as we can form a spectrum from an infinite loop space. The proof of this (in Goodwillie III) is a bit complicated, but the degree $1$ case is pretty easy. As an exercise, work out what it means for a functor to be 1-homogeneous, and then attempt to apply this to the pushout diagram that realizes the suspension of a space.

Measuring Homogeneity

We have that there is a way of measuring homogeneity, which is very much along the lines of how we can do things with polynomials. For example, we have that a function $f(x)$ is linear iff the function of two variables given by

$$g(x_1,x_2) = f(x_1 + x_2) - f(x_1) - f(x_2) + f(0)$$

is identically zero. Moreover, if $f(x)$ is degree 2, so $f(x) = ax^2 + bx + c$, then we have that

$$g(x_1,x_2)=2 a x_1 x_2,$$

so this auxiliary function measures the failure of $f$ to be linear, and if it is quadratic it recovers twice the top degree term. We can perform this procedure more generally, measuring the failure of a function to be of degree $n$ by a function of $(n+1)$ variables. This motivates the following

Definition. The $n$-th cross-effect of $F$, $\text{cr}_n F$, is the functor of $n$ spaces $X_1, \dots, X_n$ given by applying $F$ to the cube $\chi (n \setminus T) = \vee_{s \in S} X_s$ and taking the total homotopy fiber of that cube.

This is maybe a bit hard to parse at a first glance, but the $n=2$ case is easy to see.

Example: $n=2$.

$\text{cr}_2 F(X_1,X_2)$ is given by the total homotopy fiber of the diagram

 

or, put another way, “$(F(X \vee X_2) - F(X_1) ) - (F(X_2) - F(\ast))$”.

Proposition. If $F$ is $n$-excisive, then $\text{cr}_{m+1} F$ is $(n-m)$-excisive in each variable. In particular, if $F$ is $n$-excisive, then $\text{cr}_n F$ is symmetric multilinear (1-excisive in each slot). Moreover, if $F$ is $(n-1)$-excisive, then $\text{cr}_n F$ is trivial.

This tells us that the cross effect produces a map from homogeneous degree $n$ functors into symmetric multilinear functors of $n$ variables. Going the other way: if we have a symmetric multilinear functor of $n$ variables, $L$, then the composite functor

$$\Delta_n L := (L \circ \Delta)_{h \Sigma_n}$$

is homogeneous of degree $n$. It turns out that these two maps provide us with equivalences, and so we have that any homogeneous degree $n$ functor can be written as

$$F(X) = ((\text{cr}_n F)(X, \dots, X))_{h \Sigma_n}.$$

The necessity of the homotopy quotient is to compensate for “over counting”, like dividing by the factor of 2 that showed up in our example above of trying to recover the top degree term of a quadratic function.

The Coefficient Description

Like homogeneous polynomials, we have that homogeneous functors also admit a description by their “coefficients”. Fix a spectrum $C$, then we have that the functor given by

$$L(X_1,\dots,X_n) = C \wedge X_1 \dots \wedge X_n$$

is symmetric multilinear. On the other hand, if $L$ is symmetric multilinear, then we have a natural assembly map

$$L(S^0, \dots, S^0) \wedge X_1 \wedge \dots \wedge X_n \rightarrow L(X_1, \dots, X_n).$$

This assembly map is an equivalence for finite complexes $X_i$ or more generally if $L$ satisfies some finiteness condition (that it be finitary, or that its values are determined by finite complexes). This means that $L$ is determined by the “coefficient spectrum” $L(S^0,\dots,S^0)$, and that the corresponding homogeneous degree $n$ functor looks like

$$G(X) = (L(S^0,\dots,S^0) \wedge X^{\wedge n})_{h \Sigma_n}.$$

Adorable. This looks like $\frac{f^{(n)}(0) x^n}{n!}$, and so we think that the coefficient spectrum ought to be like the derivative evaluated at the base point $S^0$. Motivated by this, we make the

Definition. The $n$-th differential of $F$, $D^{(n)} F$ is the symmetric multilinear functor given by

$$D^{(n)} F := \text{cr}_n D_n F,$$

and the $n$-th derivative of $F$ at the base point, $(\partial^{(n)} F) (\ast)$, is given by the evaluation of this symmetric multilinear functor at $S^0$:

$$(\partial^{(n)} F) (\ast) := (D^{(n)} F)(S^0,\dots,S^0).$$

With this notation, we can then write

$$D_n F = ((\partial^{(n)} F) (\ast) \wedge X^{\wedge n})_{h \Sigma_n},$$

giving us a formula for the layers of the Taylor tower in terms of the differential. Unfortunately, (as was mentioned by Eric) this is a bit circuitous – in order to compute the derivative we need to know the differential, which comes from $D_n F$, but a theorem of Goodwillie says that we can actually do this directly from $F$. We have the following

Theorem. The $n$-th differential $D^{(n)} F$ is equivalent to the multilinearization of $\text{cr}_n F$, i.e. the effect of linearizing $\text{cr}_n F$ in each slot, which roughly looks like

$$\text{hocolim} \Omega^{k_1 + \dots + k_n} \text{cr}_n F(\Sigma^{k_1} X_1, \dots, \Sigma^{k_n} X_n)$$

where the hocolim is taken as $(k_1, \dots, k_n) \rightarrow \infty$.

A thing to note is that we have been doing everything “at the base point”, which is akin to just looking at MacLauren series for functions. Goodwillie develops a theory of derivatives taken at other points, and this is worth looking into.

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