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.
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 be such a function, we have that the -th approximation looks like (if we base everything at )
We can think of this approximation as being built up out of “layers”, and letting , we have that
These layers have some nice properties:
We’re going to keep trying to push our “functors 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 , so it makes sense to make the following
Definition. Let be a decently nice functor (where is either based spaces or spectra). The layer of the Taylor tower of* is defined to be
Our hope is that these guys behave like the layers of the Taylor approximation, namely that they be homogeneous of degree . What should that mean?
Based on these translations, we make the following
Definition. A functor is homogeneous of degree if it is -reduced and -excisive.
How well does this definition work? Well, let’s go through some 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 -th layers of the Taylor tower of a functor are -homogeneous.
This follows from
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.
Consider the functor from spaces to spectra given by , where is the -fnew smash product of spaces and is taking the suspension spectrum. Similarly, we can take the functor from spectra to spectra given by .
Proposition. Both of these functors are homogeneous of degree .
This proposition follows from the following two lemmas, which are sort of interesting in their own right.
Lemma. Let be a functor of several variables. If is -excisive in each slot, then the composite functor
is -excisive, where is the diagonal map .
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 is 1-reduced in each slot, then the composite functor is -reduced.
To obtain the proposition from those lemmas, we apply them to (for example) the functor given by
Composing with the diagonal gives us , and it satisfies the conditions of the lemmas.
Note. There are some variants of this which also produce -homogeneous functors. For example, if we smash with a fixed spectrum , the functor is -homogeneous. If there is an action of the symmetric group floating around, then taking the homotopy quotient by this action preserves -homogeneous functors (by virtue of the fact that the quotient can be defined as a homotopy colimit). For example, there is an obvious action on obtained by permuting the factors, so if has a action, the functor
is -homogeneous. This is the functorial version of !
A key property is the following, which says that space-valued homogeneous functors are actually nicer than we could have imagined.
Theorem. Let be a homogeneous, degree functor. Then for all , 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 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.
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 is linear iff the function of two variables given by
is identically zero. Moreover, if is degree 2, so , then we have that
so this auxiliary function measures the failure of 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 by a function of variables. This motivates the following
Definition. The -th cross-effect of , , is the functor of spaces given by applying to the cube and taking the total homotopy fiber of that cube.
This is maybe a bit hard to parse at a first glance, but the case is easy to see.
is given by the total homotopy fiber of the diagram
or, put another way, “”.
Proposition. If is -excisive, then is -excisive in each variable. In particular, if is -excisive, then is symmetric multilinear (1-excisive in each slot). Moreover, if is -excisive, then is trivial.
This tells us that the cross effect produces a map from homogeneous degree functors into symmetric multilinear functors of variables. Going the other way: if we have a symmetric multilinear functor of variables, , then the composite functor
is homogeneous of degree . It turns out that these two maps provide us with equivalences, and so we have that any homogeneous degree functor can be written as
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.
Like homogeneous polynomials, we have that homogeneous functors also admit a description by their “coefficients”. Fix a spectrum , then we have that the functor given by
is symmetric multilinear. On the other hand, if is symmetric multilinear, then we have a natural assembly map
This assembly map is an equivalence for finite complexes or more generally if satisfies some finiteness condition (that it be finitary, or that its values are determined by finite complexes). This means that is determined by the “coefficient spectrum” , and that the corresponding homogeneous degree functor looks like
Adorable. This looks like , and so we think that the coefficient spectrum ought to be like the derivative evaluated at the base point . Motivated by this, we make the
Definition. The -th differential of , is the symmetric multilinear functor given by
and the -th derivative of at the base point, , is given by the evaluation of this symmetric multilinear functor at :
With this notation, we can then write
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 , but a theorem of Goodwillie says that we can actually do this directly from . We have the following
Theorem. The -th differential is equivalent to the multilinearization of , i.e. the effect of linearizing in each slot, which roughly looks like
where the hocolim is taken as .
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.