## Thoughts on the Goodwillie Calculus

August 2, 2010

So I haven’t gotten to do much blogging in the past few months on account of having to finish up my dissertation and move to Lausanne, Switzerland for my first job here at the EPFL.  Now that I’m settled, I have much more time to devote to writing here and doing mathematics in general, so hopefully posts here will become much more regular.

For my first entry of the post-graduate student era, I figured I’d respond to, or at least share my thoughts about, the discussion about Goodwillie Calculus going on here and here, perhaps expanding a bit on the comments I made in my research statement.

John Baez points out, and I think rightly so, in his comments on n-Forum that the Goodwiliie Calculus is often presented in a way that seems at odds with an interpretation in terms of categorification.  If we are going to categorify linear functions into linear functors, then they should preserve colimits, and certainly the identity functor should be an example.  As it stands, the whole theory of Goodwillie Calculus seems to demand an explanation for why it must make the bizzare modifications to our understanding of ordinary calculus that it does, and I agree with John that there should be a “disentangled” version which serves as a kind of reference point for the idea of “categorified calculus”, and of which the Goodwillie Calculus can be seen as a particular example.

The situation we find ourselves in, though, seems to be that we have a particular example where something like a categorified version of calculus can be set up, but it’s already much more complicated than the situation which we ordinarily take as a starting point for ordinary calculus (namely, functions $\mathbb{R} \rightarrow \mathbb{R}$).  It’s as if we had discovered Riemann surfaces before we knew anything about the real numbers.  Let me explain.

It has been my conviction for a while now that the Goodwiliie Calculus is a categorification of the following setup:  suppose we have a holomorphic map of Riemann surfaces $f : Y \rightarrow X$.  Then people often think of a holomorphic map $h : Y \rightarrow \mathbb{C}$ as a “multi-valued” holomorphic function $g$ on $X$, the multiple values at a point $p \in X$ arising from the various values of $h$ on the elements of $f^{-1}(p)$.  One says that the holomorphic function $h$ “represents” the multi-valued function $g$.

Let’s take $Y = \mathbb{C}$ and $X = \mathbb{C}^*$.  Then we have the familiar diagram (which I realize is not a real diagram, since $\mathrm{log}$ is not defined on all of $\mathbb{C}^*$, but I can’t do dotted arrows, and you get the idea . . .)

\begin{aligned} \mathbb{C} & \overset{\quad \mathrm{id} \quad}{\to} & \mathbb{C} \\ {}^{\mathrm{exp}} \downarrow & \quad \nearrow {}_{\mathrm{log}} & \\ \mathbb{C}^* \end{aligned}

So in this setup, the identity function $\mathrm{id} : \mathbb{C} \rightarrow \mathbb{C}$ represents the logarithm.  My contention is that the Goodwillie Calculus ariese from this situation by replacing $\mathbb{C}$ with the category of spaces, $\mathbb{C}^*$ with the category of infinite loop spaces, $\mathrm{exp}$ with the “free homotopy commutative monoid” and $\mathrm{log}$ with the inclusion, and that this should ultimately be the explanation for claims like

‘The Goodwillie tower of the identity is a logarithm’.

Probably not too many readers of n-Forum will have much trouble believing that the “free homotopy commutative monoid” functor should play the role of the exponential.  A great way to see that this ought to be the case is by looking at Leinster’s work on Euler characteristics of categories.  If we let small categories represent our homotopy types, then each small category $\mathcal{C}$ determines a functor $\mathbb{B} \rightarrow \mathcal{C}at$ by

$n \mapsto \mathcal{C}^n$

and the Grothendieck construction on this functor (i.e. it’s homotopy colimit), which I will denote $P(\mathcal{C})$, is a model for the “free homotopy commutative monoid” which in this case is called the “free permutative category on $\mathcal{C}$.”  Moreover $\chi(P(\mathcal{C})) = \mathrm{exp}(\chi(\mathcal{C}))$ whenever $\chi(\mathcal{C})$ is defined.  Up to group completion, this is the homotopy theorist’s $\Omega^\infty \Sigma^\infty N(\mathcal{C})$, which should be the origin of statements like

$\Omega^\infty \Sigma^\infty$ is “like” $e^{x-1}$

modulo this whole business of understanding the basepoint.

By the statement

…the category of spectra plays the role of the tangent space to the category of spaces at the one-point space

I believe I mean something like the following (though I’m not sure.  It was my research statement, after all.) If we were to temporarily agree that the category of infinite loop spaces was “flat” in some sense, then we might agree that the tangent space at $QS^0 = \Omega^\infty \Sigma^\infty S^0$ was again just the category of infinite loop spaces.  Somehow I’d like to say that the category of spectra is what we get by “pulling back” this tangent space along “$\mathrm{exp}$“  In the ordinary, Riemann surface picture, the fiber over a point $p \in \mathbb{C}^*$ is isomorphic to $\mathbb{Z}$, so pulling back the tangent space at $p$ gives us a “$\mathbb{Z}$‘s” worth of copies of the same space.  In the homotopy world, all these copies are connected by suspension in the category of spectra, which shifts us up and down the “fiber”.  Well, this one may be a stretch.

Finally, let’s look at the statement

one can interpret this as saying that spaces have some kind of non-trivial curvature

As was already pointed out, curvature may be the wrong word to use here.  When we imagine the exponential map $\mathrm{exp} : \mathbb{C} \rightarrow \mathbb{C}^*$, most of us think of the standard picture of a giant thickened spiral covering the plane.  Of course, set theoretically, this is just in our imagination: the exponential map simply sends one point in the complex plane to another.  But another way to build the same Riemann surface is to start with the power series for $\mathrm{log}$ and analytically continue it, patching together the various branches, eventually arriving at a space homeomorphic to $\mathbb{C}$, but which lacks all the niceties, such as a canonical addition and multiplication.  My assertion would be that the category of spaces remembers that it arises in this second way, and this is the source of what we have been calling “curvature.”  (Someone remind me: isn’t it true that the spiral is flat from a geometric point of view? Maybe this could explain Goodwiliie’s remarks . . .)

I better wrap this up here, as this post is getting pretty long here.  I guess I can sumarize what I’m getting at as follows: the Goodwiliie Calculus, as it’s currently described, should be an example of a more general theory of “categorified calculus” which specializes to something like combinatorial species in the discrete case.  In this sense I agree with the thrust of Joyal’s description of the situation.  It seems to me that such a thing must intrinsically involve homotopy theory, and I have wondered for a long time how to create some kind of “toy model” which has many of the features, but not the complexity, of the Goodwiliie Calculus, but I must confess that as of yet, I don’t know what this should look like.