equivalences in/of $(\infty,1)$-categories
The operation of stabilization that sends an (∞,1)-category $C$ to the stable (∞,1)-category $Stab(C)$ does not in general extend to a functor.
We may think of this operation as the analog of linearizing a space. Turning an (∞,1)-functor $F : C \to D$ into a functor $Stab(C) \to Stab(D)$ is not unlike performing a first order Taylor expansion of a function.
This is what Goodwillie calculus studies.
Let $F: \mathcal{C} \to \mathcal{D}$ (where $\mathcal{C}$ and $\mathcal{D}$ are each either $Top_*$, the category of pointed topological spaces, or $Spec$, the category of spectra) be a pointed homotopy functor. Associate with $F$ a sequence of spectra, called the derivatives of $F$, denoted by $\partial_1 F, \partial_2 F,\cdots, \partial_n F, \cdots$, or, collectively, by $\partial_* F$. For each $n$ the spectrum $\partial_n F$ has a natural action of the symmetric group $\Sigma_n$. Thus, $\partial_* F$ is a symmetric sequence of spectra.
The derivatives of $F$ contain substantial information about the homotopy type of $F$. We can form a sequence of ‘approximations’ to $F$ together with natural transformations forming a so-called Taylor tower. This tower takes the form
with $P_n F$ being the universal n-excisive approximation to $F$. (A functor is n-excisive if it takes any $n + 1$-dimensional cube with homotopy pushout squares for faces to a homotopy cartesian cube.) For ‘analytic’ $F$, this tower converges for sufficiently highly connected $X$, that is
The fibre $D_n F$ of the map $P_n F \to P_{n-1} F$ is an n-homogeneous functor in an appropriate sense, and is determined by $\partial_n F$, via the following formula. If $F$ takes values in $Spec$ then
If $F$ takes values in $Top_*$ then one needs to prefix the right hand side with $\Omega^{\infty}$. (Arone & Ching)
Here is an overview of the relation between homotopy theory/∞-groupoid theory and differential calculus that is the starting point for Goodwillie calculus.
based on a message by André Joyal to the Category Theory Mailing list, May 12, 2010
Write $k[ [x] ]$ for the ring of formal power series in one variable over a field $k$. The ring $k[ [x] ]$ bears some resemblances with the category of pointed homotopy types (= pointed spaces up to weak homotopy equivalences). The category of pointed homotopy types is a ring (the product is the smash product and the sum is the wedge sum).
The following dictionary indicates what the correspondence between the two subjects is.
$k$ $\stackrel{corresponds to}{\mapsto}$ the category of pointed sets;
$k[ [x] ]$ $\mapsto$ the category of pointed homotopy types;
$x$ $\mapsto$ the pointed circle;
the augmentation $(k[ [x] ] \to k)$ $\mapsto$ the connected components functor $\pi_0$ : pointed homotopy types $\to$ pointed sets
the augmentation ideal $J$ $\mapsto$ the subcategory of pointed connected spaces;
the $n+1$ power of the augmentation ideal $J^{n+1}$ $\mapsto$ the subcategory of pointed $n$-connected spaces;
the product of an element in $J^{n+1}$ with an element of $J^{m+1}$ is an element of $J^{n+m+2}$ $\mapsto$ the smash product of an $n$-connected space with a $m$-connected space is $(n+m+1)$-connected;
multiplication by $x$ $\mapsto$ the suspension functor.
division by $x$ $\mapsto$ the loop space functor;
Notice here the difference: the loop functor is right adjoint to the suspension functor, not its inverse. Moreover, the loop space of a space has a special structure (it is a group).
the ideal $J=x k[ [x] ]$ is isomorphic to $k[ [x] ]$ via division by $x$ $\mapsto$ similarly, the category of pointed connected spaces is equivalent to the category of topological groups via the loop space functor (it is actually an Quillen equivalence of model categories).
More generally, the ideal $J^{n+1}$ is isomorphic to $k[ [x] ]$ via division by $x^{n+1}$. $\mapsto$ similarly, the category of $n$-connected spaces is equivalent to the category of $(n+1)$-fold topological groups (it is actually an Quillen equivalence of model categories) via the $(n+1)$-fold loop space functor.
the quotient ring $k[ [x] ]/J^{n+1}$ $\mapsto$ the category of $n$-truncated homotopy types (=homotopy n-types)
The sequence of approximations of a formal power series $f(x)=a_0+a_1x+ \cdots$
$a_0$
$a_0+a_1 x$
$a_0+a_1 x + a_2 x^2$
$\cdots$
$\mapsto$
the Postnikov tower of a pointed homotopy type $X$:
$[\pi_0(X)]$
$[\pi_0(X); \pi_1(X)]$
$[\pi_0(X); \pi_1(X); \pi_2(X)]$
$\cdots$
Here, $\pi_0(X)$ is the set of connected components of $X$, $[\pi_0(X); \pi_1(X)]$ is the fundamental groupoid of $X$, $[\pi_0(X); \pi_1(X); \pi_2(X)]$ is the fundamental 2-groupoid of $X$, etc.
The differences between $f(x)$ and its successives approximations
$\mapsto$
the Whitehead tower of $X$,
$C_0=[0;\pi_1(X), \pi_2(X), \pi_3(X), \cdots]$
$C_1=[0;0, \pi_2(X), \pi_3(X), \cdots]$
$C_2=[0;0, 0, \pi_3(X), \cdots]$
Here, $C_0$ is the connected component of $X$ at the base point, $C_1$ is the universal cover of $X$ constructed by from paths starting at the base point, $C_2$ is the universal 2-cover of $X$ constructed from paths starting the base point, etc.
Division by $x$ is shifting down the coefficients of a power series.
If $f(x)=a_1 x+a_2 x^2 + \cdots$, then $f(x)/x= a_1+a_1 x^2+ \cdots$
Similarly, the loop space functor is shifting down the homotopy groups of a pointed space:
if $X=[a_0,a_1,a_2,...]$ then $\Omega(X)=[a_1,a_2,....]$.
Unfortunately, the suspension functor does not shift up the homotopy groups of a space. It is however shifting the first $2n$ homotopy groups of an $n$-connected space $X$ $(n \geq 1)$ by the Freudenthal suspension theorem
For example, if $X=[0;0,a_2, a_3,...]$ then $\Sigma X=[0;0,0,a_2,a_3...]$, and if $X=[0;0,0, a_3, a_4, a_5,...]$ then $\Sigma X=[0;0,0, 0, a_3, a_4, a_5,...]$.
In other words, the canonical map $X \to \Omega X$ is a $2n$-equivalence if $X$ is $n$-connected $(n \geq 1)$. If $X[2n]$ denotes the $2n$-type of $X$ (the $2n$-truncation of $X$), then we have a homotopy equivalence
$X[2n] \to \Omega \Sigma X[2n] \simeq \Omega \Sigma X[2n+1]$.
…
From a blog discussion
Arone Kankaanrinta 95 write
The Goodwillie tower of the identity…is a tower of functors and natural transformations, which starts with stable homotopy and converges to unstable homotopy. (p. 1)
…the Goodwillie tower is an inverse to stable homotopy in the same way as logarithm is an inverse to exponential. (p. 1)
It is the point of this paper that the Goodwillie tower is the homotopy theoretic analog of logarithmic expansion, rather than of Taylor series. (p. 6)
What’s going on, they say, is like finding a function of the form $a^{x - 1}$ which best approximates $x$. This is when $a = e$.
The functor from spaces to spaces which sends $X$ to the infinite loop space underlying its suspension spectrum
sends coproducts to products and is supposed to be like $e^{x - 1}$. (The ”$-1$” comes about from issues to do with basepoints.)
A homogeneous linear functor is defined to be one sending coproducts to products, so it is like an exponential. Compared to an exponential, the identity functor is like a logarithm, so it has a non-trivial Taylor series.
…our point of view is that stable homotopy is analogous to the function $e^{x - 1}$ rather than to a linear function, and the Goodwillie tower is an infinite product, rather than an infinite sum, namely it is analogous to the product
(p. 2)
Gregory Arone in an MO answer
Covariant functors from the category of pointed sets to the category of pointed topological spaces are sometimes called $\Gamma$-spaces, and they have been important in algebraic topology. One reason is that $\Gamma$-spaces model infinite loop spaces (and therefore connective spectra) and are very helpful for understanding stable homotopy theory.
$\Gamma$-spaces also serve as a model for particularly well-behaved covariant functors from the category of pointed topological spaces to itself. Of course, these functors play an important role in topology as well. I like to think of Goodwillie’s Calculus of Homotopy Functors (and also of Michael Weiss’s Orthogonal Calculus) as a kind of “sheaf theory for covariant functors”. In these theories, covariant functors are analogous to presheaves and linear functors are analogous to sheaves (The definition of a linear functor is essentially a homotopy-invariant version of the definition of a sheaf). The process of approximating a general functor by a linear one is analogous to sheafification, and so forth. These theories provide methods for studying certain types of functors, but of course they also tell you something about the category of spaces itself.
Eric Finster’s research statement
One tantalizing aspect of the Goodwillie calculus is that it suggests the possibility of thinking geometrically about the global structure of homotopy theory. In this interpretation, the category of spectra plays the role of the tangent space to the category of spaces at the one-point space. Moreover, the identity functor from spaces to spaces is not linear…and one can interpret this as saying that spaces have some kind of non-trivial curvature.
However, Goodwillie remarks in the report (p. 905) on a Oberwolfach meeting.:
Rhetorical question: If the first derivative of the identity is the identity matrix, why is the second derivative not zero? Answer: Some of the terminology of homotopy calculus works better for functors from spaces to spectra than for functors from spaces to spaces. Specifically, since “linearity” means taking pushout squares to pullback squares, the identity functor is not linear and the composition of two linear functors is not linear.
Attempted cryptic remark: Unlike the category of spectra, where pushouts are the same as pullbacks, the category of spaces may be thought of has having nonzero curvature.
Correction: After the talk Boekstedt asked about that remark. We discussed the matter at length and found more than one connection on the category of spaces, but none that was not flat. In fact curvature is the wrong thing to look for. There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed.
This is from the report (p. 905) on a Oberwolfach meeting. The table on p. 900 also makes comparisons to differential geometry.
Chris Schommer-Pries
Any linear functor from spaces to spaces is a generalized cohomology theory. More precisely, there is a model category on the functors from spaces to spaces called the model category of W-spaces. Really I should be using pointed spaces here. This model category is one of the standard models for the category spectra and so the fibrant objects can be thought of as the (co)homology theories. The fibrant objects are precisely those functors which are linear in Goodwillie’s sense. The example $S P^{\infty}$ corresponds to ordinary cohomology (well there is a $\pi_0$ issue, but let’s ignore that). In general evaluating the linear functor $E$ on a space $X$ gives you a space which should be thought of as the smash product of $E$ and $X$.
So now why should spectra/cohomology theories be thought of as linear functors? Well if you think of spectra as analogous to abelian groups, then applying a spectrum to a space (i.e. smashing with it) is a linearization of that space.
Following this analogy, if we now have any old functor from space to spaces we can take its fibrant replacement in $W$-spaces. This is a linear functor which is the best approximation to the original functor. So it is like taking a derivative of a function. Goodwillie’s insight was to extend this analogy to encompass the rudiments of calculus. There is in fact a whole series of model categories on functors from spaces to spaces where the fibrant objects are Goodwillie’s polynomial functors of degree $n$.
See also Eric Finster’s blog post Thoughts on the Goodwillie Calculus
For each $n$, the collection of polynomial (∞,1)-functors of degree $n$ from bare homotopy types to bare homotopy types is an (infinity,1)-topos.
( Joyal 08, 35.5, with Georg Biedermann) See also at tangent (infinity,1)-category.
Surveys and introductions include
Tom Goodwillie, The differential calculus of homotopy functors, ICM contribution (pdf)
Brian Munson, Introduction to the manifold calculus of Goodwillie-Weiss (arXiv:1005.1698)
Nicholas J. Kuhn, Goodwillie towers and chromatic homotopy: an overview, Proceedings of the Nishida Fest (Kinosaki 2003), 245–279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007.
2012 Talbot Workshop (Talk schedule, Notes)
Original articles include
Calculus. I. The first derivative of pseudoisotopy theory, K-Theory 4 (1990), no. 1, 1-27. MR 1076523 (92m:57027);
Calculus. II. Analytic functors, K-Theory 5 (1991/92), no. 4, 295-332. MR 1162445 (93i:55015);
Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711 (journal, arXiv:math/0310481))
Andrew Mauer-Oats, Algebraic Goodwillie calculus and a cotriple model for the remainder, Trans. Amer. Math. Soc. 358 (2006), no. 5, 1869–1895 journal, math.AT/0212095
A model category presentation for n-excisive functors is given in
Georg Biedermann, Boris Chorny, Oliver Röndigs, Calculus of functors and model categories, Adv. Math., 214, n. 1 (2007) 92–115, doi, math.AT/0601221
Gregory Arone and Michael Ching, Operads And Chain Rules For The Calculus Of Functors, preprint
Gregory Arone, Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Inventiones mathematicae February 1999, Volume 135, Issue 3, pp 743-788 (pdf)
Gregory Arone, Marja Kankaanrinta, The Goodwillie tower of the identity is a logarithm, 1995 (pdf, web)
A discussion of the theory in light of (∞,1)-category theory and stable (∞,1)-categories is in
This is now section 7 of
See also
Luis Pereira, A general context for Goodwillie Calculus (arXiv:1301.2832)
André Joyal, Notes on Logoi, 2008 (pdf)