nLab Goodwillie calculus



(,1)(\infty,1)-Category theory

Stable Homotopy theory

Goodwillie calculus



Goodwillie calculus is an (∞,1)-categorification of differential calculus, where smooth functions between Cartesian spaces are replaced by topological functors between categories of pointed topological spaces.

As an approximation to stabilization of a functor

The operation of stabilization that sends an (∞,1)-category CC to the stable (∞,1)-category Stab(C)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:CDF \colon C \to D into a functor Stab(C)Stab(D)Stab(C) \to Stab(D) is not unlike performing a first order Taylor expansion of a function, whence one speaks here of Goodwillie-Taylor towers.

This is what Goodwillie calculus studies.

Let F:𝒞𝒟F: \mathcal{C} \to \mathcal{D} (where 𝒞\mathcal{C} and 𝒟\mathcal{D} are each either Top *Top_*, the category of pointed topological spaces, or SpecSpec, the category of spectra) be a pointed homotopy functor. Associate with FF a sequence of spectra, called the derivatives of FF, denoted by 1F, 2F,, nF,\partial_1 F, \partial_2 F,\cdots, \partial_n F, \cdots, or, collectively, by *F\partial_* F. For each nn the spectrum nF\partial_n F has a natural action of the symmetric group Σ n\Sigma_n. Thus, F\partial_\bullet F is a symmetric sequence of spectra.

The Goodwillie-derivatives of FF contain substantial information about the homotopy type of FF. We can form a sequence of ‘approximations’ to FF together with natural transformations forming a Goodwillie-Taylor tower. This tower takes the form

FP nFP n1FP 0F F \to \cdots \to P_n F \to P_{n-1} F \to \cdots\to P_0 F

with P nFP_n F being the universal n-excisive approximation to FF. (A functor is n-excisive if it takes any n+1n + 1-dimensional cube with homotopy pushout squares for faces to a homotopy cartesian cube.) For ‘analytic (∞,1)-functorsFF, this tower converges for sufficiently highly connected XX, that is

F(X)holimnP nF(X). F(X) \simeq \underset{n}{holim} P_n F(X).

The fibre D nFD_n F of the map P nFP n1FP_n F \to P_{n-1} F is an n-homogeneous functor in an appropriate sense, and is determined by nF\partial_n F, via the following formula. If FF takes values in SpecSpec then

D nF(X)( nFX n) hΣ n. D_n F(X) \simeq (\partial_n F \wedge X^{\wedge n})_{h \Sigma_n}.

If FF takes values in Top *Top_* then one needs to prefix the right hand side with Ω \Omega^{\infty}. (Arone & Ching)

Analogy between homotopy theory and calculus

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]]k[ [x] ] for the ring of formal power series in one variable over a field kk. The ring k[[x]]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.

  • kk corresponds to\stackrel{\text{corresponds to}}{\mapsto} the category of pointed sets;

  • k[[x]]k[ [x] ] \mapsto the category of pointed homotopy types;

  • xx \mapsto the pointed circle;

  • the augmentation (k[[x]]k)(k[ [x] ] \to k) \mapsto the connected components functor π 0\pi_0 : pointed homotopy types \to pointed sets

  • the augmentation ideal JJ \mapsto the subcategory of pointed connected spaces;

  • the n+1n+1 power of the augmentation ideal J n+1J^{n+1} \mapsto the subcategory of pointed nn-connected spaces;

  • the product of an element in J n+1J^{n+1} with an element of J m+1J^{m+1} is an element of J n+m+2J^{n+m+2} \mapsto the smash product of an nn-connected space with a mm-connected space is (n+m+1)(n+m+1)-connected;

  • multiplication by xx \mapsto the suspension functor.

  • division by xx \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=xk[[x]]J=x k[ [x] ] is isomorphic to k[[x]]k[ [x] ] via division by xx \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+1J^{n+1} is isomorphic to k[[x]]k[ [x] ] via division by x n+1x^{n+1}. \mapsto similarly, the category of nn-connected spaces is equivalent to the category of (n+1)(n+1)-fold topological groups (it is actually an Quillen equivalence of model categories) via the (n+1)(n+1)-fold loop space functor.

  • the quotient ring k[[x]]/J n+1k[ [x] ]/J^{n+1} \mapsto the category of nn-truncated homotopy types (=homotopy n-types)

  • The sequence of approximations of a formal power series f(x)=a 0+a 1x+f(x)=a_0+a_1x+ \cdots

    a 0a_0

    a 0+a 1xa_0+a_1 x

    a 0+a 1x+a 2x 2a_0+a_1 x + a_2 x^2



    the Postnikov tower of a pointed homotopy type XX:

    [π 0(X)][\pi_0(X)]

    [π 0(X);π 1(X)][\pi_0(X); \pi_1(X)]

    [π 0(X);π 1(X);π 2(X)][\pi_0(X); \pi_1(X); \pi_2(X)]


    Here, π 0(X)\pi_0(X) is the set of connected components of XX, [π 0(X);π 1(X)][\pi_0(X); \pi_1(X)] is the fundamental groupoid of XX, [π 0(X);π 1(X);π 2(X)][\pi_0(X); \pi_1(X); \pi_2(X)] is the fundamental 2-groupoid of XX, etc.

  • The differences between f(x)f(x) and its successives approximations

    R 0=f(x)a 0 =a 1x+a 2x 2+a 3x 3+ R 1=f(x)(a 0+a 1x) =a 2x 2+a 3x 3+a 4x 4+ R 2=f(x)(a 0+a 1x+a 2x 2) =a 3x 3+a 4x 4+a 5x 5+ \begin{aligned} R_0 = f(x)-a_0 &= a_1 x+a_2 x^2+a_3 x^3+ \cdots \\ R_1 = f(x)-(a_0+a_1 x) &= a_2 x^2 + a_3 x^3 + a_4 x^4+ \cdots \\ R_2 = f(x)-(a_0+a_1x+a_2x^2) &= a_3 x^3 + a_4 x^4 + a_5 x^5 +\cdots \end{aligned}


    the Whitehead tower of XX,

    C 0=[0;π 1(X),π 2(X),π 3(X),]C_0=[0;\pi_1(X), \pi_2(X), \pi_3(X), \cdots]

    C 1=[0;0,π 2(X),π 3(X),]C_1=[0;0, \pi_2(X), \pi_3(X), \cdots]

    C 2=[0;0,0,π 3(X),]C_2=[0;0, 0, \pi_3(X), \cdots]

    Here, C 0C_0 is the connected component of XX at the base point, C 1C_1 is the universal cover of XX constructed by from paths starting at the base point, C 2C_2 is the universal 2-cover of XX constructed from paths starting the base point, etc.

  • Division by xx is shifting down the coefficients of a power series.

    If f(x)=a 1x+a 2x 2+f(x)=a_1 x+a_2 x^2 + \cdots, then f(x)/x=a 1+a 1x 2+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,...]X=[a_0,a_1,a_2,...] then Ω(X)=[a 1,a 2,....]\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 2n2n homotopy groups of an nn-connected space XX (n1)(n \geq 1) by the Freudenthal suspension theorem

    For example, if X=[0;0,a 2,a 3,...]X=[0;0,a_2, a_3,...] then ΣX=[0;0,0,a 2,a 3...]\Sigma X=[0;0,0,a_2,a_3...], and if X=[0;0,0,a 3,a 4,a 5,...]X=[0;0,0, a_3, a_4, a_5,...] then ΣX=[0;0,0,0,a 3,a 4,a 5,...]\Sigma X=[0;0,0, 0, a_3, a_4, a_5,...].

    In other words, the canonical map XΩXX \to \Omega X is a 2n2n-equivalence if XX is nn-connected (n1)(n \geq 1). If X[2n]X[2n] denotes the 2n2n-type of XX (the 2n2n-truncation of XX), then we have a homotopy equivalence

    X[2n]ΩΣX[2n]ΩΣX[2n+1]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 x1a^{x - 1} which best approximates xx. This is when a=ea = e.

The functor from spaces to spaces which sends XX to the infinite loop space underlying its suspension spectrum

Ω Σ X=colimΩ nΣ nX \Omega^{\infty}\Sigma^{\infty} X = colim \Omega^n \Sigma^n X

sends coproducts to products and is supposed to be like e x1e^{x - 1}. (The “1-1” 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 x1e^{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

e x1e (x1) 22e (x1) 33=e ln(1+(x1))=x. e^{x - 1} \cdot e^{-\frac{(x - 1)^2}{2}} \cdot e^{\frac{(x - 1)^3}{3}} \ldots = e^{ln(1 + (x - 1))} = x.

(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 Γ\Gamma-spaces, and they have been important in algebraic topology. One reason is that Γ\Gamma-spaces model infinite loop spaces (and therefore connective spectra) and are very helpful for understanding stable homotopy theory.

Γ\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 SP S P^{\infty} corresponds to ordinary cohomology (well there is a π 0\pi_0 issue, but let’s ignore that). In general evaluating the linear functor EE on a space XX gives you a space which should be thought of as the smash product of EE and XX.

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 WW-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 nn.

See also Eric Finster’s blog post Thoughts on the Goodwillie Calculus


\infty-Toposes of polynomial (,1)(\infty, 1)-functors

For each nn, the collection of n-excisive (∞,1)-functors from bare homotopy types to bare homotopy types is an (infinity,1)-topos, the jet topos.

due to ( Joyal 08, 35.5, with Georg Biedermann) See also at tangent (infinity,1)-category, and Charles Rezk, appears as (Lurie, remark


Goodwillie derivatives of the identity functor

The Goodwillie derivatives of the identity functor on pointed topological spaces, form an operad in spectra (Ching 05).

See at Goodwillie derivatives of the identity functor

Goodwillie derivatives of exponential modality

The ( \infty -)functor

Σ Ω :SpectraSpectra \Sigma^\infty \circ \Omega^\infty \;\colon\; Spectra \to Spectra

induced from the stabilization-adjunction is (see there) the exponential modality in linear homotopy type theory (or rather the un-pointed version is). Indeed, in Goodwillie calculus it behaves like an exponential function (“minus 1”), for instance in that

  • all its higher Goodwillie derivatives are the unit sphere spectrum, and

  • on suspension spectra 𝕏Σ + X\mathbb{X} \equiv \Sigma^\infty_+ X we have

    Σ Ω 𝕏n1𝕏 nSym(n). \Sigma^\infty \Omega^\infty \mathbb{X} \;\simeq\; \underset{n \geq 1}{\oplus} \mathbb{X}^{\wedge n} \sslash Sym(n) \,.

(Arone & Ching 2019, Exp. 2.6, using Ahearn & Kuhn 2002, Cor. 1.3)

Goodwillie derivatives of mapping spaces

See at stable splitting of mapping spaces.


The original articles:

Surveys and introductions:

A model category presentation for n-excisive functors:

See also:

Discussion in the context of chromatic homotopy theory:

Discussion via (∞,1)-categories and stable (∞,1)-categories:

This is now section 7 of

See also

Generalization from (infinity,1)-categories to (infinity,n)-categories and relation to rational homotopy theory is discussed in

Relation to chromatic homotopy theory is discussed in

  • Greg Arone, Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Inventiones mathematicae, 1999, Volume 135, Issue 3, pp 743-788 (pdf)

  • Nicholas Kuhn, Goodwillie towers and chromatic homotopy: an overview, Geom. Topol. Monogr. 10 (2007) 245-279 (arXiv:math/0410342)

On the relation to configuration spaces:

  • Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) 833-934 (arXiv:math/0501429)

  • Michael Ching, Calculus of Functors and Configuration Spaces, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (pdf)

Discussion in terms of spectral Mackey functors

Discussion in an equivariant setting is in

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