HARMONIC ANALYSIS ON SYMMETRIC MONOIDAL CATEGORIES

MARK BEHRENS

In this document we attempt to understand the various manifestations of Goodwillie’s calculus of functors in a unified setting. Consider process of “categorifying”:

{category of sets, functions}

Categorify

{2-category of categories, functors, and natural transformations}

For instance, the categorification of a commutative monoid is a symmetric monoidal category.

We shall categorify harmonic analysis, that is, Fourier analysis on groups (compare with the Fourier-Mukai transform). However, we shall only work with monoids, because the notion of inverses is too restrictive for the class of symmetric monoidal categories we wish to consider. The calculus of functors will be cast in this framework. Because classical harmonic analysis involves functions which take values in a ring, we shall only investigate calculus of functors which take values in a closed symmetric monoidal stable model category.

This document is pure philosophy, it contains no precise statements. It is motivated by recent work of Dwyer-Rezk, Arone-Bauer-Christensen-Isaksen, Ching, and Walter. It would be nice to prove some theorems in the general setting suggested in this note. I’d really like to relate the categories of characters that appear with the symmetric monoidal categories that appear in the recent work of Arone-Lesh.

1. Classical harmonic analysis

Let G be an abelian group with some notion of Haar measure. Let ℂ be a field. Define a new group G^{∨} by

G^{∨} = Hom(G, (ℂ, ·)) := {χ : G → ℂ : χ(x · y) = χ(x) · χ(y), χ(e) = 1}

Here, the hom set is the set of maps of monoids, where we are regarding ℂ as a monoid under multiplication. The product in G^{∨} is defined by

(χ · ψ)(x) = χ(x) · ψ(x).

The group G^{∨} is the group of characters of G (valued in ℂ^{×}).

There are Fourier transforms

Map(G, ℂ) ⇆ Map(G^{∨}, ℂ).

Given a map f : G → ℂ, define its Fourier transform to be the map

𝔽(f) : G^{∨} → ℂ, χ ↦ ∫_{x∈G} f(x) · χ^{−1}(x).

Given a map g : G^{∨} → ℂ, define

𝔽(g) : G → ℂ, x ↦ ∫_{χ∈G^{∨}} g(χ) · χ(x).

We hope these transforms are inverses on certain subsets of Map(G, ℂ) and Map(G^{∨}, ℂ).

Let H be a subgroup of G^{∨}. We define a relative Fourier transform to be the composite

𝔽_{H} : Map(H, ℂ) ⟶^{Extend} Map(G^{∨}, ℂ) ⟶^{𝔽} Map(G, ℂ).

where the first map is extension by zero. We define a Fourier transform in the other direction by taking the composite

𝔽_{H} : Map(G, ℂ) ⟶^{𝔽} Map(G^{∨}, ℂ) ⟶^{Restrict} Map(H, ℂ).

2. Categorified harmonic analysis

Replace G with a symmetric monoidal category (G, ⊠). Replace ℂ with a closed symmetric monoidal stable model category (C, ⊗).

By a (C-valued) character of G we shall mean a pair (χ, η) where χ is a functor

χ : G → C

and η is a natural isomorphism

η_{X,Y} : χ(X ⊠ Y) ⟶^{≅} χ(X) ⊗ χ(Y)

(the naturality of η is in X and Y). Define a morphism of characters

α : (χ, η) → (χ′, η′)

to be a natural transformation α : χ → χ′ that makes the following diagram commute for all X and Y.

\xymatrix{χ(X ⊠ Y) \ar[r]^{α_{X⊠Y}} \ar[d]_{η_{X,Y}} & χ′(X ⊠ Y) \ar[d]^{η′_{X,Y}} \\ χ(X) ⊗ χ(Y) \ar[r]_{α_{X}⊗α_{Y}} & χ′(X) ⊗ χ′(Y)}

I haven’t spelled out how the functor χ behaves on the associating and commuting transformations that are part of the data of a symmetric monoidal category. It is supposed to behave beautifully. That is, we require that (χ, η) be the data of a lax symmetric monoidal functor. Let G^{∨} denote the category of C-valued characters. The category G is symmetric monoidal, under ⊗:

(χ, η) ⊗ (χ′, η′) = (χ ⊗ χ′, η ⊗ η′).

Let Funct(−, −) denote the functor category. We now define Fourier transform functors.

Funct(G, C) ⇆_{𝔽}^{𝔽} Funct((G^{∨})^{op}, C).

Given a functor F : G → C, define its Fourier transform to be the homotopy coend:

𝔽(F) : (G^{∨})^{op} → C, χ ↦ ∫ ^{X∈G} F (X) ⊗ D(χ(X)).

Here D : C → C is the duality functor

D(Z) = \underline{Hom}_{C}(Z, 1)

where \underline{Hom}_{C} is the internal Hom in C and 1 is the unit object for ⊗. The appearance of the dual character in the Fourier transform is absolutely necessary to make sense of the end. It mimics the appearance of the inverse character in the classical Fourier transform.

Going the other way, given a functor C : (G^{∨})^{op} → C, define its Fourier transform to be the homotopy coend:

𝔽(C) : G → C, X ↦ ∫ ^{χ∈G^{∨}} C(χ) ⊗ χ(X)

We also describe a dual pair of Fourier transforms.

Funct(G, C) ⇆_{𝔽^{d}}^{𝔽^{d}} Funct(G^{∨}, C).

The dual Fourier transform is obtained by replacing ends with coends and tensors with cotensors. Given a functor F : G → C, define its dual Fourier transform to be the homotopy end:

𝔽^{d}(F) : G^{∨} → C, χ ↦ ∫_{X∈G} \underline{Hom}(F (X), χ(X)).

Given a functor C : G^{∨} → C, define its dual Fourier transform to be the homotopy end:

𝔽^{d}(C) : G → C, X ↦ ∫_{χ∈G^{∨}} \underline{Hom}(C(χ), χ(X))

Remark 2.1. One could reformulate the dual Fourier transforms in terms of natural transformation objects instead. In this language we have

𝔽^{d}(F)(χ) = Nat_{X}(F (X), χ(X))
𝔽^{d}(C)(X) = Nat_{χ}(C(χ), χ(X))

This language conforms more to that of Arone-Bauer-Christensen-Isaksen.

We introduce a modification of the Fourier transform so that we don’t have to contend with the whole category G^{∨}. Let H be a subcategory of G^{∨}. The subcategory H will often be assumed to be submonoidal. We define relative Fourier transforms

Funct(G, C) ⇆ Funct(H^{op}, C).

Given a functor F : G → C, define

𝔽_{H}(F) : (H)^{op} → C, χ ↦ ∫ ^{X∈G} F (X) ⊗ D(χ(X)).

Given a functor C : (H)^{op} → C, define

𝔽_{H}(C) : G → C, X ↦ ∫ ^{χ∈H} C(χ) ⊗ χ(X)

We similarly define dual relative Fourier transforms F_{H}^{d}.

3. Forms of calculus

Goodwillie calculus, in all of the cases I know of, is a formal procedure which associates to a functor

F : G → C

a collection of n-excisive approximations P_{n}(F) and a compatible natural transformations

\xymatrix{ & ⋮ \ar[d] \\  & P_{n}(F) \ar[d] \\ F \ar[r] \ar[ur] & P_{n−1}(F) \ar[d] \\  & ⋮}

The functor F is said to be analytic if the map

F → P_{∞}(F) = lim_{n} P_{n}(F)

is an equivalence.

The layers of the Goodwillie tower are the fibers

D_{n}(F) → P_{n}(F) → P_{n−1}(F).

The power of the theory lies in the fact that the homogeneous layer D_{n}(F) admits a canonical description. Namely, the category of n-homogeneous functors is typically shown to be equivalent to another (simpler) functor category

Funct(G, C)_{n-homo} ≃ Funct((G(n))^{op}, C).

Thus associated to the functor D_{n}(F) we have its derivative

∂_{n}(F) : (G(n))^{op} → C.

There is a functor

χ_{n} : G(n) × G → C

which is an n-excisive character in the second variable. The functor D_{n}(F) is recovered as the coend (or end in some situations)

D_{n}(F)(X) = ∫ ^{G(n)} ∂_{n}(F) ∧ χ_{n}(X).

We wish to describe how the calculus perspective interacts with the Fourier perspective.

4. Goodwillie’s homotopy calculus

4.1. The set-up. In this section we have:

• (G, ⊠) = (Spaces, ∧), the category of pointed spaces under smash product.

• (C, ⊗) = (Spectra, ∧), the category of spectra with smash product.

• (H, ⊗) = (Epi^{op}, ⨿), the opposite category of the category of finite sets and epimorphisms. The category Epi is symmetric monoidal under disjoint union.

Let χ_{n} be the spectrum-valued character on spaces given by

χ_{n}(X) = Σ^{∞}X^{∧n}.

Let \underline{n} denote the set with n elements. There is an inclusion of categories

Epi^{op} ↪ Spaces^{∨}

given by

\underline{n} ↦ χ_{n}

The relative Fourier transforms

Funct(Spaces, Spectra) ⇆ Funct(Epi, Spectra)

“are” the formulations of calculus studied by Arone-Bauer-Christensen-Isaksen and Walter. The idea is that the Fourier transform gives an equivalence between the homotopy category of analytic functors from spaces to spectra and the homotopy category of functors from Epi to spectra.

Let C = 𝔽(F) be the Fourier transform of F.

• Epi_{n} is the subcategory of Epi consisting sets of size less that or equal to n.

• The nth polynomial approximations for F are given by

P_{n}(F)(X) = ∫ ^{\underline{i}∈Epi_{n}} C_{i} ∧ χ_{i}(X).

• part(\underline{n}) is the category of proper partitions (aka proper equivalence relations). These are the “kernels” of proper quotients \underline{n} ↠ \underline{n}/p by equivalence relations p on \underline{n}.

• G(n) = Σ_{n}, the category with one object and whose morphisms are given by the symmetric group Σ_{n}.

• The nth derivative of F is given by the Σ_{n}-equivariant fiber sequence

∂_{n}F → C\underline{n} → holim_{p∈part(\underline{n})} C_{\underline{n}/p}.

• The nth layer of the Goodwillie tower is given by

D_{n}(F)(X) = (∂_{n}F ∧ χ_{n}(X))_{hΣ_{n}}.

Example 4.1. Arone’s thesis. All spaces K and X are finite.

F (X) = Σ^{∞} Map_{∗}(K, X)

Then • C_{i} = D(Σ^{∞}K^{∧i}).

• P_{n}(F) = ∫^{\underline{i}∈Epi} D(Σ^{∞}K^{∧i}) ∧ Σ^{∞}X^{∧i}

• holim_{p∈part(\underline{n})} C_{\underline{n}/p} = D(Σ^{∞}Δ_{n}(K)), where Δ_{n}(K) ⊂ K^{∧n} is the “fat diagonal.”

• ∂_{n}F = D(Σ^{∞}K^{∧n}/Δ_{n}(K))

5. Spaces as structured spectra

Question: is every spectrum-valued character on spaces of the form χ_{i}? I can’t seem to come up with any others. We can also ask, is there an equivalence

(Spaces^{∨})^{∨} ≃ Spaces?

As a possible variant, we could ask, to what degree is the category of spaces given by H^{∨} = (Epi^{op})^{∨}? Suppose that

ψ : Epi → Spectra

is a character. This is the data of a commutative coalgebra Y in spectra. Indeed, let Y = ψ(\underline{1}). Then the diagonal is given by the composite

Y = ψ(\underline{1}) → ψ(\underline{2}) ≅ ψ(\underline{1} ⨿ \underline{1}) ≅ ψ(\underline{1}) ∧ ψ(\underline{1}) = Y ∧ Y

induced by the map \underline{2} ↠ \underline{1} and the character isomorphism. Arone says that this data, together with some sort of “Tate data” is enough to recover a space X such that Y = Σ^{∞}X.

Specifically, according to Arone, a (connective) spectrum Y is a suspension spectrum precisely if it admits a Σ^{∞}Ω^{∞}-coalgebra structure. The space X is recovered by the comonadic cobar construction

X = Ω(Ω^{∞}, Σ^{∞}Ω^{∞}, Y).

Now, calculus on the functor Σ^{∞}Ω^{∞} shows that its derivatives are given by the sphere spectrum (with trivial action)

∂_{∗}Σ^{∞}Ω^{∞} = S_{∗}.

Now, if Y = Σ^{∞}X, then the Goodwillie tower splits at Y (Snaith splitting). Therefore, Y is simply a coalgebra over the commutative cooperad, which is the same as it being a commutative coalgebra. However, this splitting depends on the choice of X which stabilizes to Y. That is, we need to choose a splitting of the Goodwillie tower for Σ^{∞}Ω^{∞} at Y, and this extra data is precisely the “Tate data” we need (Bauer-McCarthy).

6. Ching’s duality

This section is a summary of a description Haynes Miller gave to me of the emerging picture that he, Michael Ching, and Ben Walter have on how to use operads to understand functors from spaces to spaces from functors from spaces to spectra. Much of what I say here is conjectural (and the conjectures are not mine!).

First, let F be a functor from spaces to spectra. It’s Fourier transform is a functor

C : Epi → Spectra.

This is the same thing as saying that C_{∗} is a right comodule over the sphere (commutative) cooperad S_{∗}. The construction I gave earlier to get the derivatives ∂_{∗}F from C is nothing but the operadic cobar construction.

∂_{∗}F = Ω(C_{∗}, S_{∗}, 1)

Ching proves that this is a right module over the Koszul dual operad, which is the derivatives of the identity

ΩS_{∗} = ∂_{∗}Id

Thus the derivatives of a functor from spaces to spectra is a right ∂_{∗}Id-algebra. This structure conjecturally gives the k-invariants of the Goodwillie tower. There are action maps

ξ_{i} : Σ^{−1}∂_{n}F ≃ ∂_{n}F ∧ ∂_{1}Id ∧ ⋯ ∧ ∂_{2}Id ∧ ⋯ ∂_{1}Id → ∂_{n+1}F

for ∂_{2}Id in the ith position. Note that ∂_{1}Id = S^{0} and ∂_{2}Id = S^{−1}. These assemble to give Σ_{n+1}-equivariant maps

ξ : Σ^{−1}∂_{n}F ∧_{Σ_{n}} Σ_{n+1}_{+} → ∂_{n+1}F

and thus maps

Σ^{−1}D_{n}(F)(X) → D_{n+1}(F)(X).

These should be compared to those investigated by Ben Walter.

Suppose that F = Σ^{∞}G, where

G : Spaces → Spaces.

Then F is a Σ^{∞}Ω^{∞}-coalgebra, and thus the ”chain rule” should give an S_{∗}-coaction map

∂_{∗}F → ∂_{∗}(Σ^{∞}Ω^{∞}F) → S_{∗} ∘ ∂_{∗}F.

Then the derivatives of G are (conjecturally) obtained from the derivatives of F by applying a comonadic cobar construction

∂_{∗}G ≃ Ω(1, S_{∗}, ∂_{∗}F).

Again, Ching’s duality curiously shows that ∂_{∗}G is a left module over ∂_{∗}Id.

How are we supposed to use this information to assemble the k-invariants in the Goodwille tower for G?

Example 6.1. Let G = Id.

• F = Σ^{∞}.

• The Fourier transform C = 𝔽(Σ^{∞}) must be constant: C_{i} = S^{0}

• The derivatives of F are given by

∂_{∗}Σ^{∞} = Ω(S_{∗}, S_{∗}, 1_{∗}) = 1_{∗}

where 1_{∗} is the unit for the composition product of symmetric sequences. Thus 1_{∗} is a sphere concentrated in the 1-component.

• The derivatives of G are given by

∂_{∗}Id = Ω(1, S_{∗}, 1)

as proved by Ching. Compare with the limit over the partition category.

7. McCarthy’s dual calculus

In this section we describe the dual Fourier transform in the same setting as that used in Goodwillie calculus:

• (G, ⊠) = (Spaces, ∧)
• (C, ⊗) = (Spectra, ∧)
• (H, ⊗) = (Epi^{op}, ⨿)

We associate to \underline{n} ∈ Epi^{op} the character χ_{n} as before. Then the dual relative Fourier transforms

Funct(Spaces, Spectra) ⇆ Funct(Epi^{op}, Spectra)

are really more closely related to Arone-Bauer-Christensen-Isaksen, and are supposed to give McCarthy’s dual calculus. Since I don’t understand dual calculus, I can’t really say yet.

Let B = 𝔽^{d}(F) be the dual Fourier transform of F.

• Epi_{n} is the subcategory of Epi consisting sets of size less that or equal to n.

• The nth polynomial approximations for F are given by

P_{n}(F)(X) = ∫_{\underline{i}∈Epi_{n}} F (B_{i}, χ_{i}(X)) = Nat\underline{i}(B_{i}, χ_{i}(X)).

• G(n) = Σ_{n}, as before.

• Is the dual derivative given by the following fiber sequence?

hocolim_{p∈part(\underline{n})} B_{\underline{n}/p} → B_{n} → ∂^{n}F

• Then the nth layer of the dual tower should be given by

D_{n}(F)(X) = F (∂^{n}F, χ_{n}(X))^{hΣ_{n}}.

Example 7.1. Arone’s thesis, again. This is the easier formulation.

F (X) = Σ^{∞} Map_{∗}(K, X)

Then • B_{i} = Σ^{∞}K^{∧i}.

• P_{n}(F) = Nat_{i}(Σ^{∞}K^{∧i}, Σ^{∞}X^{∧i})

• hocolim_{p∈part(\underline{n})} B_{\underline{n}/p} = Σ^{∞}Δ_{n}(K).

• ∂^{n}F = Σ^{∞}(K^{∧n}/Δ_{n}(K))

• The Σ_{n} homotopy orbits equals the Σ_{n} homotopy fixed points using the Adams isomorphism (the space K/Δ_{n}(K) has free Σ_{n}-action away from the basepoint).

8. Weiss’s orthogonal calculus

In this section we have:

• (G, ⊠) = (Vect, ⊕), the category of finite dimensional real inner product spaces V with inner product. The morphisms of this category are given by linear isometries.

• (C, ⊗) = (Spectra, ∧) We shall call a linear epimorphism of inner product spaces π : W_{1} ↠ W_{2} an orthogonal epimorphism if there exists a linear isomorphism α such that the following diagram commutes:

\xymatrix{W_{1} \ar@{->>}[rr]^{π} \ar[dr]_{p} &  & W_{2} \\  & (ker π)^{⊥} \ar@{.>}[ur]_{α}}

Here, p is the orthogonal projection.

• (H, ⊗) = (O(Epi)^{op}, ⨿), the category of finite dimensional inner product

spaces, whose morphisms consist of orthogonal epimorphisms. Let χ_{W} be the spectrum-valued character on Vect given by

χ_{W} (V) = S^{V ⊗W} .

There is an inclusion of categories

O(Epi)^{op} ↪ Vect^{∨}

given by

W ↦ χ_{W}

There are Fourier transforms

Funct(Vect, Spectra) ⇆ Funct(O(Epi), Spectra).

Let C = 𝔽(F) be the Fourier transform of F : Vect → Spectra.

• O(Epi_{n}) is the subcategory of vector spaces of dimension less that or equal to n.

• The nth polynomial approximations for F are given by

P_{n}(F)(V) = ∫ ^{W ∈O(Epi_{n})} C_{W} ∧ χ_{W} (V)

• G(n) = O(n), the topological category with one object and whose morphisms are given by the orthogonal group O(n).

• The nth derivative of F is given by the O(n)-equivariant fiber sequence

∂_{n}F → C_{ℝ^{n}} → holim_{0≠W ≤ℝ^{n}} C_{W^{⊥}}.

• The nth layer of the Weiss tower is given by

D_{n}(F)(V) = (∂_{n}F ∧ χ_{ℝ^{n}}(V))_{hO(n)}.