# Monoidal Structures in Orthogonal Calculus

Leon Hendrian

###### Abstract

Orthogonal Calculus, first developed by Weiss in 1991 [39], provides a
“calculus of functors” for functors from real inner product spaces to spaces.
Many of the functors to which Orthogonal Calculus has been applied since carry
an additional lax symmetric monoidal structure which has so far been ignored.
For instance, the functor $V\mapsto\text{BO}(V)$ admits maps

$\text{BO}(V)\times\text{BO}(W)\to\text{BO}(V\oplus W)$

which determine a lax symmetric monoidal structure.

Our first main result, Corollary 4.2.0.2, states that the Taylor
approximations of a lax symmetric monoidal functor are themselves lax
symmetric monoidal. We also study the derivative spectra of lax symmetric
monoidal functors, and prove in Corollary 5.4.0.1 that they admit
$O(n)$-equivariant structure maps of the form

$\Theta^{n}F\otimes\Theta^{n}F\to D_{O(n)}\otimes\Theta^{n}F$

where $D_{O(n)}\simeq S^{\text{Ad}_{n}}$ is the Klein-Spivak dualising
spectrum of the topological group $O(n)$.

As our proof methods are largely abstract and $\infty$-categorical, we also
formulate Orthogonal Calculus in that language before proving our results.

This article is largely identical to the authors PhD thesis [20].

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 What is Orthogonal Calculus?
      1. 1.2.1 Functor Calculus in General
      2. 1.2.2 Orthogonal Calculus
    3. 1.3 Monoidal Structures
      1. 1.3.1 The Pointwise Monoidal Structure
      2. 1.3.2 The Day Convolution Monoidal Structure
  2. 2 Other Work on Orthogonal Calculus
    1. 2.1 Formalisms for Orthogonal Calculus
      1. 2.1.1 The Original Version (Weiss)
      2. 2.1.2 Model Categories (Barnes-Oman)
      3. 2.1.3 $\infty$-categorical Orthogonal Calculus
    2. 2.2 Some Known Calculations in Orthogonal Calculus
      1. 2.2.1 Weiss’s Original Article
      2. 2.2.2 Arone’s Work on BO and BU
      3. 2.2.3 $\Theta^{2}\text{btop}$ (Krannich-Randal-Williams)
      4. 2.2.4 Pontryagin Classes etc. (Reis-Weiss et al.)
    3. 2.3 Analyticity Conditions
      1. 2.3.1 Barnes-Eldred
      2. 2.3.2 Taggart
    4. 2.4 Various
      1. 2.4.1 Unitary Calculus
      2. 2.4.2 Comparison with Goodwillie Calculus
  3. 3 $\infty$-Categories for Orthogonal Calculus
    1. 3.1 Basic Definitions of Relevant Categories and Functors
    2. 3.2 Polynomial Functors
    3. 3.3 Homogeneous Functors and Derivative Spectra
    4. 3.4 Some Topos-Theoretical Remarks
  4. 4 Orthogonal Calculus and Monoidal Structures
    1. 4.1 Setup / Preliminaries
      1. 4.1.1 Symmetric Monoidal $\infty$-Categories
      2. 4.1.2 Day Convolution and the Internal Hom Functor
        1. 4.1.2.1 The Day Convolution Monoidal Structure
        2. 4.1.2.2 The Internal Hom Functor
      3. 4.1.3 Generalities on Monoidal Localisations
    2. 4.2 A Monoidal Localisation Theorem for Taylor Approximations in Orthogonal Calculus
  5. 5 Monoidality and Derivative Spectra
    1. 5.1 Pointed and Unpointed Functors
    2. 5.2 Monoidality of Homogeneous Layers
    3. 5.3 Classification of Homogeneous Layers and the Internal Hom
      1. 5.3.1 Calculation of $\Theta^{n}\mathcal{F}_{*}(A,B)$ for $A,B$ $n$-homogeneous
      2. 5.3.2 Calculation of $\Theta^{n}(A\circledast B)$ for $A,B$ $n$-homogeneous
    4. 5.4 Final Result on Derivative Spectra of Monoidal Functors
  6. 6 Applications and Open Questions
    1. 6.1 Examples of Monoidal Functors
    2. 6.2 A Potential Alternative Description of the Multiplication On $\Theta^{1}F$
    3. 6.3 Further Open Questions

### Acknowledgements

I am grateful to my advisor Michael Weiss for all the support I received
during my time as a PhD student. Further thanks go out to Thomas Nikolaus for
showing me the way to $\infty$-land.
The author was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 427320536 – SFB 1442, as well as under
Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster:
Dynamics–Geometry–Structure.

##  Chapter 1 Introduction

###  1.1 Overview

As already mentioned in the abstract, the main results are

  * •

Theorem 4.2.0.1 and Corollary 4.2.0.2, which state that the polynomial
approximation functors $T_{n}$ are lax symmetric monoidal and hence that
$T_{n}F$ of a lax symmetric monoidal functor $F$ is lax symmetric monoidal.

  * •

Corollary 5.4.0.1, which states that the derivative spectra $\Theta^{n}F$ of a
lax symmetric monoidal functor $F\colon\mathcal{J}\to(\mathcal{S}_{*},\times)$
admit structure maps

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes D_{O(n)}$

The content is structured into chapters as follows:

  * •

In Chapter 1, we first briefly introduce Orthogonal Calculus (and functor
calculus in general) in Section 1.2 and then explain the central motivation
for this work in Section 1.3, particularly in Section 1.3.2.

  * •

In Chapter 2, we mention some interesting work on or involving Orthogonal
Calculus in the literature. This chapter is not essential to the thesis, but
as no existing survey on Orthogonal Calculus is known to the author, it might
be useful to a reader who wants to know more about Orthogonal Calculus in
general.

  * •

In Chapter 3, we go over essential features of Orthogonal Calculus in
$\infty$-categorical language. For the proofs of the basic properties, we
mostly refer back to [39] and then explain how the $\infty$-categorical
formulation follows.

  * •

In Chapter 4, we examine the Day Convolution symmetric monoidal structure on
$\text{Fun}(\mathcal{J},\mathcal{S})$ in detail. A particularly important
feature is the existence of an internal hom functor

$\mathcal{F}(-,-)\colon\text{Fun}(\mathcal{J},\mathcal{S})\times\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

whose properties will play a major role in Chapter 4 and Chapter 5. At the end
of this chapter we prove Theorem 4.2.0.1.

  * •

In Chapter 5, we show that the homogeneous layers

$L_{n}F=\text{fib}(T_{n}F\to T_{n-1}F)\colon\mathcal{J}\to\mathcal{S}_{*}$

of a lax symmetric monoidal functor

$F\colon\mathcal{J}\to\mathcal{S}_{*}$

are also lax symmetric monoidal. Note that $\mathcal{S}_{*}$ is equipped with
the cartesian product monoidal structure here (although the same would be true
for the smash product monoidal structure). We then use the equivalence

$\displaystyle Sp^{\text{BO}(n)}$
$\displaystyle\to\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$
$\displaystyle\Theta$
$\displaystyle\mapsto\left(V\mapsto\Omega^{\infty}\left(S^{nV}\wedge\Theta\right)_{hO(n)}\right)$

to calculate the derivatives of the internal hom functor and the convolution
product of homogeneous functors. This enables us to prove Corollary 5.4.0.1,
i.e. the existence of the maps

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes D_{O(n)}$

  * •

In the final chapter, Chapter 6, we mention a few possible examples as well as
a (conjectured) alternative description of a multiplication on $\Theta^{1}$.
We finish by stating some open questions.

###  1.2 What is Orthogonal Calculus?

####  1.2.1 Functor Calculus in General

In homotopy theory, the objects of interest often are, or can be viewed as,
functors $F\colon\mathcal{C}\to\mathcal{D}$. There are multiple theories which
are (or could be) referred to as Functor Calculus. Among these are

  * •

Goodwillie Calculus studies functors from spaces to spaces111 More generally,
one can also apply it to functors $F\colon\mathcal{C}\to\mathcal{D}$ where
$\mathcal{C}$ is an $\infty$-category with finite colimits and $\mathcal{D}$
admits finite limits as well as sequential colimits such that the formation of
sequential colimits commutes with finite limits. See [24, Chapter 6]. . (The
original sources are [17, 18, 15]. See also [24, Chapter 6] for a modern
generalisation using $\infty$-categories.)

  * •

Embedding Calculus and Manifold Calculus ([16, 37]) study functors from a
category of manifolds (possibly restricted to submanifolds of a given
manifold) and embeddings to spaces.

  * •

Orthogonal Calculus, the subject of this work, studies functors from the
category of real vector spaces with inner products and linear isometric maps
to spaces. ([39] is the original source, a model category version has been
developed by Barnes and Oman [5].)

The guiding principle of any of these calculi is the following:

  * •

There are categories222 sometimes very concrete, sometimes less so, usually
enriched categories or $\infty$-categories $\mathcal{C}$ and $\mathcal{D}$.
One is interested in functors (usually specific ones)
$F\colon\mathcal{C}\to\mathcal{D}$.

  * •

There is a notion of $n$-polynomiality of these functors, in particular of the
subcategories of functors which are polynomial of degree $\leq n$

$\text{Poly}^{\leq
n}(\mathcal{C},\mathcal{D})\subset\text{Fun}(\mathcal{C},\mathcal{D})$

Any functor $F$ admits universal $n$-polynomial approximations denoted333 The
notation used in the literature is unfortunately not consistent across
calculi. In Goodwillie Calculus, the approximations are usually called
$P_{n}F$, in Orthogonal Calculus $T_{n}F$. To make matters worse, $T_{n}F$ is
also used in the Goodwillie Calculus literature for something that is called
$\tau_{n}F$ in Orthogonal Calculus. We stick to the conventions from
Orthogonal Calculus. $F\to T_{n}F$ for any $n\in\mathbb{N}$. These $T_{n}$
determine left-adjoints to the inclusions, i.e.

${{\text{Poly}^{\leq
n}(\mathcal{C},\mathcal{D})}}$${{\text{Fun}(\mathcal{C},\mathcal{D})}}$$\scriptstyle{T_{n}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$

$T_{n}F$ is also called the $n$-th Taylor approximation of $F$.

  * •

There are inclusions

$\text{Poly}^{\leq n-1}(\mathcal{C},\mathcal{D})\subset\text{Poly}^{\leq
n}(\mathcal{C},\mathcal{D})$

Hence, the $n$-polynomial approximations assemble into a diagram of the form

${{T_{\infty}F:=\lim_{n}T_{n}F}}$${{\vdots}}$${{T_{1}F}}$${{F}}$${{T_{0}F}}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\lim_{n}\eta_{n}}$

This tower is called the Taylor tower444 Depending on the type of functor
calculus, other names are also used, e.g. Goodwillie tower in Goodwillie
Calculus, Orthogonal tower or Weiss tower in Orthogonal Calculus.  of $F$.

  * •

A functor $F$ is said to be $n$-homogeneous if it is $n$-polynomial and
$T_{n-1}F$ vanishes555 This means that it is equivalent to the constant
functor to the terminal object. . The most important general examples are the
fibres of the maps $T_{n}F\to T_{n-1}F$, which are always $n$-homogeneous.
These are called the homogeneous layers of $F$.

  * •

An important but often difficult question is if a functor $F$ is analytic in
the sense that

$F\cong\lim_{n}T_{n}F$

i.e. if it can be reconstructed from its Taylor approximations. Known criteria
usually involve connectivity estimates for the maps in the image of $F$.

  * •

In Goodwillie and Orthogonal Calculus, there is a classification theorem for
$n$-homogeneous functors in terms of spectra with actions of the symmetric
group $\Sigma_{n}$ resp. the orthogonal group $O(n)$. The spectrum classifying
the fibre of $T_{n}F\to T_{n-1}F$ is called the $n$-th derivative of $F$.

######  Remark 1.2.1.1 (Analogy to calculus of real functions).

Functor calculi are partially motivated by a surprisingly strong analogy with
calculus for smooth functions $f\colon\mathbb{R}\to\mathbb{R}$. Under this
analogy, $n$-polynomial functors correspond to polynomial functions of degree
at most $n$. The $n$-polynomial approximation $F\to T_{n}F$ corresponds to
replacing a function $f$ by its $n$-th Taylor polynomial. The homogeneous
layers of $F$ correspond to the difference of the $n$-th and the $n-1$-th
Taylor polynomial (at $0$), i.e. the expression

$\frac{f^{(n)}(0)}{n!}\cdot x^{n}$

which is obviously closely related to the derivative $f^{(n)}(0)$.

Even the division by $n!$ corresponds to something in functor calculus: In
Goodwillie calculus, $n$-homogeneous functors have the form

$X\mapsto\Omega^{\infty}\left(\left(\partial_{n}(F)\otimes X^{\wedge
n}\right)_{h\Sigma_{n}}\right)$

where $\partial_{n}(F)$ is a spectrum with an action of the symmetric group
$\Sigma_{n}$. The homotopy orbit construction $(-)_{h\Sigma_{n}}$ can be
thought of as dividing out the action. As $|\Sigma_{n}|=n!$, this can be seen
as analogous to dividing by $n!$.

A similar thing is true in Orthogonal Calculus. $n$-homogeneous functors take
the form

$V\mapsto\Omega^{\infty}\left(\left(S^{V\otimes\mathbb{R}^{n}}\wedge\Theta^{n}F\right)_{hO(n)}\right)$

where $\Theta^{n}F$ is a spectrum with an action of the orthogonal group
$O(n)$. These spectra $\partial_{n}(F)\in Sp^{B\Sigma_{n}}$ and
$\Theta^{n}F\in Sp^{\text{BO}(n)}$ are referred to as the derivative spectra
of $F$.

####  1.2.2 Orthogonal Calculus

Orthogonal Calculus was first introduced by Michael Weiss in 1995 ([39]). It
is a type of functor calculus in the previously discussed sense:

  * •

$\mathcal{J}$ denotes the (topological666 or, in later versions, simplicially
enriched category, or quasicategory, via the appropriate nerve constructions )
category of finite-dimensional real inner product vector spaces where the
morphism spaces are spaces of linear isometric (in particular injective) maps
(also known as Stiefel manifolds).

  * •

The functors that one studies are always defined on $\mathcal{J}$, typically
with codomain the category of spaces777 Parts of the literature are ambiguous
if this means topological spaces, a good category of spaces, or something
else, and in which sense the functors are continuous. In the later parts of
this work, we will consider functors of $\infty$-categories to the
$\infty$-category $\mathcal{S}$ of spaces.  or the category of pointed spaces.
Other possibilities for the target category are the category of spectra
(implicit in [39]) or localisations of it [3, 32].

  * •

A functor

$F\colon\mathcal{J}\to\mathcal{S}$

is called $n$-polynomial if the natural maps

$F(V)\to\lim_{0\neq U\subseteq\mathbb{R}^{n+1}}F(V\oplus U)$

which are induced by the inclusions $V\hookrightarrow V\oplus U$ for
$U\subseteq\mathbb{R}^{n+1},U\neq 0$ are homotopy equivalences.
The limit on the right-hand side is not indexed by a (discrete) poset, but is
a (homotopy) limit over the space / simplicial set of subspaces $0\neq
U\subseteq\mathbb{R}^{n+1}$. In the $\infty$-categorical world, the meaning of
this is clear. In the more classical world, most homotopy limits are formed
over a indexing diagram which is a $1$-category and not a space, so some care
has to be taken (see e.g. [39, Comment to Def. 5.1]).

  * •

An arbitrary functor $F$ can be made $n$-polynomial by replacing $F$
iteratively by $\tau_{n}F:=\lim_{0\neq U\subseteq\mathbb{R}^{n+1}}F(V\oplus
U)$, i.e. by

$T_{n}F=\operatorname{colim}\displaylimits\left(F\to\tau_{n}F\to\tau_{n}(\tau_{n}F)\to\ldots\right)$

which spells out to the colimit over the sequential diagram

$\lim_{0\neq U_{1}\subseteq\mathbb{R}^{n+1}}F(V\oplus U)\to\lim_{0\neq
U_{2}\subseteq\mathbb{R}^{n+1}}\lim_{0\neq
U_{1}\subseteq\mathbb{R}^{n+1}}F(V\oplus U_{1}\oplus U_{2})\to\ldots$

  * •

An important role is played by the $n$-homogeneous functors, which are
functors $F$ which are $n$-polynomial and such that $T_{n-1}$ vanishes. They
occur in particular as the layers888i.e. fibres of the maps $T_{n}F\to
T_{n-1}F$ in the tower, and correspond to spectra with $O(n)$-actions in the
following way

######  Theorem 1.2.2.1 (Weiss [39], Theorem 7.3.).

Any $n$-homogeneous functor to $\mathcal{S}_{*}$, in particular the layers of
the tower, is of the form

$\Omega^{\infty}\left((S^{\mathbb{R}^{n}\otimes
V}\wedge\Theta^{n}F)_{hO(n)}\right)$

for some spectrum with $O(n)$-action $\Theta^{n}F\in Sp^{BO(n)}$.

  * •

The derivative spectra $\Theta^{n}F$ can also be constructed more explicitly
out of a functor $F$. A very brief summary of this is as follows:

    * –

There are categories $\mathcal{J}_{0}\subset\ldots\subset\mathcal{J}_{n}$
enriched in pointed spaces, and $F^{(n)}\colon\mathcal{J}_{n}\to\mathcal{S}$,
the “unstable $n$-th derivative” of $F$, is defined as a Kan extension of
$F\colon\mathcal{J}_{0}\to\mathcal{S}$ along
$\mathcal{J}_{0}\to\mathcal{J}_{n}$, and
$\text{Map}_{\mathcal{J}_{0}}(V,W)=\text{Map}_{\mathcal{J}}(V,W)_{+}$. (This
is in [39], mainly section 2.)

    * –

$F^{(n)}$ relates to the derivative spectra $\Theta^{n}F$ by
$(\Theta^{n}F)_{nl}=F^{(n)}(\mathbb{R}^{l})$. (See e.g. [39, p.2, p.3, p.10,
p.12].)

    * –

There are fibre sequences

$F^{(n)}(V)\to F(V)\to\lim_{0\neq U\subseteq\mathbb{R}^{n}}F(U\oplus V)$

([39], Proposition 5.3)

A much more detailled description of Orthogonal Calculus follows in Chapter 3.

###  1.3 Monoidal Structures

Our central question is

Which “product rules” hold in Orthogonal Calculus?

To make sense of the question, one first has to specify the products involved.
They take the form of symmetric monoidal structures.

######  Observation 1.3.0.1.

There are the following relevant symmetric monoidal structures:

  * •

On $\mathcal{J}$, there is a “direct sum” symmetric monoidal structure
“$\oplus$”, given by direct sum of underlying vector spaces and block sum of
scalar products.999This is not actually a coproduct in $\mathcal{J}$, as
morphism spaces in $\mathcal{J}$ do not consist of all linear maps but only of
isometric ones. See 4.1.1.3.

  * •

On $\mathcal{S}$, there is the cartesian monoidal structure given by products
of spaces.

  * •

On $\mathcal{S}_{*}$, there is the cartesian product, as well as the smash
product, of pointed spaces.

  * •

On $Sp$, there is the tensor product (also known as smash product) of spectra.

Let $(\mathcal{T},\otimes)$ be one of the monoidal categories
$\\{(\mathcal{S},\times),(\mathcal{S},\times),(\mathcal{S}_{*},\wedge),(Sp,\otimes)\\}$.
There are (at least) two natural induced monoidal structures on
$\text{Fun}(\mathcal{J},\mathcal{T})$. We refer to them as the “pointwise
monoidal structure” and the “Day convolution monoidal structure” and shall
explain them shortly.

The case of pointwise products has been treated (with some restrictions) by
Hahn and Yuan [19, Section 4]. We will quickly explain their result, before
discussing the case of convolution products, which is the focus of our work.

####  1.3.1 The Pointwise Monoidal Structure

Using only the monoidal structure on $\mathcal{T}$ (and ignoring that on
$\mathcal{J}$), there is a monoidal structure given by

$(F\otimes_{\text{Fun}(\mathcal{J},\mathcal{T})}G)(V)=F(V)\otimes_{\mathcal{T}}G(V)$

(For formal details, see for example [24, Example 3.2.4.4] (and possibly a
good explanation at [26]).)
Monoids for this monoidal structure can be identified with functors valued in
monoid objects in $\mathcal{T}$

$\mathcal{J}\to\text{CAlg}(\mathcal{T})$

i.e. functors $F\colon\mathcal{J}\to\mathcal{T}$ equipped with compatible maps

$F(V)\otimes_{\mathcal{T}}F(V)\to F(V)$

as well as

$1_{\mathcal{T}}\to F(V)$

for any $V\in\mathcal{J}$, where $1_{\mathcal{T}}$ denotes the monoidal unit
in $\mathcal{T}$.

###### The results of Hahn-Yuan on the pointwise monoidal structure in
Orthogonal Calculus

The result (and approach) we are describing here is for functors with values
in spectra, so $\mathcal{T}=Sp$, and $\text{Fun}(\mathcal{J},Sp)$ is
understood to be equipped with the pointwise monoidal structure. It is [19,
Theorem 4.10].

######  Definition 1.3.1.1.

The category of towers in $\text{Fun}(\mathcal{J},\mathcal{T})$ is

$\text{Fun}(\mathbb{Z}_{\geq 0}^{op},\text{Fun}(\mathcal{J},\mathcal{T}))$

On this category, there is a convolution monoidal structure, given by Day
convolution, i.e.

$(X\circledast Y)_{n}=\operatorname{colim}\displaylimits_{p+q\geq
n}X_{p}\otimes Y_{q}\simeq\oplus_{p+q=n}X_{p}\otimes_{\text{pointwise}}Y_{q}$

######  Proposition 1.3.1.2 ([19], Construction 4.4.).

The polynomial approximations $T_{n}$ ($n\in\mathbb{N}$) assemble into a
functor

$\mathrm{Tow}\colon\mathcal{T}^{\mathcal{J}}\to\text{Fun}(\mathbb{Z}_{\geq
0}^{op},\text{Fun}(\mathcal{J},\mathcal{T}))$

such that $\mathrm{Tow}(F)(n)=T_{n}F$.

######  Proposition 1.3.1.3 ([19], Lemma 4.5).

$\mathrm{Tow}$ is oplax symmetric monoidal.

Recall that an oplax structure on a functor $F\colon\mathcal{C}\to\mathcal{D}$
consists of structure maps

$F(A\otimes_{\mathcal{C}}B)\to F(A)\otimes_{\mathcal{D}}F(B)$

In order for these oplax structure maps to be equivalences (and thus to obtain
a symmetric monoidal structure from this oplax symmetric monoidal structure)
they restrict to the following subclass of functors.

######  Definition 1.3.1.4 ([19], Definition 4.6.).

Let $F\in Sp^{\mathcal{J}}$ be a functor. Call $F$ rapidly convergent if $F$
takes values in connective spectra and there exist real numbers $c,\alpha>0$
such that the natural map $F(W)\to T_{n}F(W)$ is $(\alpha n)\dim W-c$ \-
connected. Denote by $Sp^{\mathcal{J}}_{conv}$ the category of rapidly
convergent functors.

######  Theorem 1.3.1.5 ([19], Theorem 4.10).

The functor

$\mathrm{Tow}\colon Sp^{\mathcal{J}}_{conv}\to\text{Fun}(\mathbb{Z}_{\geq
0}^{op},Sp^{\mathcal{J}}_{conv})$

is symmetric monoidal.

######  Remark 1.3.1.6.

Their result should also be true in Goodwillie calculus (even without a rapid
convergence hypothesis).

####  1.3.2 The Day Convolution Monoidal Structure

For nice symmetric monoidal categories $\mathcal{C}$ and $\mathcal{D}$
($\mathcal{D}$ cocomplete) the functor category
$\text{Fun}(\mathcal{C},\mathcal{D})$ can be equipped with the Day convolution
product. This product is given on functors $F,G$ by

$(F\circledast
G)(C):=\operatorname{colim}\displaylimits_{C_{1}\otimes_{\mathcal{C}}C_{2}\to
C}F(C_{1})\otimes_{\mathcal{D}}G(C_{2})$

Day convolution goes back to Brian Day in the 1-categorical setting ([11, 9])
and has been made available in the $\infty$-categorical world by Saul Glasman
([14]) and later in greater generality by Jacob Lurie ([24, Section 2.2.6]).

Another description of $F\circledast G$ is as the left Kan extension of
$\otimes_{\mathcal{D}}\circ(F\times G)$ along $\otimes_{\mathcal{C}}$ as shown
in the diagram

${\mathcal{C}\times\mathcal{C}}$${\mathcal{D}\times\mathcal{D}}$${\mathcal{C}}$${\mathcal{D}}$$\scriptstyle{F\times
G}$$\scriptstyle{\otimes_{\mathcal{C}}}$$\scriptstyle{\otimes_{\mathcal{D}}\circ(F\times
G)}$$\scriptstyle{\otimes_{\mathcal{D}}}$$\scriptstyle{F\circledast
G:=Lan_{(\otimes_{\mathcal{D}}\circ(F\times G))}\otimes_{\mathcal{C}}}$

where the upper triangle commutes and the lower triangles commutes up to a
universal natural transformation

$\otimes_{\mathcal{D}}\circ(F\times
G)\Rightarrow(Lan_{(\otimes_{\mathcal{D}}\circ(F\times
G))}\otimes_{\mathcal{C}})\circ\otimes_{\mathcal{C}}$

A key property of Day convolution is that monoid objects in this monoidal
category are precisely lax symmetric monoidal functors, i.e. functors $F\colon
C\to D$ equipped with structure maps of the form

$F(C_{1})\otimes_{D}F(C_{2})\to F(C_{1}\otimes_{C}C_{2})$

as well as unit maps

$1_{\mathcal{D}}\to F(1_{\mathcal{C}})$

The aforementioned fact that monoids for the Day convolution structure are
precisely lax symmetric monoidal functors is our original reason for
considering it. Many functors of interest such as

  * •

$V\mapsto S^{V}$

  * •

$V\mapsto O(V)$

  * •

$V\mapsto\text{BO}(V)$

  * •

$V\mapsto\text{BTOP}(V)$

are lax symmetric monoidal101010 The required maps $O(V)\times O(W)\to
O(V\oplus W)$ are given by block sum of matrices. The maps for the other
functors are analogous or derived from this one. , and one can ask if the same
holds for their derivatives and Taylor approximations. This would be implied
by the statement that
$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{T})\to\text{Fun}(\mathcal{J},\mathcal{T})$
“restricts” to a functor

${{\text{Fun}(\mathcal{J},\mathcal{T})}}$${{\text{Fun}(\mathcal{J},\mathcal{T})}}$${{\text{Fun}^{\text{lax
symm.\ monoidal}}(\mathcal{J},\mathcal{T})}}$${{\text{Fun}^{\text{lax symm.\
monoidal}}(\mathcal{J},\mathcal{T})}}$${{\text{CAlg}(\text{Fun}(\mathcal{J},\mathcal{T}),\circledast)}}$${{\text{CAlg}(\text{Fun}(\mathcal{J},\mathcal{T}),\circledast)}}$$\scriptstyle{T_{n}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The “best” explanation for a functor between the categories of commutative
algebra objects in symmetric monoidal categories to exist is a lax symmetric
monoidal functor between the symmetric monoidal categories (and not just
between the monoid objects). This motivates the following central question,
which we will answer positively.

######  Question 1.3.2.1.

Are the functors

$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{T})\to\text{Fun}(\mathcal{J},\mathcal{T})$

lax symmetric monoidal for the Day convolution monoidal structure?

######  Question 1.3.2.2.

What can be said about the derivative spectra $\Theta^{n}F$ of a lax symmetric
monoidal functor $F$?

In Theorem 4.2.0.1, we will answer the first question positively, i.e. show
that the $T_{n}$ are indeed lax symmetric monoidal. In particular, the Taylor
approximations of lax symmetric monoidal functors are themselves lax symmetric
monoidal.
Furthermore, in Chapter 5, we examine the implications for derivative spectra.
The main result is Corollary 5.4.0.1, which states the derivative spectra of
any lax symmetric monoidal functor $F\colon\mathcal{J}\to\mathcal{S}_{*}$
admit maps

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes D_{O(n)}$

where $D_{O(n)}\simeq S^{\text{Ad}_{n}}$ is the dualising spectrum of the
topological group $O(n)$.

##  Chapter 2 Other Work on Orthogonal Calculus

This chapter is a collection of several versions of, calculations with and
statements about Orthogonal Calculus that are in the literature. We hope that
it might serve as a useful starting point for the reader who wishes to delve
deeper into the subject.

###  2.1 Formalisms for Orthogonal Calculus

Orthogonal Calculus has been developed and used in different formalisms. We
briefly mention some of them and their differences.

####  2.1.1 The Original Version (Weiss)

In Weiss’s original paper [39], the category $\mathcal{J}$ of finite-
dimensional real inner-product spaces and linear isometric embeddings is a
$1$-category enriched in (topological) spaces, and the functors $F$ which are
studied are supposed to be continuous in the sense that the evaluation maps

$\text{mor}(V,W)\times F(V)\to F(W)$

are continuous. If the category of spaces occuring as the target of the
functors is understood to be a convenient category of topological space in the
sense that it is cartesian closed (e.g. CGWH spaces), then one can
equivalently ask for the maps

$\text{mor}(V,W)\to\text{mor}(F(V),F(W))$

to be continuous, hence the setting is that of categories enriched in nice
topological spaces and enriched functors between them.

In this framework, one has to be careful about the meaning of

$\lim_{0\neq U\subseteq\mathbb{R}^{n+1}}F(U\oplus V)$

It is a holim, and extra care has to be taken as the holim is not indexed over
a set or a finite diagram, but over a space. See also [39, Comments to
Definition 5.1].

In defining the derivatives, an important role is played by categories called
$\mathcal{J}_{n}$. These are categories enriched in pointed spaces with the
same objects as $\mathcal{J}$ and

$\text{mor}_{\mathcal{J}_{n}}(V,W)=\text{Th}(\gamma_{n}(V,W))$

where $\gamma_{n}\to\text{mor}_{\mathcal{J}}(V,W)$ is the $n$-th tensor power
of the orthogonal complement bundle on $\text{mor}_{\mathcal{J}}(V,W)$ and Th
denotes the Thom space. Starting with a functor $F$ from $\mathcal{J}$ to
spaces, one obtains a functor $F^{(n)}$ on $\mathcal{J}_{n}$ by left Kan
extension. $F^{(n)}$ might be called the unstable $n$-th derivative of $F$.
The derivative spectrum $\Theta^{n}F$ is related to $F^{(n)}$ by
$(\Theta^{n}F)_{nk}=F^{(n)}(\mathbb{R}^{k})$. An inductive construction using
fibre sequences for $F^{(n)}$ is also available (see e.g. [39], p.2), but
obscures the $O(n)$-action.

A slight difference in the (statement of) properties of Orthogonal Calculus is
that in this enriched, $1$-categorical setup, the subcategory of polynomial
functors $\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\subset\text{Fun}(\mathcal{J},\mathcal{S})$ is not
quite a reflective subcategory. (See [39, Chapter 6] for details.)

####  2.1.2 Model Categories (Barnes-Oman)

Orthogonal Calculus has been re-developed in the language of model categories
by Barnes and Oman ([5]). In their description, there are several model
structures on the category of functors from $\mathcal{J}$ to spaces. More
concretely, there is a model category $n$-poly-$\mathcal{E}_{0}$ which is a
Bousfield localisation of the standard projective model structure on
$\text{Fun}(\mathcal{J},\mathcal{S})$ such that the fibrant objects of
$n$-poly-$\mathcal{E}_{0}$ are precisely the $n$-polynomial functors. This
means that the $n$-polynomial approximation functor is the fibrant replacement
functor in $n$-poly-$\mathcal{E}_{0}$.

An application of their work is given in ([5, Section 11]) where they provide
a rigorous construction of Orthogonal Calculus for functors with values in
(orthogonal) spectra. Similar model-categorical techniques also play an
important role in the later work of Taggart (e.g. [33, 34]) who constructed
Unitary Calculus as well as Calculus with Reality, which are functor calculi
similar to Orthogonal Calculus for functors defined on the category of complex
vector spaces resp. vector spaces with Reality.

####  2.1.3 $\infty$-categorical Orthogonal Calculus

It is possible to express the methods and results of Orthogonal Calculus in
the framework of $\infty$-categories, and we will use this version in much of
this thesis. As we explain it in detail in Chapter 3, we don’t say much here.

Many basic statements and properties can be very easily formulated in the
language of $\infty$-categories. As an example, the discussion in ([39,
Chapter 6]) mentions the problem that $\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\subseteq\text{Fun}(\mathcal{J},\mathcal{S})$ is
not quite a reflective subcategories. In $\infty$-land, this problem
disappears, and we get

######  Proposition 2.1.3.1 (Corollary 3.2.0.10).

Let $\mathcal{J}$ be the $\infty$-category of finite dimensional
$\mathbb{R}$-vector spaces and linear isometric maps111 A priori, this
describes a category enriched in (good) topological spaces. Applying Sing
yields a category enriched in Kan complexes, from which the homotopy coherent
nerve produces an $\infty$-category. . Then for each $n\geq 0$ there are the
subcategories $\text{Poly}^{\leq n}\subset\text{Fun}(\mathcal{J},\mathcal{S})$
of $n$-polynomial functors. Their inclusions admit left-exact left-adjoints
$T_{n}$

${{\text{Poly}^{\leq
n}}}$${{\text{Fun}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{\iota}$$\scriptstyle{T_{n}}$$\scriptstyle{\dashv}$

A similar approach to Goodwillie Calculus has been given by Lurie in [24,
Chapter 6], and our approach is indeed inspired by his.

This perspective has also been used and partially described by Hahn and Yuan
[19].

###### An Alternative Topos-theoretic Construction

A completely alternative construction using topos-theoretic methods has been
announced by Anel, Biedermann, Finster and Joyal ([1]). They associate a
Generalised Goodwillie tower to any left-exact localisation of an
$\infty$-topos, and claim that Orthogonal Calculus can be recovered by
considering the example

$\text{Fun}(\mathcal{J},\mathcal{S})\to\mathcal{S}$ $F\mapsto
F(\mathbb{R}^{\infty})=\operatorname{colim}\displaylimits_{n}F(\mathbb{R}^{n})$

See also Section 3.4 for some further comments on this perspective.

###  2.2 Some Known Calculations in Orthogonal Calculus

In this section, we collect some calculations and applications of Orthogonal
Calculus from the literature.

####  2.2.1 Weiss’s Original Article

The following are some of the examples given in the original article ([39]).

######  Example 2.2.1.1 ([39], Introduction, Example 2.7, Example 10.6).

For

$E(V)=\text{BO}(V)$

the first three derivative spectra are given by

$\Theta^{1}E\simeq\mathbb{S}$ $\Theta^{2}E\simeq\Omega\mathbb{S}$
$\Theta^{3}E\simeq\Omega^{2}\mathbb{S}/3$

######  Example 2.2.1.2 ([39], Introduction).

For

$E(V)=\text{BTOP}(V)$

the first derivative spectrum is

$\Theta^{1}E\simeq A(*)$

where $A(*)$ is Waldhausens A-theory spectrum of a point, i.e. $A(*)\simeq
K(\mathbb{S})$.

######  Remark 2.2.1.3.

Recall that for a manifold $M$, one possible description of $A(M)$ is

$A(M)=K(\mathbb{S}[\Omega M])$

where $\Omega M$ is the loop space of $M$ (as a monoid), $\mathbb{S}[\Omega
M]$ is the spherical group ring of that monoid, and $K(\mathbb{S}[\Omega M])$
is its $K$-theory spectrum ([35]).

######  Example 2.2.1.4 ([39], Example 3.2.).

Let $M$ be a smooth compact manifold, potentially with boundary. Consider

$F(V)=B\text{Diff}^{b}_{\partial}(M\times V)$

where

$\text{Diff}^{b}_{\partial}(M\times V)$

is the group of diffeomorphisms of $M\times V$ which are the identity on
$\partial M\times V$ and which are bounded in the sense that there is a
constant $c>0$ such that $||p_{2}(f(m,v))-v||\leq c$ for all $(m,v)\in M\times
V$.

Then

$\Theta^{1}F\simeq\Omega\text{Wh}^{\text{Diff}}(M)$

where $\text{Wh}^{\text{Diff}}$ is the smooth Whitehead spectrum of $M$. This
is related to the $A$-theory spectrum $A(M)$ by
$A(M)\simeq\Omega^{\infty}S^{\infty}(M_{+})\times\text{Wh}^{\text{Diff}}(M)$
(see [35], Theorem 2, Addendum 4).

####  2.2.2 Arone’s Work on BO and BU

Recall, as mentioned in Remark 2.4.1.1, that Orthogonal Calculus works just as
well in the setting of complex vector spaces as it does in real vector spaces.

In his paper [2], Arone studies the derivative spectra of the functors

$V\mapsto\text{BO}(V)$

and

$V\mapsto\text{BU}(V)$

We outline his results briefly. Let
$\mathbb{F}\in\left\\{\mathbb{R},\mathbb{C}\right\\}$.

######  Theorem 2.2.2.1 ([2], Theorem 2).

For $n\geq 1$, the $n$-th derivative of $V\mapsto BAut(V)$ is the spectrum

$\text{Map}_{*}(L_{n},\Sigma^{\infty}S^{Ad_{n}})$

where $L_{n}$ is the unreduced suspension of the geometric realisation of the
category of non-trivial direct-sum decompositions of $\mathbb{F}^{n}$ and
$Ad_{n}$ is the adjoint representation of $Aut(n)$ (i.e. $O(n)$ resp. $U(n)$).

Note that $\Sigma^{\infty}S^{\text{Ad}_{n}}\simeq D_{O(n)}$ is the dualising
spectrum which also appears in our result Corollary 5.4.0.1.

He then goes on to study the (non-equivariant, or only partially equivariant)
homotopy type of these spectra:

######  Theorem 2.2.2.2 ([2], Theorem 3).

There is an $O(n-1)$-equivariant equivalence

$\text{Map}_{*}(L_{n}^{\mathbb{R}},\Sigma^{\infty}S^{Ad_{n}})\simeq\text{Map}_{*}(S^{1}\wedge
K_{n},\Sigma^{\infty}S^{0})\wedge_{\Sigma_{n}}O(n-1)_{+}$

and a $U(n-1)$ equivariant equivalence

$\text{Map}_{*}(L_{n}^{\mathbb{C}},\Sigma^{\infty}S^{Ad_{n}^{\mathbb{C}}})\simeq\text{Map}_{*}(S^{1}\wedge
K_{n},\Sigma^{\infty}S^{n})\wedge_{\Sigma_{n}}U(n-1)_{+}$

where $K_{n}$ denotes the unreduced suspension of the category of non-trivial
partitions of $\left\\{1,\ldots,n\right\\}$.

The key idea here is to use the fibre sequences

$S^{V}\to\text{BO}(V)\to\text{BO}(V\oplus\mathbb{R})$

respectively

$\Sigma S^{V}\to\text{BU}(V)\to\text{BU}(V\oplus\mathbb{C})$

and the close comparison between the Goodwillie tower of the identity functor
on spaces and the orthogonal or unitary derivatives of $S^{V}$, coming from
the comparison functor

$\displaystyle\mathcal{J}$ $\displaystyle\to\mathcal{S}$ $\displaystyle V$
$\displaystyle\mapsto S^{V}$

inducing a functor (by precomposition)

$\text{Fun}(\mathcal{S},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

(See also Section 2.4.2 for more on the comparison functor).

He also calculates the mod $p$ \- cohomology of the derivative spectra ([2,
p.456]), which turns out to be a free module over a certain subalgebra $A_{k}$
of the Steenrod algebra $A$. Strikingly, the $n$-th derivative spectra of
$U(n)$ resp. $O(n)$, $M_{n}^{\mathbb{C}}$, are contractible for all $n$ which
are not prime powers, and $M_{n}^{\mathbb{R}}$ is contractible for all $n$
which are not prime powers of twice a prime power. ($M_{n}$ denotes the $n$-th
derivative spectrum of $\text{Aut}(n)$.)

####  2.2.3 $\Theta^{2}\text{btop}$ (Krannich-Randal-Williams)

Three very important example functors in Orthogonal Calculus are

$\displaystyle\text{BO}(-)\colon V$ $\displaystyle\mapsto\text{BO}(V)$
$\displaystyle\text{BTOP}(-)\colon V$
$\displaystyle\mapsto\text{BTOP}(V)=\text{B}\text{Homeo}(V,V)$
$\displaystyle\text{BG}(-)\colon V$ $\displaystyle\mapsto\text{BG}(V)$

where $G(V)$ is the grouplike monoid of self-homotopy equivalences of $V$.

We already mentioned that the first derivative of $\text{BTOP}(-)$ is
$A(*)\simeq K(\mathbb{S})$. Already the second one remains somewhat
mysterious. The following partial answer has been given by Krannich and
Randal-Williams ([22]).

######  Theorem 2.2.3.1 ([22], Theorem 9.9.).

$\Theta^{2}\text{BTOP}(-)\to\Theta^{2}\text{BG}(-)$ is a rational equivalence.
The latter was calculated by ([29, Prop. 3.5]) to be rationally equivalent to
$Map(S^{1}_{+},\mathbb{S}^{-1})$

####  2.2.4 Pontryagin Classes etc. (Reis-Weiss et al.)

In [29], Reis and Weiss tried to use Orthogonal Calculus to see that that the
square of the euler class is the Pontryagin class, not just in
$H^{*}(BSO(2m))$ but also in $BSTOP(2m)$.

Later, Weiss showed in [41] that this is not true.

Even later, Galatius-Randal-Williams [12] showed that the topological
Pontryagin classes are in fact algebraically independent in $BTOP(2m)$.

######  Corollary 2.2.4.1.

$Bo\to Btop$ does not admit a rational left inverse.

###### Proof.

According to Reis-Weiss, the existence of such a left inverse would be
equivalent to $e^{2}=p_{\frac{n}{2}}\in H^{2n}(BSTOP(n);\mathbb{Q})$. But this
is not true, so there is no such splitting. ∎

###  2.3 Analyticity Conditions

Recall that a functor $F\colon\mathcal{J}\to\mathcal{S}$ is called analytic if
it can be recovered from its Taylor tower, i.e. if

$F\simeq\lim_{n}T_{n}F$

Since limits of functors are computed pointwise, this implies that

$F(V)\simeq\lim_{n}T_{n}F(V)$

for any $V\in\mathcal{J}$.

Unfortunately, this notion of analyticity is quite hard to check in practice.
However, one can often show weaker versions, as we will discuss shortly.

####  2.3.1 Barnes-Eldred

######  Definition 2.3.1.1 ([4], Def. 2.25).

A functor $F$ is weakly $\rho$-analytic if and only if

$F(V)\to T_{\infty}F(V)$

is an equivalence for all $V$ of dimension $\geq\rho$.

######  Proposition 2.3.1.2 ([4], Thm. 4.1).

$\text{BO}(-)$ is weakly $2$-analytic. $\text{BU}(-)$ is weakly $1$-analytic.

###### Proof.

Their proof uses their comparison to Goodwillie Calculus and the fact that the
first unstable derivative of $\text{BO}(-)$ is $V\mapsto S^{V}$. ∎

####  2.3.2 Taggart

######  Definition 2.3.2.1 ([31] Definition 3.4.6.).

A functor $F$ is weakly $(\rho,n)$-polynomial if the map

$\eta\colon F(U)\to T_{n}F(U)$

is an agreement of order $n$ for all $U$ with $\dim U\geq\rho$. A functor is
weakly polynomial if there exists $\rho$ such that it is weakly
$(\rho,n)$-polynomial for all $n\geq 0$.

where

######  Definition 2.3.2.2.

A map $p\colon F\to G$ is an order $n$ agreement if there is some
$\rho\in\mathbb{N}$ and $b\in\mathbb{Z}$ such that $p_{U}\colon F(U)\to G(U)$
is $((n+1)\dim U-b)$-connected for all $U$ satisfying $\dim U\geq\rho$. We say
that $F$ agrees with $G$ to order $n$ if there is an order $n$ agreement
between them.

The point of these notions is the following connection to convergence of the
Taylor tower

######  Lemma 2.3.2.3 ([31] Lemma 3.4.5.).

If for all $n\geq 0$ the map $F\to T_{n}F$ is an order $n$ agreement then the
Taylor tower of $F$ converges to $F(V)$ at $V$ for $\dim V\geq\rho$.

######  Example 2.3.2.4 ([31, Examples 3.4.9], [33, Section 9]).

The following functors are weakly polynomial:

  * •

$V\mapsto S^{V}$

  * •

$\mapsto BO(V)$ ($\rho=2$)

  * •

$V\mapsto BU(V)$ (in unitary calculus)($\rho=1$, [33] Example 9.16.)

  * •

$V\mapsto O(V)$

  * •

Representable functors $V\mapsto\text{Map}_{\mathcal{J}}(U,V)$ ([33], Theorem
9.17.)

###  2.4 Various

####  2.4.1 Unitary Calculus

######  Remark 2.4.1.1 (Unitary Calculus).

There is a complex analogue of Orthogonal Calculus called “Unitary Calculus”.
In complete analogy with Orthogonal Calculus, it studies functors

$\mathcal{J}\to\mathcal{S}$

where now, $\mathcal{J}$ denotes the category of finite-dimensional vector
spaces over $\mathbb{C}$ with a positive definite inner product.

That this analogue exists has been known from the beginning (e.g. [39, Example
10.4]). A relatively modern treatment (using model-categorical techniques) has
recently been given by Taggart in [33] and subsequently been used in his
“Unitary Calculus with Reality” ([34]), a unitary version which takes into
account the complex conjugation action on the complex vector spaces and is
able to specialise to both Orthogonal and Unitary Calculus.

Other places in the literature where Unitary Calculus made an appearance
include:

  * •

Arone ([2]) calculating the derivatives of the orthogonal functor $V\mapsto
BO(V)$ as well the unitary functor $V\mapsto BU(V)$.

  * •

Hahn and Yuan ([19]) using unitary calculus of the functor (valued in spectra)
$V\mapsto\Sigma^{\infty}_{+}\Omega\mathcal{J}(V,V\oplus W)$ to study the
Mitchell-Richter filtration/splitting on $\Sigma_{+}^{\infty}\Omega SU(n)$.

####  2.4.2 Comparison with Goodwillie Calculus

Goodwillie Calculus and Orthogonal Calculus are closely related. Some explicit
statements are

######  Theorem 2.4.2.1 (Barnes-Eldred [4, Prop. 3.2]).

Precomposition with the one-point compactification functor
$\mathcal{J}\to\mathcal{S}$ induces a functor

$\mathcal{S}^{\mathcal{S}}\to\mathcal{S}^{\mathcal{J}}$

which maps $n$-excisive functors (in the sense of Goodwillie) to
$n$-polynomial functors (in the sense of Weiss).

and

######  Theorem 2.4.2.2 (Barnes-Eldred [4, Section 5], in particular
commutativity of the squares 1,2,3).

Let $F_{GW}\colon\mathcal{S}\to\mathcal{S}$ be some functor and let
$F_{W}\colon\mathcal{J}\to\mathcal{S}$ be its composition with the one-point
compactification functor $\mathcal{J}\to\mathcal{S}$. Then the $n$-th
derivative spectrum of $F_{W}$ (in Orthogonal Calculus) is obtained from the
$n$-th derivative spectrum of $F_{GW}$ (in Goodwillie Calculus) by induction
along $\Sigma_{n}\to O(n)$, i.e.

$\Theta^{n}(F_{W})\simeq\text{ind}_{\Sigma_{n}}^{O(n)}\Theta^{n}_{GW}(F_{GW})\simeq
O(n)_{+}\wedge_{\Sigma_{n}}\Theta^{n}_{GW}(F_{GW})$

where $\Theta^{n}(F_{W})$ denotes the derivative spectrum in Orthogonal
Calculus of $F_{W}$ and $\Theta^{n}_{GW}$ denotes the derivative spectrum of
$F_{GW}$ in the sense of Goodwillie calculus.

##  Chapter 3 $\infty$-Categories for Orthogonal Calculus

When Orthogonal Calculus was first introduced by Weiss in 1995 ([39]), the
theory of quasicategories / $\infty$-categories as developed later by Lurie
([25, 24]) was not yet available. As we will make essential use of
$\infty$-categorical constructions later, we describe the essential features
of Orthogonal Calculus in the language of $\infty$-categories in this chapter.
The key justification for the passage to $\infty$-land111 also referred to as
$\infty$-land, see e.g. Thomas Nikolaus’ talk at ICM 2022 ([27]). will be
Corollary 3.2.0.10.

This perspective on Orthogonal Calculus has not found widespread use in the
literature yet. It has appeared in a paper of Hahn-Yuan ([19]), and a new
construction of “Generalised Goodwillie towers”, which includes Orthogonal
Calculus, has been announced by Anel, Biedermann, FInster and Joyal ([1]).

###  3.1 Basic Definitions of Relevant Categories and Functors

######  Definition 3.1.0.1.

Let ${\mathcal{J}_{top}}$ be the category enriched in topological spaces whose
objects are finite-dimensional real inner product spaces $V$ with morphism
spaces $\text{mor}(V,W)$ given by the spaces of linear isometric embeddings

$\text{mor}(V,W):=\\{f\colon V\to W\mid f\text{ linear and }\langle
v_{1},v_{2}\rangle_{V}=\langle f(v_{1}),f(v_{2})\rangle_{W}\\}$

with the obvious composition of maps.

######  Remark 3.1.0.2.

Choosing identifications $V\cong\mathbb{R}^{k}$ und $W\cong\mathbb{R}^{n}$
leads to an identification

$\text{mor}_{\mathcal{J}_{top}}(V,W)\cong V_{k}(\mathbb{R}^{n})\cong
O(n)/O(n-k)$

where $V_{k}(\mathbb{R}^{n})$ is the Stiefel manifold of orthonormal
$k$-frames in $\mathbb{R}^{n}$ and $O(n)/O(n-k)$ is the space of cosets of
$O(n-k)$ in $O(n)$.

######  Definition 3.1.0.3.

Let $\mathcal{J}$ denote the $\infty$-category corresponding to the enriched
category $\mathcal{J}_{\text{top}}$ of Definition 3.1.0.1. More precisely,
$\mathcal{J}$ is obtained by applying the homotopy-coherent nerve construction
(see e.g. [25, Def. 1.1.5.5, Cor. 1.1.5.12]) to the category enriched in
simplicial sets (even in Kan complexes) $\mathcal{J}_{enr}$, which itself
arises by taking nerves of the hom-spaces of ${\mathcal{J}_{top}}$. (This
composition is also called the “topological nerve”.)

###### Functors on $\mathcal{J}$

Orthogonal Calculus primarily studies functors from $\mathcal{J}$ to

  * •

$\mathcal{S}$, the category of spaces.

  * •

$\mathcal{S}_{*}$, the category of pointed spaces.

  * •

$Sp$, the category of spectra. ([5, Section 11])

######  Remark 3.1.0.4.

Orthogonal Calculus can also be done for functors with values in localisations
of $\mathcal{S}$, as is shown in [32]. As far as we know, the question “What
is the highest generality, especially concerning the target category, for
which Orthogonal Calculus can be done?” is still open. We note that Lurie has
a notion of “differentiable $\infty$-category” ([24], 6.1.1.6), which might be
a candidate answer to this question.

Some important examples of functors on $\mathcal{J}$ are

######  Example 3.1.0.5.

  1.   2. a)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto S^{V}\cong V^{c}$

  3. b)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto O(V)$

  4. c)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto\text{B}O(V)$

  5. d)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto\text{B}TOP(V)$

  6. e)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto\text{B}G(V)$

  7. f)

For fixed $W\in\mathcal{J}$,

$F_{W}\colon\mathcal{J}\to Sp$
$V\mapsto\Sigma_{+}^{\infty}\Omega(\text{mor}_{\mathcal{J}}(W,W\oplus V))$

Note that these in fact define functors to pointed spaces as well.

###### Proof of functoriality.

Since

$\mathcal{J}\simeq N_{hcoh}(\text{Sing}_{*}(\mathcal{J}_{top}))$

where

$\text{Sing}_{*}\colon\text{Cat}_{top}\to\text{Cat}_{Kan}$

is the functor which associates to a category enriched in topological spaces
the corresponding category enriched in Kan complexes and

$N_{hcoh}\colon\text{Cat}_{Kan}\to\text{Cat}_{\infty}$

is the homotopy-coherent nerve which associates to a category enriched in Kan
complexes the corresponding $\infty$-category, it is sufficient to describe
the corresponding topologically enriched functors from
$\mathcal{J}_{top}\to\text{Top}$ (by functoriality of $\text{Sing}_{*}$ and
$N_{hcoh}$).

To do that, let’s examine the case $V\mapsto O(V)$ first. Let $f\colon V\to W$
be a morphism in $\mathcal{J}_{top}$. Then any $\varphi\in O(V)$ can be
extended by the identity on the orthogonal complement of $f(V)$ in $W$, i.e.

$O(V)\to O(V\oplus W)$ $\varphi\to\begin{pmatrix}\varphi&0\\\ 0&id_{W\ominus
f(V)}\end{pmatrix}$

The continuity is obvious. The checks required for the other examples work
similarly. ∎

######  Remark 3.1.0.6.

We observe that the functoriality checks only makes use of the definition of
$\mathcal{J}$, in particular the isometricity condition in the mapping spaces
and the inner product, insofar as the (natural) orthogonal complements are
used to extend automorphisms on subspaces to the whole vector space. One can
wonder whether this is a starting point for a generalisation for Orthogonal
Calculus. See also 6.3.0.4.

###  3.2 Polynomial Functors

The central notion to functor calculus is that of polynomial functors.

Recall from ordinary calculus that for a polynomial function $f$ of degree at
most $n$, the interpolation polynomial at $n+1$ distinct points is $f$ itself.

In Orthogonal Calculus, a similar interpolation property is turned into a
definition.

######  Definition 3.2.0.1.

A functor $F\colon\mathcal{J}\to\mathcal{S}$ is called polynomial of degree
$\leq n$ or simply $n$-polynomial if the canonical natural transformation

$F(-)\to\lim_{0\neq U\subseteq\mathbb{R}^{n+1}\text{ linear
subspace}}F(-\oplus U)$

is an equivalence.

A functor $F\colon\mathcal{J}\to\mathcal{S}_{*}$ is polynomial of degree $\leq
n$ if its composition with the forgetful map $\mathcal{S}_{*}\to\mathcal{S}$
is.

We denote by $\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\subseteq\text{Fun}(\mathcal{J},\mathcal{S})$ and
$\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S}_{*})$ the full subcategories of
functors which are polynomial of degree $\leq n$.

######  Remark 3.2.0.2.

The limit

$\lim_{0\neq U\subseteq\mathbb{R}^{n+1}\text{ linear subspace}}F(-\oplus U)$

appearing in the previous definition is formed over the slice category in
$\mathcal{J}$ over $\mathbb{R}^{n+1}$. Note that, from a classical viewpoint,
this is not simply the poset of sub-vector spaces of $\mathbb{R}^{n+1}$, but
takes into account the topology.

######  Example 3.2.0.3.

Examining Definition 3.2.0.1 for $n=0$, we see that a $0$-polynomial functor
$F$ is equivalent to the constant functor with value $F(0)$.

######  Example 3.2.0.4.

Let $n=1$. Then the indexing category $0\neq U\subseteq\mathbb{R}^{2}$ takes
the shape of a cone over $\mathbb{R}P^{1}$, as there is an $\mathbb{R}P^{1}$
worth of $1$-dimensional subspaces of $\mathbb{R}^{2}$, and a single
$2$-dimensional subspace, into which all of them include.

######  Example 3.2.0.5.

Let $\Theta\in Sp^{\text{BO}(n)}$. Then the functor

$V\mapsto\Omega^{\infty}\left(\left(S^{nV}\wedge\Theta\right)_{hO(n)}\right)$

is polynomial of degree $n$.

######  Definition/Proposition 3.2.0.6.

Let $F\in\text{Fun}(\mathcal{J},\mathcal{S})$. Define

$\tau_{n}F\colon\mathcal{J}\to\mathcal{S}$ $V\mapsto\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}\text{ linear subspace}}F(V\oplus U)$

By definition, $F\in\text{Poly}^{\leq n}$ iff the natural map $\rho\colon
F\to\tau_{n}F$ is an equivalence.

Define furthermore $T_{n}F$ to be the colimit of

${F}$${\tau_{n}F}$${\tau_{n}^{2}F}$${\cdots}$$\scriptstyle{\rho}$$\scriptstyle{\tau_{n}(\rho)}$$\scriptstyle{\tau_{n}^{2}(\rho)}$

The inclusion into the first term induces a natural transformation

$\eta_{n}\colon\text{id}_{\text{Fun}}(\mathcal{J},\mathcal{S})\to T_{n}$

Weiss showed in [39], Theorem 6.3 that $T_{n}F$ is always $n$-polynomial.

###### Polynomial Functors Form A Reflective Subcategory

As Michael Weiss already remarked in the original article, $\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\subset\text{Fun}(\mathcal{J},\mathcal{S})$ “wants”
to be a reflective subcategory, but could not be in the $1$-categorical world.
In $\infty$-land, we now show that this problem disappears and
$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})$ is a left adjoint to the inclusion, and
$\eta_{n}$ is the unit of this adjunction.

######  Definition 3.2.0.7 ([25], 5.2.7.9).

A full subcategory $\mathcal{C}^{0}$ of an $\infty$-category $\mathcal{C}$ is
called reflective subcategory if the inclusion
$\mathcal{C}^{0}\hookrightarrow\mathcal{C}$ admits a left adjoint.

Much of the work needed to establish an $\infty$-categorical version of
Orthogonal Calculus has already been done in the original article [39] (and
the erratum [38]), especially in Theorem 6.3. We repeat it here, noting that
we have already adapted the notation.

######  Theorem 3.2.0.8 ([39, Thm. 6.3], [38]).

For any $n\geq 0$, there exists a functor

$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

(constructed as in 3.2.0.6) taking equivalences to equivalences, and a natural
transformation

$\eta_{n}\colon id_{\text{Fun}(\mathcal{J},\mathcal{S})}\to T_{n}$

with the properties

  1. 1.

$T_{n}(E)$ is polynomial of degree $\leq n$ for all
$E\colon\mathcal{J}\to\mathcal{S}$

  2. 2.

If $E$ is already polynomial of degree $\leq n$, then

$\eta_{n,E}\colon E\to T_{n}E$

is an equivalence.

  3. 3.

For every $E$, the map

$T_{n}(\eta_{n,E})\colon T_{n}E\to T_{n}T_{n}E$

is an equivalence.

This theorem covers much of what one needs to know to use the
$\infty$-categorical language for Orthogonal Calculus. The reason is that
these conditions (as was already noted in [39, Obs. 6.1]) precisely
characterise reflective subcategories. In $\infty$-land, this is the following
proposition

######  Proposition 3.2.0.9 ([25, Prop. 5.2.7.4]).

Let $\mathcal{C}$ be an $\infty$-category and let
$L\colon\mathcal{C}\to\mathcal{C}$ be a functor with essential image
$L\mathcal{C}\subseteq\mathcal{C}$. The following conditions are equivalent:

  1. (a)

There exists a functor $F\colon\mathcal{C}\to\mathcal{D}$ with a fully
faithful right adjoint $G\colon D\to C$ and an equivalence between $G\circ F$
and $L$.

  2. (b)

When regarded as a functor from $\mathcal{C}$ to $L\mathcal{C}$, $L$ is left
adjoint to the inclusion $L\mathcal{C}\subseteq\mathcal{C}$.

  3. (c)

There exists a natural transformation

$\alpha\colon\mathcal{C}\times\Delta^{1}\to\mathcal{C}$

from $id_{\mathcal{C}}$ to $L$ such that, for every object $C$ of
$\mathcal{C}$, the morphisms

$L(\alpha(C))\colon L(C)\to L(L(C))$

and

$\alpha(LC)\colon L(C)\to L(L(C))$

are equivalences.

######  Corollary 3.2.0.10.

The inclusions of the subcategories

$\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\subseteq\text{Fun}(\mathcal{J},\mathcal{S})$

of $n$-polynomial functors admit left-exact left-adjoints $T_{n}$

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$${{\text{Fun}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{T_{n}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$

###### Proof.

Apply Proposition 3.2.0.9 to the situation

$\mathcal{C}=\text{Fun}(\mathcal{J},\mathcal{S})$, $L=T_{n}$,
$L\mathcal{C}=\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$

Theorem 3.2.0.8 gives precisely the third condition.

For the left-exactness, let $\lim_{K}F$ be a finite limit. Then

${{T_{n}(\lim_{K}F)(V)}}$${{\operatorname{colim}\displaylimits\left(\lim_{K}F(V)\to\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}\lim_{K}F(U\oplus
V)\to\ldots\right)}}$${{\lim_{K}T_{n}(F)(V)}}$${{\lim_{K}\operatorname{colim}\displaylimits\left(F(V)\to\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}F(U\oplus
V)\to\ldots\right)}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The two right-hand sides agree since $\lim_{K}$ commutes both with the other
involved limits and the filtered colimit (as $\mathcal{S}$ is a filtered
category). ∎

######  Remark 3.2.0.11.

In [19] (esp. Remark 4.2.), Hahn-Yuan used the same argument in order to use
Orthogonal Calculus in $\infty$-categorical language.

######  Remark 3.2.0.12.

Since a functor from $\mathcal{J}$ to pointed spaces is polynomial of degree
$\leq n$ if and only if the same functor to unpointed spaces is polynomial of
degree $\leq n$ (by definition), many statements we prove about unpointed
functors apply verbatim pointed functors.

######  Lemma 3.2.0.13.

Any $n$-polynomial functor is also ($n+1$)-polynomial, i.e. there are
inclusions

$\displaystyle\text{Poly}^{\leq n-1}(\mathcal{J},\mathcal{S})$
$\displaystyle\hookrightarrow\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$
$\displaystyle\text{Poly}^{\leq n-1}(\mathcal{J},\mathcal{S}_{*})$
$\displaystyle\hookrightarrow\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})$

They induce factorisations

${{\text{Fun}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n-1}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{T_{n}}$$\scriptstyle{T_{n-1}}$

and

${{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}}$${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})}}$${{\text{Poly}^{\leq
n-1}(\mathcal{J},\mathcal{S}_{*})}}$$\scriptstyle{T_{n}}$$\scriptstyle{T_{n-1}}$

###### Proof.

For the inclusions $\text{Poly}^{\leq n-1}\hookrightarrow\text{Poly}^{\leq
n}$, we refer back to [39], Proposition 5.4. The factorisations follows by
uniqueness of left adjoints from Corollary 3.2.0.10. ∎

For a functor $F$, the collection of $T_{n}F$ together with the comparison
maps is usually referred to as the Taylor tower (or “Orthogonal tower”, or
“Weiss tower”) of $F$.

######  Corollary 3.2.0.14.

The $n$-polynomial approximation functors assemble into the following tower of
categories

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$${\vdots}$${{\text{Poly}^{\leq
1}(\mathcal{J},\mathcal{S})}}$${{\text{Fun}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{0}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{T_{0}}$$\scriptstyle{T_{1}}$$\scriptstyle{T_{n}}$$\scriptstyle{{T_{1}}_{|\text{Poly}^{\leq
1}(\mathcal{J},\mathcal{S})}}$

For any single functor $F$, the units of the adjunctions assemble into the
Taylor-Weiss tower

${{T_{n}F}}$${\vdots}$${{T_{1}F}}$${F}$${{T_{0}F}}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{n}}$

###  3.3 Homogeneous Functors and Derivative Spectra

If one imagines $F\to T_{n}F$ to be the “$n$-th Taylor polynomial of $F$”, the
natural next question is

“What is the $n$-th derivative of $F$?”

Sticking to the analogy with real calculus for just a moment, for
$f\colon\mathbb{R}\to\mathbb{R}$, we observe

$(T_{n}f-T_{n-1}f)(x)=x^{n}\frac{f^{(n)}(0)}{n!}$

In functor calculus, the role corresponding to the term on the right-hand side
of this equation is played by $n$-homogeneous functors.

######  Definition 3.3.0.1 ([39], Definition 7.1.).

A functor $F\colon\mathcal{J}\to\mathcal{S}$ or
$F\colon\mathcal{J}\to\mathcal{S}_{*}$ is $n$-homogeneous if it is polynomial
of degree $\leq n$ and all values of $T_{n-1}F$ are contractible.
Denote by

$\displaystyle\text{Homog}^{n}(\mathcal{J},\mathcal{S})$
$\displaystyle\hookrightarrow\text{Fun}(\mathcal{J},\mathcal{S})$
$\displaystyle\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$
$\displaystyle\hookrightarrow\text{Fun}(\mathcal{J},\mathcal{S}_{*})$

the full subcategories on $n$-homogeneous functors.
These categories can also be characterised by pullback diagrams of
$\infty$-categories

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$${{\text{Homog}^{n}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n-1}(\mathcal{J},\mathcal{S})}}$${{*}}$$\scriptstyle{\lrcorner}$

and

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})}}$${{\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})}}$${{\text{Poly}^{\leq
n-1}(\mathcal{J},\mathcal{S}_{*})}}$${{*}}$$\scriptstyle{\lrcorner}$

where the maps $*\to\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$ and
$*\to\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S}_{*})$ map to the constant
functor with value the one point space, which is $0$-polynomial and thus
$n$-polynomial for any $n$.

In the pointed case, it is easy to associate $n$-homogeneous functors to any
$F\colon\mathcal{J}\to\mathcal{S}_{*}$ in the following way.

######  Definition/Proposition 3.3.0.2.

Let $F\colon\mathcal{J}\to\mathcal{S}_{*}$ and $n\in\mathbb{N}$. For
$V\in\mathcal{J}$, denote by $L_{n}F(V)$ the fibre in the fibre sequence

$L_{n}F(V)\to T_{n}F(V)\to T_{n-1}F(V)$

They assemble into a functor $L_{n}F\colon\mathcal{J}\to\mathcal{S}_{*}$. This
construction defines a functor
$L_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S}_{*})\to\text{Fun}(\mathcal{J},\mathcal{S}_{*})$.

######  Proposition 3.3.0.3.

$L_{n}$ is right adjoint to the inclusion
$\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})\hookrightarrow\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})$, i.e.

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})}}$${{\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})}}$$\scriptstyle{L_{n}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$

###### Proof.

Let $H\in\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$ and
$P\in\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S}_{*})$. We want to show that

$\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}(H,L_{n}P)\simeq\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}(H,P)$

Using the definition and the adjunction of Corollary 3.2.0.10, we calculate

$\displaystyle\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}(H,T_{n-1}P)$
$\displaystyle\simeq\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}(T_{n-1}H,T_{n-1}P)$
$\displaystyle\simeq\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}(*,T_{n-1}P)$
$\displaystyle\simeq*$

hence by the fibre sequence

$L_{n}P\to T_{n}P\simeq P\to T_{n-1}P$

the claim follows. ∎

We can include the homogeneous layers in the Taylor-Weiss tower

${{T_{n}F}}$${{L_{n}F}}$${\vdots}$${{T_{1}F}}$${{L_{1}F}}$${F}$${{T_{0}F}}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{n}}$

Phrased differently, any functor $F$ is the successive extension of
$n$-homogeneous functors.

######  Remark 3.3.0.4.

Even though Definition 3.3.0.1 defines homogeneous functors to pointed spaces
and to spaces in the same way, constructing an $n$-homogeneous functor out of
an arbitrary functor $F$ is more complicated in the unpointed case, as a
choice of basepoints is needed to take the fibres in 3.3.0.2. For our
purposes, the pointed case will suffice. (See Section 5.1.)

Homogeneous functors to pointed spaces have been classified in the following
way.

######  Definition/Proposition 3.3.0.5.

For any spectrum with $O(n)$-action $\Theta\in Sp^{\text{BO}(n)}$, there is a
functor

$\displaystyle F\colon\mathcal{J}$ $\displaystyle\to\mathcal{S}_{*}$
$\displaystyle V$
$\displaystyle\mapsto\Omega^{\infty}\left(\left(S^{V\otimes\mathbb{R}^{n}}\wedge\Theta\right)\right)_{hO(n)}$

$O(n)$ acts diagonally on the smash product: on
$S^{V\otimes\mathbb{R}^{n}}\simeq{(S^{V})}^{\wedge n}$ by the canonical action
on $\mathbb{R}^{n}$ as well as on $\Theta$ by its action on $\Theta$.
This defines a functor

$\displaystyle Sp^{\text{BO}(n)}$
$\displaystyle\to\text{Fun}(\mathcal{J},\mathcal{S}_{*})$
$\displaystyle\Theta$
$\displaystyle\mapsto\left\\{V\mapsto\Omega^{\infty}\left(\left(S^{V\otimes\mathbb{R}^{n}}\wedge\Theta\right)_{hO(n)}\right)\right\\}$

######  Theorem 3.3.0.6 ([39], Theorem 7.3.222Actually, this is the “more
precise version of theorem 7.3.” that Weiss mentions after stating his actual
Theorem 7.3.).

The essential image of this functor is
$\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$, and the functor is an
equivalence

$Sp^{\text{BO}(n)}\to\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$

######  Definition/Proposition 3.3.0.7.

Thus, the composition of

$L_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$

with an inverse to this equivalence defines a functor which we call

$\Theta^{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to Sp^{\text{BO}(n)}$

For a functor $F$, we call $\Theta^{n}F$ its $n$-th derivative spectrum.

######  Remark 3.3.0.8.

This “a posteriori definition” is quite far from the original definition of
$\Theta^{n}$ in [39], because it only makes sense once the properties of
orthogonal calculus such as Theorem 3.3.0.6 have been established.
In the original article, the derivative spectrum $\Theta^{n}F$ of a functor
$F$ is constructed long before homogeneous functors are classified.
However, as we do not use the original construction much333 It will appear
once in the proof of Lemma 5.3.1.5 and is also mentioned in Section 2.1.1. ,
we do not describe it in detail.

###  3.4 Some Topos-Theoretical Remarks

This short section contains some remarks about the categories
$\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$ from a topos-theoretical
viewpoint. It is not essential to the rest of the thesis and could be safely
skipped.

A trivial observation is that since

$\text{Fun}(\mathcal{J},\mathcal{S})=\text{Fun}((\mathcal{J}^{\text{op}})^{\text{op}},\mathcal{S})=\text{PSh}(\mathcal{J}^{\text{op}})$

is a presheaf category (valued in spaces), it is an example of an
$\infty$-topos.

Recall the definition:

######  Definition 3.4.0.1 (Definition 6.1.0.4 of [25]).

An $\infty$-category $\mathcal{X}$ is called an $\infty$-topos if there exists
a small $\infty$-category $\mathcal{C}$ and an accessible left exact
localisation functor $\mathcal{P}(\mathcal{C})\to\mathcal{X}$.

In $1$-category theory, this definition of a topos is equivalent to the
following other definition:

######  Definition 3.4.0.2 ([25, Prop. 6.1.0.1]).

In classical category theory, a category $\mathcal{C}$ is called a topos if
the following equivalent conditions are satisfied

  * •

$\mathcal{C}$ is equivalent to the category of sheaves of sets on a
Grothendieck site.

  * •

$\mathcal{C}$ is equivalent to a left exact localisation of the category of
presheaves of sets on some small category $\mathcal{C}_{0}$.

  * •

Giraud’s axioms are satisfied.

######  Proposition 3.4.0.3.

For every $n\in\mathbb{N}$, $\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$ is
an $\infty$-topos and

$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Poly}^{\leq n}$

is a left-exact localisation.

###### Proof.

This follows directly from Corollary 3.2.0.10 and the definition. ∎

In contrast to the $1$-categorical case, not every $\infty$-topos arises from
a Grothendieck topology. Recall that there are two classes of localisations of
$\infty$-topoi:

######  Definition 3.4.0.4 ([25], 6.5.2.17).

Let $X$ be an $\infty$-topos and let $Y\subseteq X$ be an accessible left
exact localisation of $X$. $Y$ is a cotopological localisation of $X$ if the
localisation $X\to Y$ does not invert any monomorphisms which were not already
equivalences in $X$.

######  Definition 3.4.0.5 ([25], 6.2.1.4).

Let $X$ be an $\infty$-topos and let $Y\subseteq X$ be an accessible left
exact localisation of $X$. $Y$ is a topological localisation of $X$ if any
morphism that is inverted by the localisation $X\to Y$ is a monomorphism in
$X$.

Any (accessible) left exact localisation of $\infty$-topoi can be factored
into a topological localisation followed by a cotopological localisation
([25], 6.5.2.19). Topological localisations of $\text{PSh}(\mathcal{C})$ are
precisely given by the categories of sheaves for a Grothendieck topology on
$\mathcal{C}$ ([25], 6.2.2.17).

Thus, it makes sense to ask the following question.

######  Question 3.4.0.6.

Is there a Grothendieck topology on $\mathcal{J}^{\text{op}}$ such that

$\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\hookrightarrow\text{PSh}(\mathcal{J}^{\text{op}})$

is the corresponding category of sheaves, i.e. is it a topological
localisation?

######  Answer 3.4.0.7.

No. As a side result in upcoming work of Anel, Biedermann, Finster, Joyal on
“Generalised Goodwillie Towers” ([1]), the $n$-polynomial localisations in any
generalised Goodwillie calculus, such as Goodwillie Calculus or Orthogonal
Calculus, is purely cotopological.

######  Remark 3.4.0.8.

This is in contrast to the situation in Manifold calculus, where
$n$-polynomial functors are given as sheaves for the Grothendieck topology
$\mathcal{J}_{n}$, where a cover $\\{U_{i}\to U\\}_{i\in I}$ consists of a
family of maps with the property that for any subset $X\subseteq U$ of
cardinality $\leq n$, there exists an $i\in I$ such that
$X\subseteq\text{im}(U_{i})$. (This is the main result of [7].)

In particular, Manifold Calculus is thus not a Generalised Goodwillie calculus
in the sense of Anel, Biedermann, Finster and Joyal.

##  Chapter 4 Orthogonal Calculus and Monoidal Structures

The goal of this chapter is to prove Theorem 4.2.0.1, i.e. that the
$n$-polynomial approximation functors

$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

are lax symmetric monoidal.

Before, we will need some preliminaries:

  * •

In Definition 4.1.1.2 we briefly recall some fundamentals about symmetric
monoidal infinity categories and functors between them, and explain the
symmetric monoidal structure on $\mathcal{J}$.

  * •

In Section 4.1.2.1 we explain the Day convolution symmetric monoidal structure
on $\text{Fun}(\mathcal{J},\mathcal{S})$, and in Section 4.1.2.2 we prove that
in this seting, it admits an internal hom functor with particularly nice
properties.

  * •

In Section 4.1.3 we state and explain some abstract nonsense about the
question when a localisation of a monoidal category is itself monoidal.

Then we can prove the theorem in Section 4.2.

Afterwards, in Chapter 5, we will show that the result carries over
homogeneous layers, and prove that this induces “twisted multiplication maps”

$\Theta^{n}F\otimes\Theta^{n}F\to D_{O(n)}\otimes\Theta^{n}F$

on the derivative spectra of a monoidal functor
$F\colon\mathcal{J}\to\mathcal{S}$.

###  4.1 Setup / Preliminaries

####  4.1.1 Symmetric Monoidal $\infty$-Categories

In the following, we use the language and techniques of symmetric monoidal
$\infty$-categories in the sense of Lurie ([24, Chapter 2]).

We recall the most important definition:

######  Definition 4.1.1.1 ([24], 2.0.0.7).

A symmetric monoidal $\infty$-categeory is a coCartesian fibration of
simplicial sets $p\colon\mathcal{C}^{\otimes}\to N(\text{Fin}_{*})$ with the
property:

For each $n\geq 0$, the maps $\\{p_{i}\colon\langle n\rangle\to\langle
1\rangle\\}_{1\leq i\leq n}$ induce funtors
$\rho^{i}_{!}\colon\mathcal{C}^{\otimes}_{\langle
n\rangle}\to\mathcal{C}^{\otimes}_{\langle 1\rangle}$ which determine an
equivalence $\mathcal{C}^{\otimes}_{\langle
n\rangle}\simeq\left(\mathcal{C}^{\otimes}_{\langle 1\rangle}\right)^{n}$.

Any symmetric monoidal $\infty$-category is an example of an $\infty$-operad.

In this generality, a lax symmetric monoidal functor is an $\infty$-operad map
(see also [24, Remark 2.1.3.6]).

######  Definition 4.1.1.2 ([24], 2.1.2.7).

Let $\mathcal{O}^{\otimes}$ and $\mathcal{O}^{\prime\otimes}$ be
$\infty$-operads. An $\infty$-operad map from $\mathcal{O}^{\otimes}$ to
$\mathcal{O}^{\prime\otimes}$ is a map of simplicial sets
$\mathcal{O}^{\otimes}\to\mathcal{O}^{\prime\otimes}$ which commutes with the
structure maps to $N(\text{Fin}_{*})$ and which carries inert morphisms to
inert morphisms. (A morphism $\langle m\rangle\to\langle n\rangle$ is called
inert if it identifies a subset of $\langle m\rangle$ with the basepoint of
$\langle n\rangle$ and is otherwise bijective. A morphism in an
$\infty$-operad is inert if it maps to an inert morphism of
$N(\text{Fin}_{*})$.)

###### The Symmetric Monoidal Structure on $\mathcal{J}$

We want to equip $\mathcal{J}$ with the structure of a symmetric monoidal
$\infty$-category corresponding to the construction

$\left(V,\langle-,-\rangle_{V}),(W,\langle-,-\rangle_{W})\right)\mapsto(V\oplus
W,\langle-,-\rangle_{V}\oplus\langle-,-\rangle_{W})$

We simply write $V\oplus W$ for this, noting however that this is not actually
a coproduct in $\mathcal{J}$.

######  Warning 4.1.1.3.

$\mathcal{J}$ does not admit any coproducts of inner product spaces of
positive dimension.

###### Proof.

Let $V$ be an inner product space of dimension $n>0$ and assume $X$ to be a
coproduct of $V$ and $V$. $X$ admits a codiagonal map to $V$, thus $\dim X\leq
n$ because all maps are embeddings. On the other hand, consider the two
standard embeddings $f,g\colon V\to V\oplus V$. Clearly, $\dim(V\oplus V)=2n$.
By the universal property, there is an embedding $f\oplus g\colon X\to V\oplus
V$ extending $f$ and $g$, thus $\dim X=2n$.
The case of a coproduct of different inner product spaces works analogously
after choosing an embedding of one into the other. ∎

######  Proposition/Definition 4.1.1.4.

There is a symmetric monoidal $\infty$-category $(\mathcal{J}^{\oplus})$ with
underlying $\infty$-category $\mathcal{J}$ extending the construction

$(V,\langle-,-\rangle_{V})\times(W,\langle-,-\rangle_{W})\mapsto(V\oplus
W,\langle-,-\rangle_{V}\oplus\langle-,-\rangle_{W})$

###### Proof.

We will explicitly describe a cocartesian fibration
$\mathcal{J}^{\oplus}\to\text{N}(\text{Fin}_{*})$ such that
$\mathcal{J}^{\oplus}_{n}\cong(\mathcal{J}^{\oplus}_{1})^{n}$, i.e. verify
[24, Definition 2.0.0.7.].

An object of $\mathcal{J}^{\oplus}$ consists of a finite pointed set $X$
together with objects $(X_{i})_{i\in X}\in\mathcal{J}$. A morphism of
$\mathcal{J}^{\oplus}$ consists of a map of pointed sets $f\colon X\to Y$
together with maps $\tilde{f}_{k}\colon\bigoplus_{i\in f^{-1}(k)}X_{i}\to
Y_{k}$ for $k\in Y\backslash\\{*\\}$. The functor to
$\text{N}(\text{Fin}_{*})$ is the one which remembers only the finite pointed
set.

The property

$\mathcal{J}^{\oplus}_{n}\cong(\mathcal{J}^{\oplus}_{1})^{n}$

follows directly from the definition.

Next, $\mathcal{J}^{\oplus}\to\text{N}(\text{Fin}_{*})$ has to be an inner
fibration. The required lifting property follows, after unravelling the
definitions, directly from the fact that $\mathcal{J}$ is an $\infty$-category
(see also [24, Proof of 2.4.3.3.]).

Finally, $\mathcal{J}^{\oplus}\to\text{N}(\text{Fin}_{*})$ should be a
cocartesian fibration. Thus, let $f\colon X\to Y$ be a morphism in
$\text{N}(\text{Fin}_{*})$ and $\hat{X}\in\mathcal{J}^{\oplus}$ a lift of $X$.
We need to find a cocartesian lift $\hat{f}\colon\hat{X}\to\hat{Y}$. We let
$(\hat{Y})_{i}=\bigoplus_{f^{-1}(i)}\hat{X}_{i}$, and $\hat{f}$ the canonical
map (consisting of identity maps). $\hat{f}$ is cocartesian, as can be
verified using the definition of $\mathcal{J}^{\oplus}$ and [25, Prop.
2.4.4.3.]. ∎

####  4.1.2 Day Convolution and the Internal Hom Functor

#####  4.1.2.1 The Day Convolution Monoidal Structure

We will now put a monoidal structure on $\text{Fun}(\mathcal{J},\mathcal{S})$,
called the Day convolution monoidal structure, which has the important
property that monoids with respect to it are precisely lax monoidal functors.

Day convolution goes back to Brian Day in the 1-categorical setting ([11, 9])
and has been made available in the $\infty$-categorical world first by Saul
Glasman ([14]) and in greater generality by Jacob Lurie ([24, Section 2.2.6]).

We will first recall some generalities. The reader who is already familiar
with Day convolution may want to skip this section.

###### The case of symmetric monoidal 1-categories

For sufficiently nice111 i.e. $\mathcal{C}$ small, $\mathcal{D}$ admits all
small colimits symmetric monoidal categories $\mathcal{C}$ and $\mathcal{D}$
with tensor product functors

$\otimes_{\mathcal{C}}\colon\mathcal{C}\times\mathcal{C}\to\mathcal{C}$,
$\otimes_{\mathcal{D}}\colon\mathcal{D}\times\mathcal{D}\to\mathcal{D}$

and arbitrary functors

$F,G\colon\mathcal{C}\to\mathcal{D}$

their Day convolution product

$F\circledast G\in\text{Fun}(\mathcal{C},\mathcal{D})$

is defined to be the left Kan extension of $\otimes_{\mathcal{D}}\circ(F\times
G)$ along $\otimes_{\mathcal{C}}$ as shown in the diagram

${\mathcal{C}\times\mathcal{C}}$${\mathcal{D}\times\mathcal{D}}$${\mathcal{C}}$${\mathcal{D}}$$\scriptstyle{F\times
G}$$\scriptstyle{\otimes_{\mathcal{C}}}$$\scriptstyle{\otimes_{\mathcal{D}}\circ(F\times
G)}$$\scriptstyle{\otimes_{\mathcal{D}}}$$\scriptstyle{F\circledast
G:=\text{Lan}_{(\otimes_{\mathcal{D}}\circ(F\times G))}\otimes_{\mathcal{C}}}$

where the upper triangle commutes and the lower triangles commutes up to a
universal natural transformation

$\otimes_{\mathcal{D}}\circ(F\times
G)\Rightarrow(\text{Lan}_{(\otimes_{\mathcal{D}}\circ(F\times
G))}\otimes_{\mathcal{C}})\circ\otimes_{\mathcal{C}}$

######  Definition/Proposition 4.1.2.1 (Day [9], Lurie [24, Example
2.2.6.17]).

The association

$-\circledast-\colon\text{Fun}(\mathcal{C},\mathcal{D})\times\text{Fun}(\mathcal{C},\mathcal{D})\to\text{Fun}(\mathcal{C},\mathcal{D})$

determines a symmetric monoidal structure on
$\text{Fun}(\mathcal{C},\mathcal{D})$ such that monoid objects of
$(\text{Fun}(\mathcal{C},\mathcal{D}),\circledast)$ are precisely the lax
symmetric monoidal functors.

Using the pointwise formula for Kan extensions, there is a more explicit
description.

######  Proposition 4.1.2.2 (Day, Lurie [24, Remark 2.2.6.15]).

Let $F,G\colon\mathcal{C}\to\mathcal{D}$, then $F\circledast
G\colon\mathcal{C}\to\mathcal{D}$ is given on objects by

$(F\circledast
G)(C)=\operatorname{colim}\displaylimits_{C_{0}\otimes_{\mathcal{C}}C_{1}\to
C}F(C_{0})\otimes_{\mathcal{D}}G(C_{1})$

###### The case of symmetric monoidal $\infty$-categories

Analogous statements hold in the world of symmetric monoidal (or more
generally $\mathcal{O}$-monoidal) $\infty$-categories. We will use the
constructions of Lurie. References are in particular [24, Construction
2.2.6.7, Example 2.2.6.9, Example 2.2.6.17]. We reproduce the latter here for
convenience

######  Proposition 4.1.2.3 (Lurie, [24, Example 2.2.6.9]).

Let $\mathcal{C}^{\otimes}$ be a symmetric monoidal $\infty$-category and let
$\mathcal{D}^{\otimes}$ be an arbitrary $\infty$-operad. Applying Construction
2.2.6.7 of [24] in the case where
$\mathcal{O}^{\otimes}=\text{Comm}^{\otimes}$ is the commutative
$\infty$-operad, we obtain an $\infty$-operad
$\text{Fun}(\mathcal{C},\mathcal{D})^{\otimes}$ with the following features:

  * •

The underlying $\infty$-category of
$\text{Fun}(\mathcal{C},\mathcal{D})^{\otimes}$ is canonically equivalent to
the $\infty$-category $\text{Fun}(\mathcal{C},\mathcal{D})$ of functors from
$\mathcal{C}$ to $\mathcal{D}$.

  * •

The $\infty$-category $\text{CAlg}(\text{Fun}(\mathcal{C},\mathcal{D}))$ of
commutative algebra objects of $\text{Fun}(\mathcal{C},\mathcal{D})$ is
equivalent to the $\infty$-category $\text{Alg}_{\mathcal{C}}(\mathcal{D})$ of
lax symmetric monoidal functors from $\mathcal{C}$ to $\mathcal{D}$.

###### Day convolution on $\text{Fun}(\mathcal{J},\mathcal{S})$ and some
properties

As a special case of the previous discussion we obtain

######  Proposition 4.1.2.4.

$\text{Fun}(\mathcal{J},\mathcal{S})$ and
$\text{Fun}(\mathcal{J},\mathcal{S}_{*})$ are symmetric monoidal
$\infty$-categories with respect to Day convolution.

An example of functors whose convolution product is easy to understand are
representable functors.

######  Example 4.1.2.5.

$\text{mor}(V,-)\circledast\text{mor}(W,-)\simeq\text{mor}(V\oplus W,-)$

###### Proof.

By the pointwise formula (4.1.2.2), we see that

$(\text{mor}(V,-)\circledast\text{mor}(W,-))(X)\simeq\operatorname{colim}\displaylimits_{A\oplus
B\to X}\text{mor}(V,A)\times\text{mor}(W,B)$

The terms on the right-hand side admit maps

$\text{mor}(V,A)\times\text{mor}(W,B)\to\text{mor}(V\oplus W,A\oplus
B)\to\text{mor}(V\oplus W,X)$

which assemble to a map

$\operatorname{colim}\displaylimits_{A\oplus B\to
X}\text{mor}(V,A)\times\text{mor}(W,B)\to\text{mor}(V\oplus W,X)$

This map is an equivalence. An inverse is given by

$\displaystyle\text{mor}(V\oplus W,X)$
$\displaystyle\to\text{mor}(V,\text{im}_{f}(V))\times(W,\text{im}_{f}(W))$
$\displaystyle\to\operatorname{colim}\displaylimits_{A\oplus B\to
X}\text{mor}(V,A)\times\text{mor}(W,B)$ $\displaystyle(f\colon V\oplus W\to
X)$ $\displaystyle\mapsto(f_{|V},f_{|W})$

∎

Another important property of Day convolution in our setting is that it
commutes with colimits in each variable.

######  Lemma 4.1.2.6.

On $\text{Fun}(\mathcal{J},\mathcal{S})$, the convolution product
$\circledast$ commutes with colimits in each variable separately.

###### Proof.

Roughly, this follows from Proposition 4.1.2.2 by commuting colimits with
colimits and using that the (monoidal) product in $\mathcal{S}$ commutes with
colimits as $\mathcal{S}$ is closed monoidal.
Recall the definition

$F\circledast G\cong\text{Lan}_{\otimes_{J}}\otimes_{\mathcal{S}}\circ(F\times
G)$

Since $\text{Lan}_{\otimes_{\mathcal{J}}}$ is left adjoint to
$\otimes_{\mathcal{J}}^{*}\colon[\mathcal{J},\mathcal{S}]\to[\mathcal{J}\times\mathcal{J},\mathcal{S}]$,
it commutes with colimits in $\text{Fun}(J\times J,\mathcal{S})$. To have
$\circledast$ commute with colimits in each variable we thus need

$(\operatorname{colim}\displaylimits_{I}F_{i})\times
G\simeq\operatorname{colim}\displaylimits_{I}(F_{i}\times G)$

(and its analogue for $G_{i}$) in $\text{Fun}(J\times J,S)$. This holds as
colimits are computed in the codomain category and since $\mathcal{S}$ is
closed monoidal, hence $-\times G$ preserves colimits. ∎

#####  4.1.2.2 The Internal Hom Functor

######  Corollary 4.1.2.7.

$\text{Fun}(\mathcal{J},\mathcal{S})$ equipped with $\circledast$ is closed
monoidal, i.e. there is an internal hom functor

$\mathcal{F}\colon\text{Fun}(\mathcal{J},\mathcal{S})^{\text{op}}\times\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

such that for $X\in\text{Fun}(\mathcal{J},\mathcal{S})$, there is an
adjunction with left adjoint

$X\circledast-\
\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

and right adjoint

$\mathcal{F}(X,-)\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

###### Proof.

For fixed $X$, the existence of an adjoint to $X\circledast-$ follows from
Lemma 4.1.2.6 using the adjoint functor theorem (Corollary 5.5.2.9 of [25]).
That all of those adjoints assemble into a single internal hom functor
$\mathcal{F}$ follows since an adjoint exists if it exists for every object
(see e.g. [8, Prop. 6.1.11]). ∎

######  Remark 4.1.2.8.

For every $X$,
$\mathcal{F}(X,-)\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$
commutes with limits, as it is a right adjoint.

######  Proposition 4.1.2.9.

For an arbitrary functor $A\in\text{Fun}(\mathcal{J},\mathcal{S})$ and an
$n$-polynomial functor $P\in\text{Poly}^{\leq^{n}}(\mathcal{J},\mathcal{S})$,
the internal hom

$\mathcal{F}(A,P)\in\text{Fun}(\mathcal{J},\mathcal{S})$

is $n$-polynomial.

###### Proof.

By Definition 3.2.0.1, we have to check that

$\mathcal{F}(A,P)(X)\to\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}\mathcal{F}(A,P)(X\oplus U)=\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}\text{sh}_{U}^{*}\mathcal{F}(A,P)$

is an equivalence, where $\text{sh}_{U}$ denotes the functor
$\text{sh}_{U}\colon V\mapsto V\oplus U$ and
$\text{sh}_{U}^{*}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$
denotes precomposition by $\text{sh}_{U}$.

Assume for the moment that

$\text{sh}_{U}^{*}\mathcal{F}(A,P)\simeq\mathcal{F}(A,\text{sh}_{U}^{*}P)$

Then, we could calculate

$\displaystyle\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}\left(\text{sh}_{U}^{*}\mathcal{F}(A,P)\right)$
$\displaystyle\simeq\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}\mathcal{F}(A,\text{sh}_{U}^{*}P)$
$\displaystyle\simeq_{\lx@cref{creftype~refnum}{internal_hom_commutes_with_products}}\mathcal{F}\left(A,\lim_{0\neq
U\subseteq\mathbb{R}^{n+1}}\text{sh}_{U}^{*}P\right)$
$\displaystyle\simeq_{P\in\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}\mathcal{F}(A,P)$

and thus conclude that $\mathcal{F}(A,P)$ is $n$-polynomial.

It remains to prove that

$\text{sh}_{U}^{*}\mathcal{F}(A,P)\simeq\mathcal{F}(A,\text{sh}_{U}^{*}P)$

We observe that

$\text{sh}_{U}^{*}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

has a left adjoint

$(\text{sh}_{U})_{!}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

given by left Kan extension along $\text{sh}_{U}$, which exists as
$\mathcal{S}$ is cocomplete. Using the pointwise formula for Kan extensions,
it is given by

$(\text{sh}_{U})_{!}F(V)\simeq\operatorname{colim}\displaylimits\left(\text{sh}_{U}/V\to\mathcal{J}\to\mathcal{S}\right)\simeq\operatorname{colim}\displaylimits_{X\oplus
U\to V}F(X)$

Using the adjunction $(\text{sh}_{U})_{!}\dashv\text{sh}_{U}^{*}$ we can
compute

$\displaystyle\text{Nat}(A,\text{sh}_{U}^{*}\mathcal{F}(B,C))$
$\displaystyle\simeq\text{Nat}((\text{sh}_{U})_{!}A,\mathcal{F}(B,C))$
$\displaystyle\simeq\text{Nat}(((\text{sh}_{U})_{!}A)\circledast B,C)$
$\displaystyle\simeq\text{Nat}(((\text{sh}_{U})_{!}(A\circledast B),C)$
$\displaystyle\simeq\text{Nat}(A\circledast B,\text{sh}_{U}^{*}C)$
$\displaystyle\simeq\text{Nat}(A,\mathcal{F}(B,\text{sh}_{U}^{*}C))$

where we used that

$(\text{sh}_{U})_{!}A\circledast B\simeq(\text{sh}_{U})_{!}(A\circledast B)$

as Day convolution commutes with colimits and the left Kan extension
$(\text{sh}_{U})_{!}$ is given pointwise by colimits.

Thus, by the Yoneda lemma

$\text{sh}_{U}^{*}\mathcal{F}(B,C)\simeq\mathcal{F}(B,\text{sh}_{U}^{*}C)$

∎

###### A Formula for the Internal Hom Functor

In Corollary 4.1.2.7 and Proposition 4.1.2.9, we only proved and used the
existence of the internal hom functor
$\mathcal{F}\colon\text{Fun}(\mathcal{J},\mathcal{S})^{op}\times\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$,
but did not provide a formula. Luckily, $\text{Fun}(\mathcal{J},\mathcal{S})$
is a (co)presheaf category, where this is relatively easy.

######  Proposition 4.1.2.10.

For $A,B\in\text{Fun}(\mathcal{J},\mathcal{S})$ and $X\in\mathcal{J}$

$\mathcal{F}(A,B)(X)\simeq\text{Nat}(A,\text{sh}_{X}^{*}B)$

###### Proof.

Denote by

$\displaystyle\text{y}\colon\mathcal{J}^{op}$
$\displaystyle\to\text{Fun}(\mathcal{J},\mathcal{S})$ $\displaystyle V$
$\displaystyle\mapsto\text{Map}(V,-)$

the contravariant Yoneda embedding.

Observe first that there is an equivalence

$\mathcal{F}(A,B)(X)\simeq\text{Nat}(\text{y}(X)\circledast A,B)$

which is a simple consequence of the Yoneda lemma, as

$\displaystyle\mathcal{F}(G,H)(X)$
$\displaystyle\simeq\text{Map}_{PSh(\mathcal{J}^{op})}(\text{y}(X),\mathcal{F}(G,H))$
$\displaystyle\simeq\text{Map}_{PSh(\mathcal{J}^{op})}(\text{y}(X)\circledast
G,H)$ $\displaystyle\simeq\text{Nat}(\text{y}(X)\circledast G,H)$

Next, recall from Example 4.1.2.5 that

$\text{y}(X)\circledast\text{y}(Y)\simeq\text{y}(X\oplus Y)$

for $X,Y\in\mathcal{J}$. Thus, again by Yoneda,

$\mathcal{F}(\text{y}(Y),A)(X)\simeq A(Y\oplus X)$

Finally, the claim follows by calculating

$\displaystyle\mathcal{F}(A,B)(X)$
$\displaystyle\simeq\text{Map}(\text{y}(X),\mathcal{F}(A,B))$
$\displaystyle\simeq\text{Map}(\text{y}(X)\circledast A,B)$
$\displaystyle\simeq\text{Map}(A,\mathcal{F}(\text{y}(X),B))$
$\displaystyle\simeq\text{Nat}(A,\text{sh}_{X}^{*}B)$

∎

######  Remark 4.1.2.11.

One can also use this formula to give another proof of Proposition 4.1.2.9.
Recall that we had to check that

$\text{sh}_{U}^{*}\mathcal{F}(A,P)\simeq\mathcal{F}(A,\text{sh}_{U}^{*}P)$

Indeed, using Proposition 4.1.2.10, we see that

$\text{sh}_{U}^{*}\mathcal{F}(A,P)(X)\simeq\text{Nat}(A,\text{sh}_{X\oplus
U}^{*}P)$

and that

$\mathcal{F}(A,\text{sh}_{U}^{*}P)(X)\simeq\text{Nat}(A,\text{sh}_{X}^{*}\text{sh}_{U}^{*}P)$

The claim follows as $\text{sh}_{U}\text{sh}_{X}\simeq\text{sh}_{X\oplus U}$.

####  4.1.3 Generalities on Monoidal Localisations

To exhibit the localisation $T_{n}$ as monoidal, we will use the following
abstract result from category theory, which is essentially due to Brian Day
([10], Corollary 1.4) in the 1-categorical setting and has been carried over
to $\infty$-land by Lurie.

######  Definition 4.1.3.1 ([10], Definition 1.3).

Let $Z$ be a class of morphisms in a symmetric monoidal category
$\mathcal{A}$. Then $Z$ is called monoidal if

$Z=\\{s\in Z\mid\text{id}_{A}\otimes s\in Z\text{ for all }A\in\mathcal{A}\\}$

or equivalently if the class $Z$ is closed under the tensor product.

######  Proposition 4.1.3.2 ([10], Corollary 1.4).

If $Z$ is monoidal in $\mathcal{A}$, then there is a monoidal structure on
$\mathcal{A}[Z^{-1}]$ such that the projection functor
$\mathcal{A}\to\mathcal{A}[Z^{-1}]$ is a monoidal functor.

###### Proof sketch.

Under the assumption on $Z$, it is straightforward to check that the upper map
in the diagram

${{\mathcal{A}\times\mathcal{A}}}$${{\mathcal{A}}}$${{\mathcal{A}[Z^{-1}]\times\mathcal{A}[Z^{-1}]}}$${{\mathcal{A}[Z^{-1}]}}$$\scriptstyle{\otimes_{\mathcal{A}}}$$\scriptstyle{\otimes_{\mathcal{A}[Z^{-1}]}}$

sends morphisms with components in $Z$ to equivalences, hence the universal
property induces the required factorisation. ∎

In $\infty$-land, the analogous statement follows from [24, Prop. 2.2.1.9]. In
the literature, it appears as [13, Lemma 3.4]. The (equivalent) version which
we find most convenient appears in Eigil Rischel’s bachelor thesis [30,
Section 8].

######  Definition 4.1.3.3 ([24, Def. 2.2.1.6], [13, Def. 3.3],[30, Def. 8.1,
Rmk. 8.2]).

Let $\mathcal{C}^{\otimes}$ be a symmetric monoidal $\infty$-category and
$L\colon\mathcal{C}\to\mathcal{C}$ be a localisation of the underlying
$\infty$-category. Then we say that $L$ is compatible with the symmetric
monoidal structure if the following equivalent conditions are satisfied:

  * •

The tensor product of local equivalences222 Recall that morphism $f\colon
C_{1}\to C_{2}$ in $\mathcal{C}$ is a local equivalence if $L(f)$ is an
equivalence.  is a local equivalence.

  * •

Whenever $f\colon X\to Y$ is a local equivalence, and $Z$ is any object of
$\mathcal{C}$, the map

$f\otimes\text{id}_{Z}\colon X\otimes Z\to Y\otimes Z$

is also a local equivalence.

######  Proposition 4.1.3.4 ([24, Prop. 2.2.1.9], [13, Lemma 3.4], [30, Prop.
8.3]).

Let $p\colon\mathcal{C}^{\otimes}\to\text{N}(\text{Fin}_{*})$ be a symmetric
monoidal $\infty$-category and $L\colon\mathcal{C}\to\mathcal{C}$ a
localisation of $\mathcal{C}$ which is compatible with the symmetric monoidal
structure. Let $(L\mathcal{C})^{\otimes}$ be the subcategory of
$\mathcal{C}^{\otimes}$ spanned by objects of the form
$(LC_{1},\ldots,LC_{n})$. Then

  * •

The restriction $p\colon(L\mathcal{C})^{\otimes}\to\text{N}(\text{Fin}_{*})$
exhibits $(L\mathcal{C})^{\otimes}$ as a symmetric monoidal $\infty$-category
with underlying $\infty$-category $L\mathcal{C}$.

  * •

The inclusion $(L\mathcal{C})^{\otimes}\to\mathcal{C}^{\otimes}$ admits a left
adjoint $L^{\otimes}$ such that $(L^{\otimes})_{|\mathcal{C}}\simeq L$ and the
unit $\eta$ of the adjuncton can be chosen such that $p(\eta)=1$.

  * •

$L^{\otimes}$ is a (strict) symmetric monoidal functor.

  * •

The inclusion $(L\mathcal{C})^{\otimes}\subset\mathcal{C}^{\otimes}$ is a lax
symmetric monoidal functor.

###### Proof sketch.

The key step is to construct the left adjoint
$L^{\otimes}\colon\mathcal{C}^{\otimes}\to(L\mathcal{C})^{\otimes}$. This uses
that $\infty$-categorical adjoints exist once each object admits an adjoint
object333 This is where the coherences required for a functor of
$\infty$-categories are taken care of. See e.g. Proposition 6.1.11 of [8]. .
For an object $(C_{1},\ldots,C_{k})\in\mathcal{C}^{\otimes}$,

$(\eta_{C_{1}},\ldots,\eta_{C_{k}})\colon(C_{1},\ldots,C_{k})\to(LC_{1},\ldots,LC_{k})$

(where $\eta$ is the unit of the original adjunction) is a localisation with
respect to $(L\mathcal{C})^{\otimes}$, which can be checked by considering an
arbitrary object $(Y_{1},\ldots,Y_{l})$ and checking that the induced map

$\text{Map}_{\mathcal{C}^{\otimes}}((LC_{1},\ldots,LC_{k}),(Y_{1},\ldots,Y_{l}))\to\text{Map}_{\mathcal{C}^{\otimes}}((C_{1},\ldots,C_{k}),(Y_{1},\ldots,Y_{l}))$

is an equivalence, which can be checked componentwise, where it follows from
the assumption.
The remainder of the proof consists of some considerations of coCartesian
morphisms and fibrations, using [24, Lemma 2.2.1.11]. ∎

######  Corollary 4.1.3.5 ([13, Lemma 3.6, Rmk. 3.7], , [30, Cor. 8.6, Cor.
8.8]).

Let $\mathcal{C}^{\otimes}$ be a symmetric monoidal $\infty$-category equipped
with a symmetric monoidal localisation $L\colon\mathcal{C}\to L\mathcal{C}$
and let $R\colon L\mathcal{C}\to\mathcal{C}$ be the right adjoint. Then there
is an induced localisation
$L^{\prime}\colon\text{CAlg}(\mathcal{C})\to\text{CAlg}(L\mathcal{C})$ such
that the diagram

${{\text{CAlg}(\mathcal{C})}}$${{\text{CAlg}(L\mathcal{C})}}$${\mathcal{C}}$${L\mathcal{C}}$$\scriptstyle{L}$$\scriptstyle{L^{\prime}}$

commutes. Moreover, given $A\in\text{CAlg}(\mathcal{C})$, there exists a
unique commutative algebra structure on $RLA$ such that the unit map $A\to
RLA$ extends to a morphism of commutative algebras.

###### Proof.

An $E_{\infty}$-algebra in $\mathcal{C}$ is given by a section of
$p\colon\mathcal{C}^{\otimes}\to\text{N}(\text{Fin}_{*})$. Composing such a
section with $\mathcal{C}^{\otimes}\to(L\mathcal{C})^{\otimes}$ gives a
section of $(L\mathcal{C})^{\otimes}\to\text{N}(\text{Fin}_{*})$, i.e. an
$E_{\infty}$-algebra in $L\mathcal{C}$.
The second part follows from the fact that a map of algebras is an equivalence
if the underlying map is an equivalence and the universal property of the
unit. ∎

Now that we have stated the rigorous details, let’s step back for a moment to
reflect on what we just said.

######  Remark 4.1.3.6 (Informal summary).

If a localisation $L\colon\mathcal{C}\to L\mathcal{C}$ and a monoidal
structure $\otimes_{\mathcal{C}}$ on $\mathcal{C}$ are compatible in the sense
that local equivalences are closed under the tensor product, then:

  * •

There is a symmetric monoidal structure on $L\mathcal{C}$ given by

$LC_{1}\otimes_{L\mathcal{C}}LC_{2}:=L(LC_{1}\otimes_{\mathcal{C}}LC_{2})$

  * •

The lax monoidal structure on $\iota\colon L\mathcal{C}\to\mathcal{C}$ is
given by the unit of the adjunction, i.e. by the maps

$\eta_{LC_{1}\otimes_{\mathcal{C}}LC_{2}}\colon
LC_{1}\otimes_{\mathcal{C}}LC_{2}\to
L(LC_{1}\otimes_{\mathcal{C}}LC_{2})\simeq LC_{1}\otimes_{L\mathcal{C}}LC_{2}$

  * •

$L\colon\mathcal{C}\to L\mathcal{C}$ is (strict) symmetric monoidal, as under
the hypothesis, there are equivalences

$L(C_{1}\otimes_{\mathcal{C}}C_{2})\to
L(LC_{1}\otimes_{\mathcal{C}}LC_{2})\simeq LC_{1}\otimes_{L\mathcal{C}}LC_{2}$

###  4.2 A Monoidal Localisation Theorem for Taylor Approximations in
Orthogonal Calculus

Recall that Orthogonal Calculus provides left-exact localisations

${{\text{Fun}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{T_{n}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$

Our original goal was to show that whenever $F\colon\mathcal{J}\to\mathcal{S}$
is a lax monoidal functor (for example $V\mapsto\text{O}(V)$ or
$V\mapsto\text{TOP}(V)$), so is $T_{n}F$. In light of 4.1.2.3, it is enough to
show that
$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$
is lax monoidal with respect to Day convolution, as a lax monoidal functor
sends monoids to monoids. Hence, our theorem now takes the following form

######  Theorem 4.2.0.1.

The $n$-polynomial approximation functors

$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Poly}^{\leq
n}(\text{Fun}(\mathcal{J},\mathcal{S}))$

are (strict) symmetric monoidal with respect to the Day convolution monoidal
structure on $\text{Fun}(\mathcal{J},\mathcal{S})$ and an induced monoidal
structure on $\text{Poly}^{\leq n}(\text{Fun}(\mathcal{J},\mathcal{S}))$. The
inclusion

$\iota\colon\text{Poly}^{\leq
n}(\text{Fun}(\mathcal{J},\mathcal{S}))\to\text{Fun}(\mathcal{J},\mathcal{S})$

is lax symmetric monoidal. The composition444 which is often just called
$T_{n}$ as well

$\iota\circ
T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$

is lax symmetric monoidal.

######  Corollary 4.2.0.2.

The $n$-polynomial approximation $T_{n}F\colon\mathcal{J}\to\mathcal{S}$ of a
lax symmetric monoidal functor $F\colon\mathcal{J}\to\mathcal{S}$ is lax
symmetric monoidal.

Using what we explained in the previous section, it is sufficient to show that
the localisation $T_{n}$ is compatible with the Day convolution monoidal
structure on $\text{Fun}(\mathcal{J},\mathcal{S})$ in the sense of Definition
4.1.3.3.

######  Proposition 4.2.0.3.

$T_{n}$ is compatible with the Day convolution monoidal structure on
$\text{Fun}(\mathcal{J},\mathcal{S})$ in the sense of Definition 4.1.3.3.

The proof of Proposition 4.2.0.3 will be given in the next section. Assuming
it for now, the main theorem follows easily:

###### Proof of Theorem 4.2.0.1 assuming compatibility of $T_{n}$ and Day
convolution.

By Proposition 4.2.0.3, the localisation

${{\text{Fun}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{T_{n}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$

satisfies the conditions of 4.1.3.4. As a consequence, $\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})$ inherits a symmetric monoidal structure such that
$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})$ is symmetric monoidal, the inclusion
$\iota\colon\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$ is lax
symmetric monoidal, $\iota\circ
T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S})\to\text{Fun}(\mathcal{J},\mathcal{S})$
is lax symmetric monoidal. ∎

###### Proof of Corollary 4.2.0.2.

There is an equivalence between lax symmetric monoidal functors
$\mathcal{J}\to\mathcal{S}$ and monoids in the Day convolution symmetric
monoidal category (Proposition 4.1.2.3). By Theorem 4.2.0.1, $\iota\circ
T_{n}$ is lax symmetric monoidal, hence maps monoids to monoids. The diagram
looks as follows.

${{\text{CAlg}(\text{Fun}(\mathcal{J},\mathcal{S}))}}$${{\text{CAlg}(\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}))}}$${{\text{CAlg}(\text{Fun}(\mathcal{J},\mathcal{S}))}}$${{\text{Fun}^{\text{lax
monoidal}}(\mathcal{J},\mathcal{S})}}$${{\text{Fun}^{\text{lax
monoidal}}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\iota\circ
T_{n}}$$\scriptstyle{T_{n}}$$\scriptstyle{\iota}$

∎

###### Proof of Proposition 4.2.0.3 (Compatibility).

Step 1: Reduction to $\eta_{A}\colon A\to T_{n}A$
First, we claim that it is sufficient to show that for any
$A,C\in\text{Fun}(\mathcal{J},\mathcal{S})$, the transformation

$\eta_{A}\circledast\text{id}_{C}\colon A\circledast C\to T_{n}A\circledast C$

is an $n$-polynomial equivalence555One might also call such a map a
$T_{n}$-local equivalence., i.e. $T_{n}(\eta_{A}\circledast\text{id}_{C})$ is
an equivalence.
To see that, let $f\colon A\to B$ be an arbitrary $n$-polynomial equivalence.
Consider the diagram

${{A\circledast C}}$${{B\circledast C}}$${{T_{n}A\circledast
C}}$${{T_{n}B\circledast
C}}$$\scriptstyle{f\circledast\text{id}_{C}}$$\scriptstyle{\eta_{A}\circledast\text{id}_{C}}$$\scriptstyle{\eta_{B}\circledast\text{id}_{C}}$$\scriptstyle{T_{n}(f)\circledast\text{id}_{C}}$

The lower horizontal morphism is an equivalence to begin with, hence the upper
morphism is sent to an equivalence by $T_{n}$ if the same is true for the left
and the right one.
Step 2: Yoneda
Let again $A,C\in\text{Fun}(\mathcal{J},\mathcal{S})$. By Yoneda in
$\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$, it suffices to show that for
any $P\in\text{Poly}^{\leq n}(\mathcal{J},\mathcal{S})$, the upper map in

${{\text{Map}(T_{n}A\circledast C,P)}}$${{\text{Map}(A\circledast
C,P)}}$${{\text{Map}(T_{n}(T_{n}A\circledast
C),P)}}$${{\text{Map}(T_{n}(A\circledast
C),P)}}$$\scriptstyle{(\eta_{A}\circledast\text{id}_{C})^{*}}$$\scriptstyle{(T_{n}(\eta_{A}\circledast\text{id}_{C}))^{*}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

is an equivalence.

Step 3: $(\eta_{A}\circledast\text{id}_{C})^{*}$ is an equivalence:

What remains to be done is the following simple calculation using the internal
hom adjunction from Corollary 4.1.2.7, Proposition 4.1.2.9 and the adjunction
between $T_{n}$ and the inclusion of polynomial functors.

$\displaystyle\text{Nat}(A\circledast C,P)$
$\displaystyle\simeq\text{Nat}(A,\mathcal{F}(C,P))$
$\displaystyle\simeq\text{Nat}(T_{n}A,\mathcal{F}(C,P))$ $\displaystyle{\text{
\small(as }\mathcal{F}(C,P)\text{ is polynomial by
\lx@cref{creftype~refnum}{internal_hom_remains_poly})}}$
$\displaystyle\simeq\text{Nat}(T_{n}A\circledast C,P)$

where in the second step we used that $\mathcal{F}(C,P)$ is polynomial by
Proposition 4.1.2.9.

∎

######  Proposition 4.2.0.4.

Theorem 4.2.0.1 also holds for functors valued in pointed spaces, equipped
with the smash product.

###### Proof sketch.

Recall the key steps of the proof

  1. 1.

Lemma 4.1.2.6: Day Convolution commutes with colimits

  2. 2.

Corollary 4.1.2.7: The internal hom functor exists.

  3. 3.

Proposition 4.1.2.9: The internal hom into a $n$-polynomial functor is
$n$-polynomial.

  4. 4.

Section 4.2: We check the compatibility condition of Definition 4.1.3.3 to
conclude via Proposition 4.1.3.4 that the localisation is monoidal.

They are all readily adapted. In the first step, we use that
$(\mathcal{S}_{*},\wedge)$ is closed monoidal. The other steps are then quite
formal and work the same as for $\mathcal{S}$.

∎

The same proof cannot work for the categories of polynomial functors valued in
pointed spaces equipped with the cartesian product, as the cartesian product
in $\mathcal{S}_{*}$ and thus the Day convolution product in
$\text{Fun}(\mathcal{J},\mathcal{S}_{*})$ do not commute with colimits and
hence cannot admit an internal hom functor.

However, the same result is still true.

######  Proposition 4.2.0.5.

Theorem 4.2.0.1 also holds for functors valued in pointed spaces, equipped
with the cartesian product.

###### Proof.

This follows relatively easily from Theorem 4.2.0.1. Namely, consider the
localisation

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})}}$${{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}}$$\scriptstyle{T_{n}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$

By Proposition 4.1.3.4, just like in the proof of Theorem 4.2.0.1, it is
sufficient to check that for $F,G\in\text{Fun}(\mathcal{J},\mathcal{S}_{*})$,
$T_{n}(\eta_{F}\circledast\text{id}_{G})$ is an equivalence (as a natural
transformation of functors to pointed spaces). But as the cartesian product of
pointed spaces is just the cartesian product of the corresponding spaces
(equipped with the obvious basepoint), we have already shown in Theorem
4.2.0.1 that it is an equivalence of functors to spaces. To conclude, we
remember that a map between pointed spaces which is an unpointed equivalence
is a pointed equivalence. ∎

##  Chapter 5 Monoidality and Derivative Spectra

Finally, we are ready to come back to the question that started this
endeavour, namely

######  Question 5.0.0.1.

Let $F\colon\mathcal{J}\to\mathcal{S}$ be a monoidal functor. To what
structure on the $n$-th derivative spectra $\Theta^{n}F$ does the
multiplication map

$F\circledast F\to F$

correspond?

At the end of this chapter, we will have our answer.

######  Answer 5.0.0.2 (Corollary 5.4.0.1).

Let $F\colon\mathcal{J}\to\mathcal{S}$ be a monoidal functor. Then, for any
$n\in\mathbb{N}$, there are $O(n)$-equivariant maps

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes\mathbb{S}^{\text{Ad}_{\text{O}(n)}}$

###  5.1 Pointed and Unpointed Functors

In this chapter, we mostly work with functors
$F\colon\mathcal{J}\to\mathcal{S}_{*}$, i.e. functors to pointed spaces. When
we form their convolution product and internal hom, the one of the
corresponding functors to $\mathcal{S}$ is meant. These a priori unpointed
convolution products and internal homs are functors to pointed spaces as well
(as we show now). This should be understood in analogy with spaces and pointed
spaces: The product of pointed spaces as well as the space of not necessarily
basepoint-preserving maps of pointed spaces are pointed spaces, even though
there is no adjunction between them.

######  Proposition 5.1.0.1.

For functors $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$, both their convolution
product $A\circledast B\colon\mathcal{J}\to\mathcal{S}$ and their internal hom
$\mathcal{F}(A,B)\colon\mathcal{J}\to\mathcal{S}$ naturally lift to functors
to pointed spaces.

###### Proof.

First observe that a lift

${\mathcal{S}_{*}}$${\mathcal{J}}$${\mathcal{S}}$$\scriptstyle{A}$

corresponds to a natural transformation

$*\to A$

where $*$ denotes to constant functor with value the one-point space. This
functor is the monoidal unit in $\text{Fun}(\mathcal{J},\mathcal{S})$ with
respect to Day convolution.

The convolution product $A\circledast B$ is then pointed via

$*\simeq*\circledast*\to A\circledast B$

More explicitly, recall that

$(A\circledast B)(V)\simeq\operatorname{colim}\displaylimits_{V_{1}\oplus
V_{2}\to V}A(V_{1})\times B(V_{2})$

In this perspective, the basepoint is given by the class of the basepoint
(i.e. product of the basepoints) in $A(0)\times B(0)$.

For the internal hom, observe that

$\text{Nat}(*,\mathcal{F}(A,B))\simeq\text{Nat}(*\circledast
A,B)\simeq\text{Nat}(A,B)$

The basepoint in $\text{Nat}(A,B)$ is given by the constant natural
transformation

$A\to*\to B$

∎

######  Proposition 5.1.0.2 (Monoids are canonically pointed).

Let $F\colon\mathcal{J}\to\mathcal{S}$ be lax symmetric monoidal. Then $F$
canonically lifts to a functor $F\colon\mathcal{J}\to\mathcal{S}_{*}$ to
pointed spaces. The corresponding natural transformation $*\to F$ is a
homomorphism of monoidal functors.

###### Proof.

By definition, $F$ is equipped with a morphism $\varphi\colon
1_{\mathcal{S}}\to F(1_{\mathcal{J}})$. As $1_{\mathcal{S}}=\\{*\\}$ and
$1_{\mathcal{J}}=0$, this defines a basepoint in $F(0)$. As $0\in\mathcal{J}$
is initial, this also defines a basepoint in any $F(V)$. The homomorphism
property follows from the coherence axioms for lax monoidal functors.
Alternatively (but equivalently), such an $F$ is a commutative monoid in
$\text{Fun}(\mathcal{J},\mathcal{S})$ and hence receives a map from the tensor
unit of $\text{Fun}(\mathcal{J},\mathcal{S})$. In general, the tensor unit
with respect to Day convolution in $\text{Fun}(\mathcal{C},\mathcal{D})$ is
given by composing

$\text{Map}_{\mathcal{C}}(1_{\mathcal{C}},-)\colon\mathcal{C}\to\mathcal{S}$

with the unique cocontinuous functor

$\displaystyle\mathcal{S}$ $\displaystyle\to\mathcal{D}$ $\displaystyle X$
$\displaystyle\mapsto(1_{\mathcal{D}})^{\oplus X}$

In our case, as $1_{\mathcal{C}}=1_{\mathcal{J}}=0$ is initial, the tensor
unit in $\text{Fun}(\mathcal{J},\mathcal{S})$ is the constant functor with
value $*$, which also proves the claim. ∎

######  Corollary 5.1.0.3.

Let $F,G\in\text{CAlg}(\text{Fun}(\mathcal{J},\mathcal{S}))$. Then
$\mathcal{F}(F,G)$ is canonically pointed.

###### Proof.

Apply Proposition 5.1.0.1. ∎

###  5.2 Monoidality of Homogeneous Layers

There are two corollaries left to be drawn from the earlier Theorem 4.2.0.1,
which are almost but not quite the same.

######  Corollary 5.2.0.1.

For $F\colon\mathcal{J}\to\mathcal{S}$ lax symmetric monoidal, its homogeneous
layers

$L_{n}F=\text{fib}(T_{n}F\to T_{n-1}F)\colon\mathcal{J}\to\mathcal{S}$

(where the fibres are taken over the canonical basepoints provided by the
monoidal unit) are lax monoidal as well.

###### Proof.

By Theorem 4.2.0.1, $T_{n}F$ and $T_{n-1}F$ are monoidal. The natural
transformation $T_{n}F\to T_{n-1}F$ is a morphism of monoidal functors, by
commutativity of

${{\text{Fun}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$${{\text{Poly}^{\leq
n-1}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{{T_{n-1}}_{|\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})}}$$\scriptstyle{T_{n-1}}$

as the inclusion $\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S})\hookrightarrow\text{Fun}(\mathcal{J},\mathcal{S})$
is lax monoidal.

Hence, $L_{n}F$ is monoidal, as it is the kernel of a morphism of monoidal
functors which preserves the unit. (See [24, Cor. 3.2.2.5] for details.) ∎

######  Corollary 5.2.0.2.

For $F\colon\mathcal{J}\to(\mathcal{S}_{*},\times)$ lax symmetric monoidal,
its homogeneous layers

$L_{n}F=\text{fib}(T_{n}F\to T_{n-1}F)\colon\mathcal{J}\to\mathcal{S}_{*}$

are lax monoidal as well.

###### Proof.

By Corollary 5.2.0.1, they are monoidal functors to spaces. By Proposition
5.1.0.2, this is the same as being monoidal functors to pointed spaces (with
the cartesian product). ∎

###  5.3 Classification of Homogeneous Layers and the Internal Hom

The next step is to combine this with the classification of the homogeneous
layers. Recall from Theorem 3.3.0.6 that any $n$-homogeneous functor to
pointed spaces is of the form

$V\mapsto\Omega^{\infty}\left(\left(S^{V\otimes\mathbb{R}^{n}}\wedge\Phi\right)_{hO(n)}\right)$

as there is an equivalence

$\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})\simeq Sp^{\text{BO}(n)}$

Let’s denote the involved functors by

${\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})}$${{Sp^{\text{BO}(n)}}}$$\scriptstyle{\Theta^{n}}$$\scriptstyle{\Phi}$

where $\Theta^{n}$ is the $n$-th derivative spectrum (over the basepoint) as
constructed by Weiss and

$\displaystyle\Phi\colon Sp^{\text{BO}(n)}$
$\displaystyle\to\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$ $\displaystyle
A$ $\displaystyle\mapsto\left(V\mapsto\Omega^{\infty}\left((S^{nV}\wedge
A)_{hO(n)}\right)\right)$

Homogeneous functors to $\mathcal{S}_{*}$ are in particular functors to
$\mathcal{S}$, thus it makes sense to ask for a description of
$\mathcal{F}(\Phi A,\Phi B)$ for $A,B\in Sp^{\text{BO}(n)}$.

######  Proposition 5.3.0.1.

For $A,B\in Sp^{\text{BO}(n)}$ and $X\in\mathcal{J}$

$\displaystyle\mathcal{F}(\Phi A,\Phi B)(X)$
$\displaystyle\simeq\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S})}\left(\Omega^{\infty}\left((S^{nV}\wedge
A)_{hO(n)}\right),\Omega^{\infty}\left((S^{nV}\wedge S^{nX}\wedge
A)_{hO(n)}\right)\right)$

###### Proof.

A straightforward application of Proposition 4.1.2.10 gives

$\displaystyle\mathcal{F}(\Phi A,\Phi B)(X)$
$\displaystyle\simeq\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S})}\left(\Omega^{\infty}\left((S^{nV}\wedge
A)_{hO(n)}\right),\Omega^{\infty}\left((S^{n(V\oplus X)}\wedge
A)_{hO(n)}\right)\right)$

Using $S^{n(V\oplus X)}\simeq S^{nV}\wedge S^{nX}$ yields the claim. ∎

######  Remark 5.3.0.2.

Recall that as we have $\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})\simeq
Sp^{\text{BO}(n)}$, we know that

$\displaystyle\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}\left(\Omega^{\infty}\left((S^{nV}\wedge
A)_{hO(n)}\right),\Omega^{\infty}\left((S^{nV}\wedge S^{nX}\wedge
B)_{hO(n)}\right)\right)$
$\displaystyle\simeq\text{Map}_{Sp^{\text{BO}(n)}}(A,S^{nX}\wedge B)$

######  Definition 5.3.0.3.

For $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$, denote by $\mathcal{F}_{*}(A,B)$
the subfunctor of $\mathcal{F}(A,B)$ given by pointed natural transformations,
i.e.

$\mathcal{F}_{*}(A,B)(V)\simeq\text{Nat}_{\text{Fun}(\mathcal{J},\mathcal{S}_{*})}(A,\text{sh}_{V}^{*}B)$

(This is the internal hom functor in
$\text{Fun}((\mathcal{J},\oplus),(\mathcal{S}_{*},\wedge))$.)

######  Proposition 5.3.0.4.

For functors $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$, there is a natural
fibre sequence

$\mathcal{F}_{*}(A,B)\to\mathcal{F}(A,B)\to B$

which splits and hence induces an equivalence

$\Theta^{n}\mathcal{F}_{*}(A,B)\oplus\Theta^{n}B\simeq\Theta^{n}\mathcal{F}(A,B)$

###### Proof.

The second map is given by evaluation at the basepoint of $A$. More formally,
the pointing of $A$ is given by a map $p_{A}\colon*\to A$. This induces the
map

$p_{A}^{*}\colon\mathcal{F}(A,B)\to\mathcal{F}(*,B)\simeq B$

The fibre of this map over the basepoint of $B$ is then given by the
basepoint-preserving transformations, i.e. $\mathcal{F}_{*}(A,B)$. The
splitting is given by

$(A\to*)^{*}\colon\mathcal{F}(*,B)\to\mathcal{F}(A,B)$

(i.e. by sending every $a\in A(V)$ to the same chosen $b\in B(V)$ for all $V$
simultaneously).
The formula for $\Theta^{n}$ then follows as $\Theta^{n}$ preserves finite
limits, as $T_{n}$ and $T_{n-1}$ do and $L_{n}$, which determines
$\Theta^{n}$, is the fibre of $T_{n}\to T_{n-1}$. ∎

####  5.3.1 Calculation of $\Theta^{n}\mathcal{F}_{*}(A,B)$ for $A,B$
$n$-homogeneous

In this subsection, we will do the following calculation.

######  Proposition 5.3.1.1.

For $A,B\in Sp^{\text{BO}(n)}$ there is an equivalence

$\displaystyle\Theta^{n}\mathcal{F}_{*}(\Phi A,\Phi B)$
$\displaystyle\simeq\text{Map}_{Sp}(A,B)\otimes D_{O(n)}$

with $D_{O(n)}\simeq(\Sigma^{\infty}_{+}O(n))^{hO(n)}$ being the dualising
spectrum of $O(n)$ (see Remark 5.3.1.7).

On the right-hand side, $\text{Map}_{Sp}(A,B)\otimes D_{O(n)}$, $O(n)$ acts
diagonally, i.e. via the natural action on $D_{O(n)}$ and the conjugation
action on $\text{Map}_{Sp}(A,B)$. The proof is given at the end of this
section, after some preparatory calculations. The observation that gets us
going is that

$\text{Map}_{Sp^{\text{BO}(n)}}(A,B)\simeq(\text{Map}_{Sp}(A,B))^{hO(n)}$

where $O(n)$ acts on $\text{Map}_{Sp}(A,B)$ by conjugation. We shall first
consider the functor

$V\mapsto\text{Map}(A,S^{nV}\wedge B)$

and then take care of the homotopy fixed point construction.

######  Lemma 5.3.1.2.

For any (nonequivariant) spectrum $A$, the functor

$\displaystyle\mathcal{J}$ $\displaystyle\to\mathcal{S}_{*}$ $\displaystyle V$
$\displaystyle\mapsto\Omega^{\infty}(S^{nV}\wedge A)$

is $n$-homogeneous, with $n$-th derivative spectrum given by

$\Theta^{n}(\Omega^{\infty}(S^{nV}\wedge A))\simeq A\wedge O(n)_{+}$

with the $O(n)$-action being given by the canonical left action on $O(n)_{+}$.

###### Proof.

We can associate to any spectrum $X$ the $O(n)$-free spectrum

$O(n)_{+}\wedge X$

It satisfies

$(O(n)_{+}\wedge X)_{hO(n)}\simeq X$

Applying this to $X=S^{nV}\wedge A$, we obtain

$\Omega^{\infty}(S^{nV}\wedge
A)\simeq\Omega^{\infty}\left((S^{nV}\wedge(O(n)_{+}\wedge A))_{hO(n)}\right)$

To conclude that $O(n)_{+}\wedge A$ is indeed the derivative, we implicitly
use that there is a shearing isomorphism which identifies the $O(n)$-action
acting only on $O(n)_{+}$ with the one acting diagonally on $O(n)_{+}$ and on
the $\mathbb{R}^{n}$ in $S^{nV}\simeq S^{\mathbb{R}^{n}\otimes V}$. ∎

######  Corollary 5.3.1.3.

For $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$, the functor

$\displaystyle\mathcal{J}$ $\displaystyle\to\mathcal{S}_{*}$ $\displaystyle V$
$\displaystyle\mapsto\text{Map}_{Sp}(\Theta^{n}A,\Theta^{n}(\text{sh}_{V}^{*}B))\simeq
S^{nV}\wedge\text{Map}_{Sp}(\Theta^{n}A,\Theta^{n}B)$

is $n$-homogeneous with $n$-th derivative spectrum given by

$O(n)_{+}\wedge\text{Map}_{Sp}(\Theta^{n}A,\Theta^{n}B)$

The $O(n)$-structure of this derivative spectrum is given by the action only
on the $O(n)_{+}$-factor.

###### Proof.

This follows from Lemma 5.3.1.2 by setting
$A=\text{Map}_{Sp}(\Theta^{n}A,\Theta^{n}B)$. ∎

Now we want to make use of the $O(n)$-action on
$\text{Map}_{Sp}(\Theta^{n}A,\Theta^{n}B)$ and identify
$\Theta^{n}(\text{Map}_{Sp}(\Theta^{n}A,S^{nV}\wedge\Theta^{n}B))^{hO(n)}$.

######  Lemma 5.3.1.4.

Let $F\colon\mathcal{J}\to\mathcal{S}_{*}$ be an $n$-polynomial functor with
an action of a (topological) group $K$. Then $F^{hK}$ is $n$-polynomial.

###### Proof.

Being $n$-polynomial is defined by a certain map into a limit being an
equivalence. Since homotopy fixed points are a limit construction, they
commute with this limit.
$F$ being $n$-polynomial means that

$F(V)\to\lim_{0\neq U\subseteq\mathbb{R}^{n+1}}F(V\oplus U)$

is an equivalence. Thus, for $F^{hK}$, we have

${{F(V)^{hK}}}$${{\lim_{0\neq U\subseteq\mathbb{R}^{n+1}}(F(V\oplus
U)^{hK})}}$${{(\lim_{0\neq U\subseteq\mathbb{R}^{n+1}}F(V\oplus
U))^{hK}}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

as $(-)^{hK}$ is a limit and thus commutes with limits. The diagonal map is an
equivalence since it is the image of one under the functor $(-)^{hK}$, hence
so is the third one, and $F^{hK}$ is $n$-polynomial. ∎

We can calculate $\Theta^{n}(F^{hK})$ in the following way.

######  Lemma 5.3.1.5.

Let $F\colon\mathcal{J}\to\mathcal{S}_{*}$ be $n$-homogeneous and let $K$ be a
(topological) group acting on $F$. Then
$\Theta^{n}(F^{hK})\simeq(\Theta^{n}F)^{hK}$.

###### Proof.

For this, we have to make heavier use of the methods of Weiss’ original paper
than before. Recall in particular from his approach:

  * •

There are categories $\mathcal{J}_{0}\subset\ldots\subset\mathcal{J}_{n}$
enriched in pointed spaces, and $F^{(n)}\colon\mathcal{J}_{n}\to\mathcal{S}$,
the “unstable $n$-th derivative” of $F$, is defined as a Kan extension of
$F\colon\mathcal{J}_{0}\to\mathcal{S}$ along
$\mathcal{J}_{0}\to\mathcal{J}_{n}$, and
$\text{Map}_{\mathcal{J}_{0}}(V,W)=\text{Map}_{\mathcal{J}}(V,W)_{+}$. (This
is in [39, Section 2]¨.)

  * •

$F^{(n)}$ relates to the derivative spectra $\Theta^{n}F$ by
$(\Theta^{n}F)_{nl}=F^{(n)}(\mathbb{R}^{l})$. (See e.g. [39], p.2, p.3, p.10,
p.12.)

  * •

When $F$ is $n$-homogeneous, this presents $\Theta^{n}F$ as an
$\Omega$-spectrum. (See [39, Cor. 5.12].)

By [39, Prop. 5.3] there are fibre sequences

$F^{(n)}(V)\to F(V)\to\lim_{0\neq U\subseteq\mathbb{R}^{n}}F(U\oplus V)$

Since $(-)^{hK}$ is a limit these induce fibre sequences

$(F^{(n)}(V))^{hK}\to F(V)^{hK}\to\left(\lim_{0\neq
U\subseteq\mathbb{R}^{n}}F(U\oplus V)\right)^{hK}$

As limits commute with limits,

$\left(\lim_{0\neq U\subseteq\mathbb{R}^{n}}F(U\oplus
V)\right)^{hK}\simeq\lim_{0\neq U\subseteq\mathbb{R}^{n}}\left(F(U\oplus
V)^{hK}\right)$

Thus there are fibre sequences

$(F^{(n)}(V))^{hK}\to F(V)^{hK}\to\left(\lim_{U\in G_{n}}F(U\oplus
V)^{hK}\right)$

whence

$\left(F^{(n)}\right)^{hK}\simeq\left(F^{hK}\right)^{(n)}$

Since $(\Theta^{n}F)_{nl}=F^{(n)}(\mathbb{R}^{l})$ describes an
$\Omega$-spectrum, $(-)^{hK}$ (as a limit) can be computed levelwise.
Hence

$\left(\Theta^{n}\left(F^{hK}\right)\right)_{nl}\simeq\left(F^{hK}\right)^{(n)}(\mathbb{R}^{l})\simeq\left(F^{(n)}\right)^{hK}(\mathbb{R}^{l})\simeq\left(\Theta^{n}\left(F\right)^{hK}\right)_{nl}$

∎

######  Corollary 5.3.1.6.

For $A,B\in Sp^{\text{BO}(n)}$, the $n$-th derivative spectrum of the functor

$\displaystyle V\mapsto\text{Map}_{Sp^{\text{BO}(n)}}(A,S^{nV}\wedge B)$

is given by

$\displaystyle(O(n)_{+}\wedge\text{Map}_{Sp}(A,B))^{hO(n)}$

The $O(n)$-homotopy fixed points are taken with respect to the $O(n)$-action
from the right on $O(n)_{+}$ and the conjugation action on
$\text{Map}_{Sp}(A,B)$.
As an $n$-th derivative spectrum,
$(O(n)_{+}\wedge\text{Map}_{Sp}(A,B))^{hO(n)}$ is equipped with another
$O(n)$-action. This action is given by left multiplication on $O(n)_{+}$.

###### Proof.

This follows from Corollary 5.3.1.3 and Lemma 5.3.1.5. (We also explain the
identification of the actions once again in Remark 5.3.1.9.) ∎

As we soon want to replace a homotopy fixed point construction by a homotopy
orbit construction, the Norm map will play a role. Recall from algebra that
for $G$ a finite group and $M$ a $G$-module, the Norm map is given by

$\displaystyle\text{Nm}\colon M_{G}$ $\displaystyle\to M^{G}$
$\displaystyle\overline{m}$ $\displaystyle\mapsto\sum_{g\in G}gm$

There is a generalisation of this to the case where $M$ is a spectrum and $G$
is a topological group, of which we will recall some basics now.

######  Remark 5.3.1.7 (The Dualising Spectrum).

Recall that the Spivak-Klein dualising spectrum of a topological group $G$ is
the $G$-spectrum $D_{G}\in Sp^{\text{BG}}$ given by
$D_{G}=(\Sigma_{+}^{\infty}G)^{hG}=(\mathbb{S}[G])^{hG}$, where the homotopy
fixed points are taken with respect to the left action of $G$ on itself. This
spectrum has a residual right $G$-action. (Compare [21], Definition 1 for the
original reference, or [28] Definition I.4.2 for a more recent, albeit brief,
treatment111 Nikolaus and Scholze would use the notation $D_{\text{BG}}$ for
what Klein would call $D_{G}$. .). The important properties of $D_{G}$ and in
particular $D_{O(n)}$ as they pertain to this work are:

  * •

For any $E\in Sp^{\text{BG}}$, there is a natural map called the norm map

$\text{Nm}\colon(D_{\text{BG}}\otimes E)_{hG}\to E^{hG}$

([21, Section 3], [28, Theorem I.4.1 (iv)])
A nice construction of this map is given as Remark 3.1 in [21], namely as the
following composition pairing in $Sp^{\text{BG}}$, where hom denotes the
internal hom functor in $Sp^{\text{BG}}$:

$\text{hom}(\mathbb{S},\mathbb{S}[G])\otimes_{\mathbb{S}[G]}\text{hom}(\mathbb{S}[G],E)\to\text{hom}(\mathbb{S},E)$

To make sense of this, one simply needs to realise that
$\mathbb{S}[G]=\Sigma_{+}^{\infty}G$, $\text{hom}(\mathbb{S},X)\simeq X^{hG}$
and $A\otimes_{\mathbb{S}[G]}B\simeq(A\otimes B)_{hG}$ (with respect to the
diagonal action of $G$).

  * •

The cofibre of the norm map is denoted $E^{tG}$ and called the Tate
construction. Together, this forms the famous Tate fibre sequence

$(D_{\text{G}}\otimes E)_{hG}\to E^{hG}\to E^{tG}$

$(-)^{tG}$ is universally characterised by vanishing on compact objects ([28]
I.4.1 (iii)) such that the fibre of $E^{hG}\to E^{tG}$ commutes with colimits.

  * •

When $G$ is a compact Lie group such as $O(n)$ and $E$ is an induced spectrum,
i.e. of the form

$E\simeq E^{\prime}\otimes\Sigma^{\infty}_{+}G$

for a non-equivariant spectrum $E^{\prime}$, then

$E^{tG}\simeq 0$

and equivalently

$(D_{\text{BG}}\otimes E)_{hG}\simeq E^{hG}$

(This is [21, Theorem D, Cor. 10.2], or alternatively follows by identifying
the compact objects of $Sp^{\text{BG}}$.)

  * •

([21, Theorem 10.1]) For any compact Lie group $G$, there is an (equivariant)
equivalence

$D_{G}\simeq S^{\text{Ad}_{G}}$

where $S^{\text{Ad}_{G}}$ denotes the suspension spectrum of the one point
compactification of the adjoint representation $\text{Ad}_{G}\colon
G\to\text{GL}(\mathfrak{g})$ of $G$ on its Lie algebra $\mathfrak{g}$.

  * •

We observe that $S^{\text{Ad}_{O(n)}}$ is invertible as an $O(n)$-spectrum,
with inverse $S^{-\text{Ad}_{O(n)}}$. To see that, interpret
$S^{\text{Ad}_{O(n)}}$ as the Thom spectrum of a vector bundle with fibre
$\mathfrak{g}$ on $BO(n)$, take the inverse
$-\text{Ad}_{O(n)}\in\text{KO}(BO(n))$ and construct its (virtual) Thom
spectrum $S^{-\text{Ad}_{O(n)}}$. Then $S^{\text{Ad}_{O(n)}}\otimes
S^{-\text{Ad}_{O(n)}}\simeq S^{\text{Ad}_{O(n)}+(-\text{Ad}_{O(n)})}\simeq
S^{0}=\mathbb{S}$.

######  Lemma 5.3.1.8.

For $A,B\in Sp$, there is an equivalence

$((\text{Map}_{Sp}(A,B)\wedge
O(n)_{+}))^{hO(n)}\simeq\text{Map}_{Sp}(A,B)\otimes D_{O(n)}$

where the homotopy fixed points are taken with respect to $O(n)$ acting on
$O(n)_{+}$.

###### Proof.

The Norm map for the $O(n)$-spectrum

$E=\text{Map}_{Sp}(A,B)\wedge O(n)_{+}$

where $O(n)$ acts on $O(n)_{+}$ takes the form

${{(D_{O(n)}\otimes\text{Map}_{Sp}(A,B)\wedge
O(n)_{+})_{hO(n)}}}$${{(\text{Map}_{Sp}(A,B)\wedge O(n)_{+})^{hO(n)}}}$Nm

Since the action is free, Nm is an equivalence ([21, Cor. 10.2]). The left-
hand side simplifies as

$D_{O(n)}\otimes\text{Map}_{Sp}(A,B)\simeq{(D_{O(n)}\otimes\text{Map}_{Sp}(A,B)\wedge
O(n)_{+})_{hO(n)}}$

∎

At last, we can now calculate $\Theta^{n}\mathcal{F}_{*}(\Phi A,\Phi B)$:

###### Proof of Proposition 5.3.1.1.

The proof is now a straightforward combination of the previous calculations:

  * •

$\mathcal{F}_{*}(\Phi A,\Phi
B)(V)\simeq\text{Map}_{Sp^{\text{BO}(n)}}(A,S^{nV}\wedge B)$ by definition and
classification of pointed homogeneous functors.

  * •

$\text{Map}_{Sp^{\text{BO}(n)}}(A,S^{nV}\wedge
B)\simeq(\text{Map}_{Sp}(A,S^{nV}\wedge B))^{hO(n)}$ with $O(n)$ acting by
conjugation.

  * •

Ignoring the $(-)^{hO(n)}$ for a moment, the functor

$V\mapsto\text{Map}_{Sp}(A,S^{nV}\wedge B)$

is $n$-homogeneous with $n$-th derivative

$\text{Map}_{Sp}(A,B)\wedge O(n)_{+}$

by Corollary 5.3.1.3.

  * •

Note that $\text{Map}_{Sp}(A,B)\wedge O(n)_{+}$ now has two $O(n)$-actions:
One acting by multiplication from the left on $O(n)_{+}$ (this is the action
making the derivative spectrum an $O(n)$-spectrum), and a second one acting on
$\text{Map}_{Sp}(A,B)$ by “conjugation” and on $O(n)_{+}$ from the right by
multiplication with the inverse. This second action is the one coming from the
action on $\text{Map}_{Sp}(A,\text{sh}_{V}^{*}B)$ under the identifications we
made. Note that we spell them out once more in Remark 5.3.1.9. By Lemma
5.3.1.5, the homotopy fixed point spectrum with respect to the second action
calculates

$\Theta^{n}\text{Map}_{Sp^{\text{BO}(n)}}(A,S^{nV}B)\simeq(\text{Map}_{Sp}(A,B)\wedge
O(n)_{+})^{hO(n)}$

  * •

By Lemma 5.3.1.8, after equivariantly identifying $\text{Map}_{Sp}(A,B)\wedge
O(n)_{+}$ with the diagonal $O(n)$-action on both factors with the same
spectrum with $O(n)$ acting only on the second factor222 via a shearing
isomorphism of the form $A\otimes G\to A\otimes G,(a,g)\mapsto(ga,g)$ , this
is equivalent to

$D_{O(n)}\otimes\text{Map}_{Sp}(A,B)$

with the remaining $O(n)$-action given diagonally.

∎

######  Remark 5.3.1.9 (Spelling out the actions once more).

The identification

$\displaystyle S^{nV}\wedge\text{Map}_{Sp}(A,B)$
$\displaystyle\simeq(S^{nV}\wedge\text{Map}_{Sp}(A,B)\wedge O(n)_{+})_{hO(n)}$
$\displaystyle(s,f)$ $\displaystyle\mapsto[(s,f,1)]$
$\displaystyle(l^{-1}s,f)$ $\displaystyle\mapsfrom[(s,f,l)]$

identifies the $O(n)$-action on the left side which is diagonal on $S^{nV}$
and $\text{Map}_{Sp}(A,B)$, i.e. given by $(s,f)\mapsto(ks,kf)$ for $k\in
O(n)$ with the one on $\text{Map}_{Sp}(A,B)\wedge O(n)_{+}$ which is the same
action on $\text{Map}_{Sp}(A,B)$ and given by acting with $k^{-1}$ from the
right on $O(n)_{+}$.

See also

${{(o^{-1}s,f)}}$${{[(s,f,o)]}}$${{(ko^{-1}s,kf)}}$${{[(ko^{-1}s,kf,1)]=[(s,kf,ok^{-1})]}}$

Hence on

$\Theta^{n}\text{Map}_{Sp^{\text{BO}(n)}}(A,B)\simeq\text{Map}_{Sp}(A,B)\wedge
O(n)_{+}$

there are two commuting $O(n)$-actions:

  * •

The “inner” action, i.e. the one present on any $n$-th derivative spectrum,
acts from the left on $O(n)_{+}$.

  * •

The “outer” action, i.e. the one coming from the conjugation action on
$\text{Map}_{Sp}(A,B)$, acts in the usual way on $\text{Map}_{Sp}(A,B)$ and
from the right on $O(n)_{+}$.

Then, under the identification

$\left(D_{O(n)}\otimes\text{Map}_{Sp}(A,B)\otimes
O(n)_{+}\right)_{hO(n)}\simeq\left(\text{Map}_{Sp}(A,B)\wedge
O(n)_{+}\right)^{hO(n)}$

(where the homotopy fixed points and orbits are taken w.r.t. the “outer”
action), the inner action acts (only) on $O(n)_{+}$ on either side of this
description. But then, under the final identification

$\left(D_{O(n)}\otimes\text{Map}_{Sp}(A,B)\otimes
O(n)_{+}\right)_{hO(n)}\simeq D_{O(n)}\otimes\text{Map}_{Sp}(A,B)$

the “inner” $O(n)$-action on the left hand side is identified with the
$O(n)$-action on the right-hand side on both tensor factors, for the same
reason as in the first identification in this remark.

####  5.3.2 Calculation of $\Theta^{n}(A\circledast B)$ for $A,B$
$n$-homogeneous

After calculating $\Theta^{n}\mathcal{F}_{*}(A,B)$ for $n$-homogeneous
functors $A$ and $B$, we also calculate $\Theta^{n}(A\circledast B)$ for $A,B$
$n$-homogeneous. For our final result, either this or the calculation of
$\Theta^{n}\mathcal{F}_{*}(A,B)$ are sufficient on their own, but it is nice
to have both.

######  Proposition 5.3.2.1.

For $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$ $n$-homogeneous, there is an
equivalence

$\Theta^{n}(A\circledast B)\simeq\Theta^{n}A\oplus\Theta^{n}B\oplus
D_{O(n)}^{\vee}\otimes\Theta^{n}A\otimes\Theta^{n}B$

where $D_{O(n)}^{\vee}\simeq S^{-\text{Ad}_{n}}$ denotes the dual
$O(n)$-spectrum to $D_{O(n)}$.

The proof consists of two steps:

######  Lemma 5.3.2.2.

For functors $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$, there is a cofibre
sequence

$A\vee B\to A\circledast B\to A\circledast_{\text{red}}B$

where $(A\vee B)(V):=A(V)\vee B(V)$ and $A\circledast_{\text{red}}B$ denotes
the reduced convolution product of $A$ and $B$, taken with respect to the
smash product monoidal structure on $\mathcal{S}_{*}$ instead of the cartesian
product monoidal structure.
At the level of derivative spectra, the sequence splits and thus induces an
equivalence

$\Theta^{n}(A\circledast
B)\simeq\Theta^{n}A\oplus\Theta^{n}B\oplus\Theta^{n}(A\circledast_{\text{red}}B)$

######  Lemma 5.3.2.3.

For $A,B\colon\mathcal{J}\to\mathcal{S}_{*}$ $n$-homogeneous, there is an
equivalence of $O(n)$-spectra

$\Theta^{n}(A\circledast_{\text{red}}B)\simeq\Theta^{n}A\otimes\Theta^{n}B\otimes
D_{O(n)}^{\vee}$

###### Proof of Proposition 5.3.2.1.

This is immediate from Lemma 5.3.2.2 and Lemma 5.3.2.3. ∎