# The localization of orthogonal calculus with respect to homology

Niall Taggart  Mathematical Institute, Utrecht University, Budapestlaan 6,
3584 CD Utrecht, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

For a set of maps of based spaces $S$ we construct a version of Weiss’
orthogonal calculus which only depends on the $S$-local homotopy type of the
functor involved. We show that $S$-local homogeneous functors of degree $n$
are equivalent to levelwise $S$-local spectra with an action of the orthogonal
group $O(n)$ via a zigzag of Quillen equivalences between appropriate model
categories. Our theory specialises to homological localizations and
nullifications at a based space. We give a variety of applications including a
reformulation of the Telescope Conjecture in terms of our local orthogonal
calculus and a calculus version of Postnikov sections.

###### Key words and phrases:

Orthogonal calculus, Bousfield localization, homological localization,
nullification, calculus of functors

###### 2020 Mathematics Subject Classification:

55P65, 55P42, 55P60, 55N20

###### Contents

  1. 1 Introduction
  2. I Preliminaries
    1. 2 The localization of topological spaces
    2. 3 The Borel stablisation of local spaces with a $G$-action
  3. II Local orthogonal calculus
    1. 4 Input functors
    2. 5 Polynomial functors
    3. 6 Homogeneous functors
    4. 7 The derivatives
    5. 8 The classification of local homogeneous functors
  4. III Applications
    1. 9 Bousfield classes
    2. 10 Postnikov sections

###  1\. Introduction

#### Motivation

Building on work of Adams, Bousfield [BousfieldLocalSpaces] established that
on the category of (based) spaces there is a well defined localization functor
$L_{E}$ which sends a space $X$ to a space $X_{E}$ which contains all the
$E_{\ast}$-homology of $X$ but disregards any of the other homotopical
information of $X$. For example, for rational homology, the space
$X_{\mathbb{Q}}$ is the rationalisation of $X$ in the sense of Quillen
[QuillenRational]. Similar localizations exist for spectra, see
[BousfieldLocalSpectra].

The theory of localizations at homology theories are ubiquitous and have had
wide applications; of particular note is _chromatic homotopy theory_ which
among other things gives a spectrum level interpretation for the periodic
families appearing in the stable homotopy groups of spheres. An extensive
amount of effort has been geared toward understanding how localization at
homology theories–particularly the chromatic localizations–interact with
Goodwillie’s calculus of functors [AroneMahowald, KuhnTate, KuhnTAQ,
AroneKuhn], see e.g., [Kuh07] for a survey. This article is concerned with the
analogous question in orthogonal calculus although our approach is noticeably
different than those applied to Goodwillie calculus.

#### Overview

The classical theory of orthogonal calculus (and more generally functor
calculus) is a homotopy theoretic tool for studying ‘geometric’ objects by
constructing a filtration on the object the ‘quotients’ of which are typically
easier to understand and manipulate. The spaces under consideration in
orthogonal calculus naturally take the form of a $\mathcal{J}$-space: a
functor from the category of real vector spaces to the category of (based)
spaces. The calculus is constructed in such a way that it not only constructs
a filtration on each level of the $\mathcal{J}$-space, but these constructions
are natural; a linear map induces a map between filtrations; given a
$\mathcal{J}$-space $F$ the calculus assigns a tower of $\mathcal{J}$-spaces
under $F$

${F}$${\cdots}$${{T_{n}F}}$${{T_{n-1}F}}$${\cdots}$${{T_{1}F}}$${{T_{0}F}}$

called the _Weiss tower_ for $F$. The $n$-th term $T_{n}F$ acts as a
categorification of polynomial of degree less than or equal $n$ functions from
classical calculus. The $n$-th layer of the tower is the homotopy fibre
$D_{n}F$ of the map $T_{n}F\to T_{n-1}F$, and acts as the ‘quotients’ of the
filtration. Of all the flavours of functor calculus, orthogonal calculus is
synonymous with being the most computationally challenging due to the
interaction between the highly ‘geometric’ nature of the objects of study and
the highly homotopical constructions.

Given a set $S$ of maps of based spaces we produce an _$S$ -local Weiss tower_
of the following form:

${F}$${\cdots}$${{T_{n}L_{S}F}}$${{T_{n-1}L_{S}F}}$${\cdots}$${{T_{1}L_{S}F}}$${{T_{0}L_{S}F}}$${{D_{n}^{S}F}}$${{D_{n-1}^{S}F}}$${{D_{1}^{S}F}}$

To understand the $S$-local Weiss tower, we utilise the interpretation of
localizations in terms of a model structure on the category of bases spaces in
which the localization functor $L_{S}$ is a fibrant replacement. In the case
of homological localizations the weak equivalences are the homology
isomorphisms. We begin by constructing a model structure, denoted
$\mathsf{Poly}^{\leq n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$,
which captures the homotopy theory of functors which are $S$-locally
polynomial of degree less than or equal $n$. From this model structure we
construct a new model structure, denoted
$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$,
which captures the homotopy theory of functors which are $S$-locally
homogeneous of degree $n$. In particular, when the localization functor
$L_{S}$ is “nice” (in a sense which is made precise below) the $S$-local
$n$-homogeneous model structure contains the $n$-th layer of the $S$-local
Weiss tower as a bifibrant object.

We consider three classes of localizations, giving three levels of results:

  1. (1)

we give general existence theorems for these $S$-local model structures for
_any_ set $S$ of maps of based spaces;

  2. (2)

we obtain more computationally auspicious results when we assume the class of
local objects is closed under homotopy colimits111The requirement of the class
of local objects to be closed under homotopy colimits is reminiscent of
_smashing_ localizations in the stable setting.; and,

  3. (3)

we restrict to the case when the localization is a _nullification_ which
further advances the applicability of the $S$-local calculus to computations.

Through a zigzag of Quillen equivalences we characterise the $S$-local
$n$-homogeneous functors as levelwise $S$-local spectra with an action of
$O(n)$.

######  Theorem (Corollary 8.4.2).

Let $S$ be a set of maps of based spaces and $n\geq 0$. There is a zigzag of
Quillen equivalences

$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})\simeq_{Q}{{\sf{Sp}}}(L_{S}\operatorname{\sf{Top}_{\ast}})[O(n)].$

On the derived level, we obtain a computationally accessible classification
theorem for $S$-local homogeneous of degree $n$ functors.

######  Theorem (Theorem 8.6.1).

Let $S$ be a set of maps of based spaces and $n\geq 1$. Any $S$-local
homogeneous of degree $n$ functor $F$ is (up to homotopy) of the form

$V\longmapsto\Omega^{\infty}[(S^{\mathbb{R}^{n}\otimes
V}\wedge\partial_{n}^{S}F)_{hO(n)}],$

where $\partial_{n}^{S}F$ is a levelwise $S$-local spectrum with an action of
$O(n)$.

#### Applications

We envision that the applications of this local version of orthogonal calculus
are vast. For example, computations of rational derivatives of functors in
orthogonal calculus have been of interest in differential topology recently,
see e.g., [KRW2021], and a full understanding of the Weiss tower of ${\sf
BO}(-)$ has been achieved in $v_{n}$-periodic homotopy theory through complex
computations of Arone [Ar02] using compuations of Arone and Mahowald
[AroneMahowald]. A compelling application of the local orthogonal calculus
would be to recast these computations in a new light. In the last part of this
paper we give a number of intial applications of the theory.

By considering the acyclics of a localization with respect to a homology
theory, Bousfield [BousfieldLocalSpectra] described a lattice of localizations
built from _Bousfield classes_ –equivalences classes of acyclics. A similar
lattice exists for nullity classes of nullifications at based spaces, see,
e.g., [FarjounCellular]. This has a number of repercussions for the
localization of orthogonal calculus. We prove a more general result that the
stated theorem below, see Theorem 9.1.1.

######  Theorem (Corollary 9.1.2 $\&$ Corollary 9.1.3).

  1. (1)

Let $E$ and $E^{\prime}$ be spectrum. The $E$-local orthogonal calculus is
equivalent to the $E^{\prime}$-local orthogonal calculus if and only if $E$
and $E^{\prime}$ have the same Bousfield class.

  2. (2)

Let $W$ and $W^{\prime}$ be based spaces. The $W$-local orthogonal calculus is
equivalent to the $W^{\prime}$-local orthogonal calculus if and only if $W$
and $W^{\prime}$ have the same nullity class.

A major application of our study of Bousfield classes is in relation to the
Telescope Conjecture of Ravenel, [RavenelLocalizations, Conjecture 10.5]. Fix
a prime $p$, and work $p$-locally. Denote by $T(n)$ the telescope of any
$v_{n}$-self map on a finite complex of type $n$, and denote by $K(n)$ the
$n$-th Morava $K$-theory.

######  Theorem (Theorem 9.2.3).

The height $n$ Telescope Conjecture holds if and only if the $K(n)$-local
orthogonal calculus and the $T(n)$-local orthogonal calculus are equivalent.

From a computational perspective we obtain the following relation between the
Telescope Conjecture and the local Weiss spectral sequences.

######  Theorem (Lemma 9.2.4).

If the height $n$ Telescope Conjecture holds, then for all $r\geq 0$, the
$r$-th page of the $T(n)$-local Weiss spectral sequence is isomorphic to the
$r$-th page of the $K(n)$-local Weiss spectral sequence.

By considering nullifications with respect to the spheres we obtain a theory
of Postnikov sections in orthogonal calculus, which is a particular example of
Postnikov sections in arbitrary model categories. For example, we prove that
our $S^{k+1}$-local projective model structure on the category of orthogonal
functors is identical to the model structure of $k$-types in the category of
orthogonal functors in the sense of [GR17, §4].

######  Theorem (Proposition 10.2.1).

Let $k\geq 0$. The model structure of $k$-types in orthogonal functors is
identical to the $S^{k+1}$-local model structure, that is, there is an
equality of model structures,

$P_{k}{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}):=L_{W_{k}}{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})={\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}).$

As an application we produce a tower of model categories

$\cdots\longrightarrow\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{k}\operatorname{\sf{Top}_{\ast}})\longrightarrow\cdots\longrightarrow\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{1}\operatorname{\sf{Top}_{\ast}})\longrightarrow\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{0}\operatorname{\sf{Top}_{\ast}}),$

by nullification at the spheres $S^{k+1}$ as $k$ varies. By applying the
theory of homotopy limits of model categories, we show that the
$n$-homogeneous model structure of Barnes and Oman [BO13, Proposition 6.9] is
the homotopy limit of this tower, in the following sense.

######  Theorem (Corollary 10.7.3).

There is a Quillen equivalence

$\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\simeq_{Q}\underset{k}{\operatorname{\mathrm{holim}}}~{}\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{k}\operatorname{\sf{Top}_{\ast}}).$

#### Relation to other work

This work is intimately related to the rational orthogonal calculus developed
by Barnes [Ba17]. Replacing our generalised homology theory $E_{\ast}$ with
rational homology recovers the theory developed by Barnes.

Unstable chromatic homotopy theory can be described algebraically, via Heuts’
[HeutsAlgModels] _algebraic model_ for $v_{n}$-periodic spaces via an
equivalence (of $\infty$-categories) with Lie algebras in $T(n)$-local
spectra. This model indicated that there is likely a relationship between
$v_{n}$-periodic orthogonal calculus and orthogonal calculus of Heuts’ Lie
algebra models. Such an equivalence at chromatic height zero suggests a
relationship between rational orthogonal calculus and the algebraic models for
rational homotopy theory of Sullivan and Quillen [QuillenRational,
SullivanRational]. In particular this together with Barnes’ [Ba17] model for
rational $n$-homogeneous functors as torsion models over the rational
cohomology ring of ${\sf BSO}(n)$ suggests the existence of _algebraic model
calculi_. We plan to return to this in future work.

There is a strong connection between the calculus and chromatic homotopy
theory. Ravenel [RavenelLocalizations], made several conjectures relating to
the structure of chromatic homotopy theory including the Nilpotence and
Periodicity Theorems of Devinatz, Hopkins and Smith [NilpotenceI,
NilpotenceII]. These conjectures have be resolved except for one, the
Telescope Conjecture, which is trivial at height $n=0$, has been verified at
height $n=1$ by Bousfield [BousfieldLocalSpectra], Mahowald
[MahowaldboResolutions] and Miller [MillerASS], but in general is widely
believed to be false. The validity of the Telescope Conjecture would imply an
equivalence between $K(n)$-local orthogonal calculus and and $T(n)$-local
orthogonal calculus.

This work also forms part of an extensive program to go “beyond orthogonal
calculus” which was initiated in the Ph.D. thesis of the author
[Taggartthesis], together with a series of articles exploring extensions of
the orthogonal calculus and the relations between these, [TaggartUnitary,
TaggartOCandUC, TaggartReality, TaggartRealityUnitary]. The hopes of this
extensive project is to illuminate our understanding of orthogonal calculus
which (at least relative to Goodwillie calculus) remains largely unexplored.

The future applications of the homological localization of orthogonal calculus
are abound. For example in recent work of Beaudry, Bobkova, Pham and Xu
[BeaudryBobkovaPhamXu], the authors compute the $tmf$-homology of
$\mathbb{R}P^{2}$, where $tmf$ denotes the connective spectrum of topological
modular forms. It would be interesting to investigate whether their
computation for $\mathbb{R}P^{2}$ and the $tmf$-local Weiss tower for the
functor $V\mapsto\mathbb{R}P(V)$ yield a calculation of the $tmf$-homology of
$\mathbb{R}P^{k}$ for all $k$. Such a connection would, for example, feed into
a chromatic understanding of block structures, see e.g., [Ma07].

#### Conventions

We work extensively with model categories and refer the reader to the survey
article [DS95] and the textbooks [Ho99, Hi03] for a detailed account of the
theory. We further assume the reader has familiarity with orthogonal calculus,
references for which include [BO13, We95].

The category $\operatorname{\sf{Top}_{\ast}}$ will always denote the category
of based compactly generated weak Hausdorff spaces, and we will, for brevity,
call the objects of this category “based spaces”. The category of based spaces
will always be equipped with the Quillen model structure unless specified
otherwise. The weak equivalences are the weak homotopy equivalences and
fibrations are Serre fibrations. This is a cellular, proper and topological
model category with sets of generating cofibrations and acyclic cofibrations
denoted by $I$ and $J$, respectively.

Unless otherwise stated the word “spectra” is synonymous with the phrase
“orthogonal spectra”, details of which can be found in [MMSS01] in the non-
equivariant case, and [MM02] in the equivariant situation.

#### Acknowledgements

This work has benefited from helpful conversations and comments from D.
Barnes, T. Barthel, and J. Williamson. We are particularly grateful to S.
Balchin for reading an earlier version of this material. We thank the Max
Plank Institute for Mathematics for its hospitality.

##  Part I Preliminaries

###  2\. The localization of topological spaces

####  2.1. Left Bousfield localization

Given a model category $\mathcal{C}$ there is a systematic way of constructing
a new model category which has the same cofibrations but more weak
equivalences, yielding a finer homotopy theory. This process is called left
Bousfield localization.

Give objects $X$ and $Y$ in a model category $\mathcal{C}$ we denote by
$\operatorname{\mathsf{map}}_{\mathcal{C}}(X,Y)$ the homotopy function complex
in the sense of [Hi03, Chapter 17]. In many cases our model structures will be
simplicial or topological in which case the homotopy function complex may be
replaced by suitably ‘derived’ version of the internal hom arising from the
simplicial or topological structure, see e.g., [Hi03, Example 17.1.4].

######  Definition 2.1.1.

Let $\mathcal{C}$ be a model category and $S$ a class of maps in
$\mathcal{C}$.

  1. (1)

An object $Z$ in $\mathcal{C}$ is _$S$ -local_ if for every map $f\colon A\to
B$ in $S$ the map on homotopy function complexes

$\operatorname{\mathsf{map}}_{\mathcal{C}}(B,Z)\longrightarrow\operatorname{\mathsf{map}}_{\mathcal{C}}(A,Z),$

induced by precomposition with $f$ is a weak homotopy equivalence of
simplicial sets.

  2. (2)

a map $g\colon X\to Y$ is an _$S$ -local equivalence_ if for every $S$-local
object $Z$ the map on homotopy function complexes

$\operatorname{\mathsf{map}}_{\mathcal{C}}(Y,Z)\longrightarrow\operatorname{\mathsf{map}}_{\mathcal{C}}(X,Z),$

induced by precomposition with $g$ is a weak homotopy equivalence of
simplicial sets.

Under mild conditions on the model category $\mathcal{C}$ there exists a model
structure on $\mathcal{C}$ with weak equivalences the $S$-local equivalences
and fibrant objects the $S$-local objects which are fibrant in $\mathcal{C}$.
We will sometimes refer to such objects as being _$S$ -fibrant_.

######  Theorem 2.1.2 ([Hi03, Theorem 4.1.1.]).

Let $\mathcal{C}$ be a model category and $S$ a set of maps in $\mathcal{C}$.
If the model structure on $\mathcal{C}$ is left proper and cellular, then the
left Bousfield localization of $\mathcal{C}$ at the set $S$ exists. The weak
equivalences are the $S$-local equivalences, the cofibrations are the
cofibrations of $\mathcal{C}$ and the fibrant objects are the $S$-local
objects which are fibrant in $\mathcal{C}$. This model structure is cellular,
left proper and topological whenever the underlying model structure on
$\mathcal{C}$ is topological. We denote this model structure by
$L_{S}\mathcal{C}$.

As an example, consider homological localization with respect to a homology
theory $E_{\ast}$. Bousfield [BousfieldLocalSpaces] showed that there exists a
set $J_{E}$ of maps of based spaces such that the $J_{E}$-local model
structure on based spaces captures the homotopy theory of based spaces up to
$E_{\ast}$-isomorphism.

######  Theorem 2.1.3 ([BousfieldLocalSpaces, Theorem 10.2]).

Let $E_{\ast}$ be a generalised homology theory. There is a model category
structure on the category of based spaces with weak equivalences the
$E_{\ast}$-isomorphisms, cofibrations the cofibrations of the Quillen model
structure on based spaces and the fibrant objects are the $E$-local spaces.
This model structure is cellular, left proper and topological. We denote this
model structure by $L_{E}\operatorname{\sf{Top}_{\ast}}$.

####  2.2. Right Bousfield localization

There is a dual theory in which one adds more weak equivalences but fixes the
class of fibrations. This is called a Right Bousfield localization.

######  Definition 2.2.1.

Let $\mathcal{C}$ be a model category and $\mathcal{K}$ a set of cofibrant
objects in $\mathcal{C}$.

  1. (1)

A map $g\colon X\to Y$ is an _$\mathcal{K}$ -colocal equivalence_ if for every
$K\in\mathcal{K}$ the map on homotopy function complexes

$\operatorname{\mathsf{map}}_{\mathcal{C}}(K,X)\longrightarrow\operatorname{\mathsf{map}}_{\mathcal{C}}(K,Y),$

induced by postcomposition with $g$ is a weak homotopy equivalence of
simplicial sets.

  2. (2)

An object $Z$ in $\mathcal{C}$ is _$S$ -colocal_ if for every
$\mathcal{K}$-colocal equivalence $f\colon A\to B$ the map on homotopy
function complexes

$\operatorname{\mathsf{map}}_{\mathcal{C}}(Z,A)\longrightarrow\operatorname{\mathsf{map}}_{\mathcal{C}}(Z,B),$

induced by postcomposition with $f$ is a weak homotopy equivalence of
simplicial sets.

Under mild conditions on the model category $\mathcal{C}$ there exists a model
structure on $\mathcal{C}$ with weak equivalences the $\mathcal{K}$-colocal
equivalences and cofibrant objects the $\mathcal{K}$-colocal objects which are
cofibrant in $\mathcal{C}$. We will sometimes call the cofibrant objects
$\mathcal{K}$-cofibrant.

######  Theorem 2.2.2 ([Hi03, Theorem 5.1.1.]).

Let $\mathcal{C}$ be a model category and $\mathcal{K}$ a set of cofibrant
objects in $\mathcal{C}$. If the model structure on $\mathcal{C}$ is right
proper and cellular, then the right Bousfield localization of $\mathcal{C}$ at
the set $\mathcal{K}$ exists. The weak equivalences are the
$\mathcal{K}$-colocal equivalences, the fibrations are the fibrations of
$\mathcal{C}$ and the cofibrant objects are the $\mathcal{K}$-colocal objects
which are cofibrant in $\mathcal{C}$. This model structure is cellular, right
proper and topological whenever the underlying model structure on
$\mathcal{C}$ is topological. We denote this model structure by
$R_{\mathcal{K}}\mathcal{C}$.

######  Example 2.2.3.

If ${\sf Ch}(\mathbb{Z})$ is the category of (unbounded) chain complexes of
abelian groups equipped with the projective model structure, then derived
completion at a prime $p$ may be modelled by the right Bousfield localization
at the complex $\mathbb{Z}/p$.

####  2.3. Left Bousfield localization and right properness

######  Remark 2.3.1.

The process of left Bousfield localization can interfere with other model
categorical properties, for instance left Bousfield localization need not
preserve right properness. For example if $E=H\mathbb{Q}$, then the
$H\mathbb{Q}$-local model structure on based spaces is not right proper since
there is a pullback square

${{K(\mathbb{Q}/\mathbb{Z},0)}}$${P}$${{K(\mathbb{Z},1)}}$${{K(\mathbb{Q},1)}}$$\scriptstyle{\simeq_{H\mathbb{Q}}}$

in which the right hand vertical map is a fibration, $P$ is contractible and
the lower horizontal map is a $H\mathbb{Q}$-equivalence but the left hand
vertical map is not. Another example is provided by Quillen in
[QuillenRational, Remark 2.9].

The property of being right proper has many advantages including the ability
to right Bousfield localize. As such we investigate when the $S$-local model
structure is right proper. It suffices to examine when the $f$-local model
structure is right proper for some map $f\colon X\to Y$ of based spaces.

The following has motivation in [Bo01, Remark 9.11], in which Bousfield
remarks that the $f$-local model structure cannot be right proper unless the
localization functor $L_{f}$ is equivalent to a nullification. We extend
Bousfield’s remark by showing that his nullification condition is both
necessary and sufficient in a stronger sense than originally proposed by
Bousfield. This result depends on two constructions also due to Bousfield; the
first is the construction of a based space $A(f)$ associated to a map $f\colon
X\to Y$ of based spaces, see [BousfieldHomotopical, Theorem 4.4], the second
is a nullification functor
$P_{W}\colon\operatorname{\sf{Top}_{\ast}}\to\operatorname{\sf{Top}_{\ast}}$
associated to any based space $W$, see [BousfieldPeriodicity, Theorem 2.10].
This nullification functor has two key properties which we would also like to
highlight; firstly, when $W$ is connected $P_{W}$ preserves disjoint unions
and secondly, $P_{W}$ is contractible when $W$ is not connected. For example,
if $f$ is the map which induces localization with respect to integral
homology, then $P_{A(f)}$ is Quillen’s plus construction, see e.g.,
[FarjounCellular, 1.E.5].

######  Proposition 2.3.2.

Let $f\colon X\to Y$ be a map of based spaces. The $f$-local model structure
on based spaces is right proper if and only if there exists a based space
$A(f)$ and and equality of model structures

$L_{f}\operatorname{\sf{Top}_{\ast}}=P_{A(f)}\operatorname{\sf{Top}_{\ast}},$

where $P_{A(f)}\operatorname{\sf{Top}_{\ast}}$ is the Bousfield-Friedlander
localization [Bo01, Theorem 9.3], at the nullification endofunctor

$P_{A(f)}\colon\operatorname{\sf{Top}_{\ast}}\to\operatorname{\sf{Top}_{\ast}}.$

###### Proof.

By [BousfieldHomotopical, Theorem 4.4], there exists a based space $A(f)$ such
that the classes of $A(f)$-acyclic and $f$-acyclic spaces agree, and every
$P_{A(f)}$-equivalence is an $f$-local equivalence.

Assume that the $f$-local model structure is right proper. For a connected
based space $X$, the path fibration over $L_{f}X$ is an $f$-local fibration,
hence the homotopy fibre of the map $X\to L_{f}X$ is $f$-acyclic, and hence
$A(f)$-acyclic. It follows by [BousfieldPeriodicity, Corollary 4.8(i)], the
map $X\to L_{f}X$ is a $P_{A(f)}$-equivalence, hence every $f$-local
equivalences of connected spaces is a $P_{A(f)}$-equivalence. Since the
functor $P_{A(f)}$ on based spaces comes from a functor on unbased spaces
which preserves disjoint unions when $A(f)$ is connected and which takes
contractible values when $A(f)$ is not connected, every $f$-local equivalence
must be a $P_{A(f)}$-equivalence. It follows that the class of $f$-local
equivalences agrees with the class of $P_{A(f)}$-equivalences. The equality of
the model structures follows immediately since both model structures have the
same cofibrations inherited from the Quillen model structure on the category
of based spaces.

For the converse, assume that the $f$-local model structure agrees with the
$A(f)$-local model structure. The latter model structure is right proper by
[Bo01, Theorem 9.9], and since both model structures have the same weak
equivalences and fibrations, the $f$-local model structure must also be right
proper. ∎

######  Remark 2.3.3.

The property of being right proper is completely determined by the weak
equivalence class of the model structure; if two model structures have the the
same weak equivalences, then one is right proper if and only if the other is,
see e.g., [BalchinModelCats, Remark 2.5.6].

We now provide some examples of localizations which can be written as a
nullification.

######  Examples 2.3.4.

  1. (1)

Localization of spaces with respect to the map $u_{n+1}\colon\ast\to S^{n+1}$
gives the Postnikov section, that is for a space $X$,
$P_{n}X=P_{S^{n+1}}X=L_{u_{n+1}}X$, see e.g., [FarjounCellular, Example E.1].

  2. (2)

Extensive work of Bousfield [BousfieldPeriodicity, Bo01] exhibits that
$v_{n}$-periodic homotopy theory, or _unstable chromatic homotopy theory_ can
be expressed as a nullification.

###  3\. The Borel stablisation of local spaces with a $G$-action

We investigate the stablisation of $S$-local spaces for $S$ a set of maps of
based spaces and the relationship between $E$-local spectra and the
stablisation of $E$-local spaces for a homology theory $E_{\ast}$. This will
be particularly useful since homogeneous functors of degree $n$ are classified
by the _Borel stablisation_ of spaces with an $O(n)$-action, see e.g., [We95,
Theorem 7.3], and $S$-local homogeneous functors of degree $n$ will be
classified by the Borel stablisation of $S$-local spaces with an
$O(n)$-action, see Section 8.

####  3.1. The Borel stablisation of local spaces with a $G$-action

Let $G$ be a compact Lie group. Recall from [MMSS01] that orthogonal spectra
may be described as a category of topologically enriched functors
${\sf{Fun}}(\mathcal{I},\operatorname{\sf{Top}_{\ast}})$. Furthermore, spectra
with a $G$-action is the category of $G$-objects in spectra. Spectra with a
$G$-action is the ‘naive’ stabilisation of $G$-spaces, which we prefer to call
the _Borel stablisation_. This stablisation is captured by the underlying
model structure on spectra with a $G$-action. We are particularly interested
in spectra with a $G$-action, when $G$ is the orthogonal group $O(n)$.
Following the procedure in the non-equivariant setting [MMSS01], or the
equivariant setting on a trivial universe [MM02], a left Bousfield
localization of the projective model structure on spectra with a $G$-action
yields the stable model structure.

######  Lemma 3.1.1.

The category ${{\sf{Sp}}}[G]$ of spectra with a $G$-action may be equipped
with a model structure with weak equivalences the $\pi_{\ast}$-isomorphisms
and a map $f\colon X\to Y$ is a fibration if it is levelwise fibration and for
each $U\in\mathcal{I}$, the square

${{X(U)}}$${{\Omega X(U)}}$${{Y(U)}}$${{\Omega
Y(U)}}$$\scriptstyle{f}$$\scriptstyle{\Omega f}$

is a homotopy pullback. This model structure is cellular, proper, stable and
topological.

Analogous to spectra, we can preform a suitable left Bousfield localization of
the category of spectra with a $G$-action to obtain a model for the Borel
stablisation of $S$-local spaces with a $G$-action.

######  Proposition 3.1.2.

Let $S$ a set of maps of based spaces. There is a model structure on the
category of spectra with a $G$-action such that the cofibrations are the
cofibrations of the stable model structure and the fibrant objects are the
levelwise $S$-local $\Omega$-spectra. This model structure is cellular, left
proper and topological. We call this model structure the (Borel) stablisation
of $S$-local spaces with an $O(n)$-action, and it
${{\sf{Sp}}}(L_{S}\operatorname{\sf{Top}_{\ast}})[G]$.

####  3.2. $E$-local spectra with a $G$-action

For a homology theory $E_{\ast}$, it is also possible to construct a model
structure on spectra such that the weak equivalences are detected by
$E_{\ast}$-homology isomorphisms.

######  Proposition 3.2.1 (Bousfield [BousfieldLocalSpectra]).

Let $E$ be a spectrum. There is a model structure on the category of spectra
such that the weak equivalences are the $E_{\ast}$-isomorphisms, the
cofibrations are the cofibrations of the stable model structure and the
fibrant objects are the $E$-local $\Omega$-spectra. This model structure is
cellular, proper, stable and topological. We call this model structure the
$E$-local model structure and denote it ${\sf{Sp}}_{E}$.

###### Proof.

Existence of the model structure follows from Hovey’s Recognition Principle
for model categories, [Ho99, Theorem 2.1.19], see, e.g. [BarnesRoitzheimSHT,
Theorem 7.3.3] for details and [BarnesRoitzheimSHT, Corollary 7.3.6] for the
characterisation of the fibrant objects. This model structure is stable by
[BarnesRoitzheimStable, Example 4.3 $\&$ Proposition 4.6], and right proper by
[BarnesRoitzheimStable, Proposition 4.7]. ∎

This model structure can be transferred to spectra with a $G$-action.

######  Proposition 3.2.2.

There is a model structure on the category ${{\sf{Sp}}}[G]$ of spectra with a
$G$-action where the weak equivalences are the $E_{\ast}$-isomorphisms, the
cofibrations are the cofibrations of the projective model structure and the
fibrant objects are the $E$-local $\Omega$-spectra. This model structure is
cellular, right proper, stable and topological. We denote this model structure
by ${\sf{Sp}}_{E}[G]$.

###### Proof.

This is the right transfer of the $E$-local model structure on spectra through
the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
29.67065pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{+}\wedge(-):{\sf{Sp}}_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
53.67065pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.67065pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{\sf{Sp}}}_{E}[G]:i^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.67067pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
####  3.3. “Stably E-local” versus “stable and $E$-local”

We end this section with some remarks on the relationship between $E$-local
spectra and the stablisation of $E$-local spaces. Various aspects of the
relationship between these two constructions have been considered by various
groups of authors including; Mahowald and Thompson for $p$-adic $K$-theory
[MahowaldThompsonKTheory], Barthel and Bousfield for completion at a prime
[BarthelBousfieldCompletion], and by Barnes and Roitzheim in their work on
local framings [BarnesRoitzheimFramings].

######  Lemma 3.3.1 ([BarnesRoitzheimFramings, Lemma 8.3]).

Let $E$ be a spectrum. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
31.83672pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:{\sf{Sp}}(L_{E}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
55.83672pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
55.83672pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Sp}}_{E}:\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
31.83673pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction between the stablisation of $E$-local spaces and
$E$-local spectra.

######  Remark 3.3.2.

This Quillen adjunction is not always a Quillen equivalence. For example, in
the case of completion at a prime $p$, the suspension spectrum of a
$p$-complete nilpotent space is $p$-complete if and only the homotopy groups
of the space are bounded $p$-torsion [BarthelBousfieldCompletion, Theorem
4.7], hence the derived adjunction is not an equivalence of homotopy
categories.

Barnes and Roitzheim give a large class of examples for which the adjunction
is a Quillen equivalence. In fact, they give a class of examples for which the
model structures become identical.

######  Lemma 3.3.3 ([BarnesRoitzheimFramings, Lemma 8.6]).

Let $R$ be a subring of the rationals. There is an equality of model
structures

${\sf{Sp}}(L_{HR}\operatorname{\sf{Top}_{\ast}})={\sf{Sp}}_{HR}.$

##  Part II Local orthogonal calculus

With the preliminaries in place, we now construct the $S$-local orthogonal
calculus for $S$ a set of maps of based spaces. Our approach is motivated by
the model categorical perspective on orthogonal calculus of Barnes and Oman
[BO13], and Barnes’ rational orthogonal calculus [Ba17].

###  4\. Input functors

####  4.1. Input functors

We first recall the ‘input functors’ for the calculus. These are the base
functors which one wishes to study through the lens of orthogonal calculus.

######  Definition 4.1.1.

Define $\mathcal{J}$ to be the category with objects finite-dimensional inner
product subspaces of $\mathbb{R}^{\infty}$ and morphisms linear isometries.
Define $\mathcal{J}_{0}$ to be the category with the same objects and
$\mathcal{J}_{0}(U,V)=\mathcal{J}(U,V)_{+}$.

The morphism set $\mathcal{J}(U,V)$ may be topologised as the Stiefel manifold
of $\dim(U)$-frames in $V$. As such, $\mathcal{J}$ is a topologically enriched
category, and $\mathcal{J}_{0}$ is enriched in based spaces.

The category of topological222By which we mean “enriched in based spaces”.
functors from $\mathcal{J}_{0}$ to $\operatorname{\sf{Top}_{\ast}}$ is the
category of input functors for orthogonal calculus. We will refer to such
functors as _orthogonal functors_ and denote the category of such functors by
$\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$. Examples of
orthogonal functors are abound in geometry, topology and homotopy theory, and
include:

  1. (1)

the one-point compactification functor $\mathbb{S}\colon V\mapsto S^{V}$;

  2. (2)

the functor ${\sf{BO}}(-)\colon V\mapsto{\sf{BO}}(V)$ which sends an inner
product space to the classifying space of its orthogonal group;

  3. (3)

the functor ${\sf{BTOP}}(-)\colon V\mapsto{\sf{BTOP}}(V)$, which sends an
inner product space $V$ to ${\sf{BTOP}}(V)$, the classifying space of the
space of self-homeomorphisms of $V$;

  4. (4)

the functor ${\sf{BDiff}^{b}}(M\times-)\colon V\mapsto{\sf{BDiff}^{b}}(M\times
V)$, which for a fixed smooth and compact manifold $M$ sends an inner product
space $V$ to the classifying space of the group of bounded diffeomorphisms
from $M\times V$ to $M\times V$ which are the identity on $\partial M\times
V$; and,

  5. (5)

the restriction of an endofunctor on spaces spaces to evaluation on spheres
333Endofunctors of based spaces are particularly interesting from a homotopy
theoretic point of view when you restrict to the values on spheres, see e.g.,
[BehrensEHP, AroneMahowald, Ar02]. In particular for $F$ the identity functor,
the Weiss tower of $F\circ\mathbb{S}=\mathbb{S}$ and the Goodwillie tower for
$F$ agree up to weak equivalence [BE16], hence orthogonal calculus is
intimately related to understanding the (stable) homotopy groups of spheres.

The category of orthogonal functors may be equipped with a projective model
structure.

######  Proposition 4.1.2.

There is a model category structure on the category of orthogonal functors
$\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ with weak
equivalences and fibrations defined levelwise. The generating cofibrations are
of the form

$\mathcal{J}_{0}(U,-)\wedge
S^{n-1}_{+}\longrightarrow\mathcal{J}_{0}(U,-)\wedge D^{n}_{+},$

for $U\in\mathcal{J}_{0}$ and $n\geq 0$. The generating acyclic cofibrations
are of the form

$\mathcal{J}_{0}(U,-)\wedge
D^{n}_{+}\longrightarrow\mathcal{J}_{0}(U,-)\wedge(D^{n}\times[0,1])_{+},$

for $U\in\mathcal{J}_{0}$ and $n\geq 0$. This model structure is cellular,
proper and topological.

####  4.2. Local input functors

The category of topological functors from $\mathcal{J}_{0}$ to
$\operatorname{\sf{Top}_{\ast}}$ is also the category of input functors for
orthogonal calculus localized at a set of maps of based spaces $S$. The ‘base’
model structure for the $S$-local orthogonal calculus will be the $S$-local
model structure on the category of orthogonal functors.

######  Proposition 4.2.1.

Let $S$ be a set of maps of based spaces. There is model structure on the
category of topological functors from $\mathcal{J}_{0}$ to
$\operatorname{\sf{Top}_{\ast}}$ such that a map $f\colon X\to Y$ is a weak
equivalence or fibration if for all $V\in\mathcal{J}_{0}$, the induced map
$f(V)\colon X(V)\to Y(V)$ is a $S$-local equivalence or a $S$-local fibration
of based spaces, respectively. This model structure is cellular, left proper
and topological. We call this model structure the $S$-local projective model
structure and it by
${\sf{Fun}}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$.

###### Proof.

This model structure is an instance of a projective model structure on a
category of functors, see e.g., [Hi03, Theorem 11.6.1]. ∎

As a particular example, we obtain the following model structure for the
localization with respect to a homology theory $E_{\ast}$.

######  Corollary 4.2.2.

Let $E$ be a spectrum. There is a model structure on the category of
topological functors from $\mathcal{J}_{0}$ to
$\operatorname{\sf{Top}_{\ast}}$ such that a map $f\colon X\to Y$ is a weak
equivalence or fibration if for all $V\in\mathcal{J}_{0}$ the induced map
$f(V)\colon X(V)\to Y(V)$ is a $E_{\ast}$-isomorphism or a fibration in the
model category of $E$-local based spaces, respectively.

###  5\. Polynomial functors

####  5.1. Polynomial functors

Polynomial functors behave in many ways like polynomial functions from
classical calculus, e.g., a functor which is polynomial of degree less than or
equal $n$, is polynomial of degree less than or equal $n+1$. We give only the
necessary details here and refer the reader to [We95] or [BO13] for more
details on polynomial functors in orthogonal calculus.

######  Definition 5.1.1.

An orthogonal functor $F$ is _polynomial of degree less than or equal $n$_ if
for each $U\in\mathcal{J}_{0}$, the canonical map

$F(U)\longrightarrow\underset{0\neq
V\subseteq\mathbb{R}^{n+1}}{\operatorname{\mathrm{holim}}}F(U\oplus
V)=:\tau_{n}F(U),$

is a weak homotopy equivalence. Functors which are polynomial of degree less
than or equal $n$ will sometimes be referred to as _$n$ –polynomial_ functors.

######  Remark 5.1.2.

Given an orthogonal functor $F$ and an inner product space $U$ we can restrict
the orthogonal functor $F(U\oplus-)$ to a functor

$F(U\oplus-)\colon\mathcal{P}(\mathbb{R}^{n+1})\longrightarrow\operatorname{\sf{Top}_{\ast}},$

where $\mathcal{P}(\mathbb{R}^{n+1})$ is the poset of finite-dimensional inner
product subspaces of $\mathbb{R}^{n+1}$. Such functors are deserving of the
name _$\mathbb{R}^{n+1}$ -cubes_ by analogy with cubical homotopy theory. The
orthogonal functor $F$ being $n$-polynomial is equivalent to asking that for
each $U$ this restricted functor is homotopy cartesian. Informally speaking,
orthogonal calculus can be thought of a calculus built from _$\mathbb{R}^{n}$
-cubical homotopy theory_ in a similar way to how Goodwillie calculus is built
from cubical homotopy theory, see e.g., [MV15].

There is a functorial assignment of a universal (up to homotopy)
$n$-polynomial functor to any orthogonal functor $F$. It is the _$n$
-polynomial approximation_ of $F$, and is defined as

$T_{n}F(U)=\operatorname{\mathrm{hocolim}}(F(U)\longrightarrow\tau_{n}F(U)\longrightarrow\cdots\longrightarrow\tau_{n}^{k}F(U)\longrightarrow\cdots).$

In [BO13, Proposition 6.5 $\&$ Proposition 6.6], Barnes and Oman construct a
localization of the projective model structure on the category of orthogonal
functors which captures the homotopy theory of $n$-polynomial functors. The
$n$-polynomial approximation functor is a fibrant replacement in this model
structure, which implies that any map $F\to F^{\prime}$ with $F^{\prime}$
$n$-polynomial factors (up to homotopy) through the $n$-polynomial
approximation of $F$. There are two equivalent ways to consider this model
structure; as the Bousfield-Friedlander localization of
$\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ at the
$n$-polynomial approximation endofunctor

$T_{n}\colon\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\longrightarrow\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}),$

or, as the left Bousfield localization at the set of maps

$\mathcal{S}_{n}=\\{S\gamma_{n+1}(U,V)_{+}\longrightarrow\mathcal{J}_{0}(U,V)\
\mid\ U,V\in\mathcal{J}_{0}\\},$

where $S\gamma_{n+1}(U,V)$ is the sphere bundle of the $(n+1)$-st complement
bundle $\gamma_{n+1}(U,V)$ over the space of linear isometries
$\mathcal{J}(U,V)$. The fibre over a linear isometry $f$ in the $(n+1)$-st
complement bundle is $\mathbb{R}^{n+1}\otimes f(U)^{\perp}$, where
$f(U)^{\perp}$ is the orthogonal complement of $f(U)$ in $V$.

######  Proposition 5.1.3 ([BO13, Proposition 6.5]).

There is a model category structure on the category of orthogonal functors
with weak equivalences the $T_{n}$-equivalences444A map $f\colon X\to Y$ is a
$T_{n}$-equivalence if $T_{n}(f)\colon T_{n}X\to T_{n}Y$ is a levelwise weak
equivalence. and fibrations those levelwise fibrations $f\colon X\to Y$ such
the square

${X}$${T_{n}X}$${{Y}}$${{T_{n}Y}}$

is a homotopy pullback. This model structure is cellular, proper and
topological. We call this the $n$-polynomial model structure and denote it by
$\mathsf{Poly}^{\leq n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$.

####  5.2. Local polynomial functors

######  Definition 5.2.1.

Let $S$ be a set of maps of based spaces. An orthogonal functor is _$S$
-locally $n$-polynomial_ if it is levelwise $S$-local and $n$-polynomial.

We have the following useful reduction using the $S$-local Whitehead’s
Theorem, [Hi03, Theorem 3.2.13(1)].

######  Lemma 5.2.2.

Let $S$ be a set of maps of based spaces, and $F$ an orthogonal functor. If
$F$ is levelwise $S$-local, then $F$ is $n$-polynomial if and only if the
canonical map

$F(U)\longrightarrow\underset{0\neq
V\subseteq\mathbb{R}^{n+1}}{\operatorname{\mathrm{holim}}}F(U\oplus V),$

is a $S$-local equivalence.

In our search for a model structure which captures the homotopy theory of
$S$-locally $n$-polynomial functors we start with a general existence theorem
which is an iterated left Bousfield localization involving the set
$\mathcal{S}_{n}$ and the set

$J_{S}=\\{\mathcal{J}_{0}(U,-)\wedge j\mid U\in\mathcal{J},,j\in
J_{L_{S}\operatorname{\sf{Top}_{\ast}}}\\}.$

Without extra assumptions on our localizing set (or homology theory) we are
unable to obtain clear descriptions of the weak equivalences.

######  Proposition 5.2.3.

Let $S$ be a set of maps of based spaces. There is model category structure on
the category of orthogonal functors with cofibrations the projective
cofibrations, and fibrant objects the $S$-locally $n$-polynomial functors.
This model structure is cellular, left proper and topological. We call this
model structure the $S$-local $n$-polynomial model structure and denote it by
$\mathsf{Poly}^{\leq n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$.

###### Proof.

The process of left Bousfield localizations may be iterated and it follows
that the $J_{S}$-localization of the $n$-polynomial model structure on and the
$\mathcal{S}_{n}$-localization of the $S$-local projective model structure are
identical, and have as cofibrations the projective cofibrations. It remains to
characterise the fibrant objects.

For the fibrant objects, notice that the model structure is equivalently
described as the left Bousfield localization of the projecitve model structure
with respect to the set of maps $\mathcal{S}_{n}\cup J_{S}$. By definition an
object $X$ is $\mathcal{S}_{n}\cup J_{S}$-local if and only if it is both
$\mathcal{S}_{n}$-local and $J_{S}$-local, and hence the fibrant objects are
precise those $S$-locally $n$-polynomial functors. ∎

The $S$-local $n$-polynomial model structure behaves precisely like the
$S$-localization of the $n$-polynomial model structure in the following sense.

######  Lemma 5.2.4.

Let $S$ be a set of maps of based spaces. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
39.65852pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
63.65852pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
63.65852pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.65852pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction.

###### Proof.

Since the cofibrations of both model structures are the projective
cofibrations the identity functor

$\mathds{1}\colon\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\to\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}}),$

preserves cofibrations. On the other hand, since an $S$-locally $n$-polynomial
functor is in particular $n$-polynomial, the identity functor

$\mathds{1}\colon\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})\to\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}),$

preserves fibrant objects. ∎

Our focus now turns to understanding the $S$-local $n$-polynomial model
structure better. The weak equivalences of the $S$-local $n$-polynomial model
structure are the $\mathcal{S}_{n}\cup S$-local equivalences, i.e., detected
by objects which are both $\mathcal{S}_{n}$-local and $S$-local, hence the
class of which contains both the $T_{n}$-equivalences and $S$-local
equivalences but may contain maps which are neither. To understand these
equivalences better we place a condition on the local objects analogous to the
condition that a homological localization (on spectra) is smashing, see e.g.,
[BarnesRoitzheimSHT, Proposition 7.4.3].

######  Proposition 5.2.5.

Let $S$ be a set of maps of based spaces. If the class of $S$-local objects is
closed under sequential homotopy colimits, then the weak equivalences of the
$S$-local $n$-polynomial model structure are those maps $f\colon X\to Y$ such
that the induced map

$T_{n}L_{S}f\colon T_{n}L_{S}X\to T_{n}L_{S}Y,$

is a $S$-local equivalence. In particular, The composite $T_{n}L_{S}$ is a
functorial fibrant replacement in the $S$-local $n$-polynomial model
structure.

###### Proof.

We show $(1)$, the proof of $(2)$ is similar. We apply [Ba17, Lemma 5.5] which
shows that a map $f\colon X\to Y$ is weak equivalence in the iterated left
Bousfield localization if and only if $L_{S}f\colon L_{S}X\to L_{S}Y$ is a
$\mathcal{S}_{n}$-local equivalence. This last is equivalent to $L_{S}f\colon
L_{S}X\to L_{S}Y$ being a $T_{n}$-equivalence, i.e., $T_{n}L_{S}f\colon
T_{n}L_{S}X\to T_{n}L_{S}Y$ being a levelwise weak equivalence. Since both
domain and codomain of this map are $S$-local checking this map is a levelwise
weak equivalence is equivalent to checking that it is an $S$-local equivalence
by the $S$-local Whitehead’s Theorem. ∎

######  Remark 5.2.6.

Let $S$ be a set of maps of based spaces. To ease notation, we will denote the
composite $T_{n}L_{S}$ by $T_{n}^{S}$. In particular, for $E$ a spectrum we
denote the composite functor $T_{n}L_{E}$ by $T_{n}^{E}$. In general,
$T_{n}^{S}$ need not be $S$-local, but will be when the class of $S$-local
objects is closed under sequential homotopy colimits.

######  Examples 5.2.7.

  1. (1)

For a finite cell complex $W$, $T_{n}^{W}F$ is $W$-local (or $W$-periodic) for
all orthogonal functors $F$.

  2. (2)

For localization at the Eilenberg-Maclane spectrum associated to a subring $R$
of the rationals, $T_{n}^{HR}F$ is $HR$-local for all orthogonal functors $F$.

####  5.3. Characterisations for nullifications

Bousfield [BousfieldPeriodicity, BousfieldUnstable], and Farjoun
[FarjounCellular], among others, see e.g., [CasacubertaUnstable], have
extensively studied the nullifcation of the category of based spaces at a
based space $W$. This nullification is functorial giving a functor

$P_{W}\colon\operatorname{\sf{Top}_{\ast}}\longrightarrow\operatorname{\sf{Top}_{\ast}},$

which extends levelwise to a functor

$P_{W}\colon\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\longrightarrow\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}).$

The nullified model structure on spaces (see, e.g., [Bo01, §§9.8]) extends in
a canonical way to give the Bousfield-Frielander localization of the category
of orthogonal functors at the functor $P_{W}$, which we denote by
${\sf{Fun}}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$. We show
that the $W$-local $n$-polynomial model structure is precisely the model
structure obtained by Bousfield-Friedlander localization at the composite

$T_{n}\circ
P_{W}\colon{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\longrightarrow{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}).$

We begin with a lemma which deals with fibrant objects in the Bousfield-
Friedlander localization of orthogonal functors at the endofunctor $P_{W}$,
which we call the $W$-periodic projective model structure.

######  Lemma 5.3.1.

For a finite cell complex $W$ and an orthogonal functor $F$, the functor
$T_{n}P_{W}F$ is fibrant in the Bousfield-Friedlander localization of the
category of orthogonal functors at the functor $P_{W}$. In particular, the map

$\omega_{T_{n}P_{W}F}\colon T_{n}P_{W}F\longrightarrow P_{W}T_{n}P_{W}F,$

is a levelwise weak homotopy equivalence.

###### Proof.

The Bousfield-Friedlander localization of based spaces at the endofunctor
$P_{W}$ is identical to the left Bousfield localization of based spaces at the
map $\ast\to W$, since both model structures have the same cofibrations and
fibrant objects. It follows that the Bousfield-Friedlander localization of the
category of orthogonal functors at the endofunctor $P_{W}$ is identical to the
$W$-local projective model structure. In particular, we see that $P_{W}F$ is
fibrant and hence $\tau_{n}P_{W}F$ is also fibrant, since the class of
$W$-local objects is closed under homotopy limits. The result follows since
local objects for a nullification are closed under sequential homotopy
colimits by [FarjounCellular, 1.D.6]. ∎

######  Proposition 5.3.2.

For a finite cell complex $W$ the Bousfield-Friedlander localization of the
category of orthogonal functors at the endofunctor

$T_{n}\circ
P_{W}\colon\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\longrightarrow\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}),$

exists. This model structure is proper and topological. We call this the
$W$-periodic $n$-polynomial model structure and denote it by
$\mathsf{Poly}^{\leq n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$.

###### Proof.

We verify the axioms of [Bo01, Theorem 9.3]. First note that since $P_{W}$ and
$T_{n}$ both preserve levelwise weak equivalences so does their composite,
hence verifying [Bo01, Theorem 9.3(A1)].

The natural transformation from the identity to the composite $T_{n}\circ
P_{W}$ is given in components as the composite

$F\xrightarrow{\omega_{F}}P_{W}F\xrightarrow{\eta_{P_{W}F}}T_{n}P_{W}F,$

where $\omega\colon\mathds{1}\to P_{W}$ and $\eta\colon\mathds{1}\to T_{n}$,
hence at $T_{n}P_{W}F$, we obtain the composite

$T_{n}P_{W}F\xrightarrow{\omega_{T_{n}P_{W}F}}P_{W}T_{n}P_{W}F\xrightarrow{\eta_{P_{W}T_{n}P_{W}F}}T_{n}P_{W}T_{n}P_{W}F.$

Since the domain is fibrant in the $W$-periodic projective model structure the
first map in the composite is a levelwise weak equivalence, see Lemma 5.3.1.
The second map is also a weak equivalence. To see this, note that since
$T_{n}P_{W}F$ is polynomial of degree less than or equal $n$, the functor
$P_{W}T_{n}P_{W}F$ is also polynomial of degree less than or equal $n$ by the
commutativity of the diagram

${{T_{n}P_{W}F}}$${{\tau_{n}T_{n}P_{W}F}}$${{P_{W}T_{n}P_{W}F}}$${{\tau_{n}P_{W}T_{n}P_{W}F}}$

and the fact that homotopy limits preserve levelwise weak equivalences. It
follows that the natural transformation $\eta\colon T_{n}P_{W}F\to
T_{n}P_{W}T_{n}P_{W}F$ is a levelwise weak equivalence, as a composite of two
levelwise weak equivalences.

The map $T_{n}P_{W}(\eta)\colon T_{n}P_{W}F\to T_{n}P_{W}T_{n}P_{W}F$ is also
a levelwise weak equivalence. To see this, note that there is a commutative
diagram,

${F}$${{P_{W}F}}$${{T_{n}P_{W}F}}$${{(1)}}$${{(2)}}$${{P_{W}F}}$${{P_{W}P_{W}F}}$${{P_{W}T_{n}P_{W}F}}$${{(3)}}$${{(4)}}$${{T_{n}P_{W}F}}$${{T_{n}P_{W}P_{W}F}}$${{T_{n}P_{W}T_{n}P_{W}}}$$\scriptstyle{\omega_{F}}$$\scriptstyle{\eta_{P_{W}F}}$$\scriptstyle{\omega_{F}}$$\scriptstyle{P_{W}\omega_{F}}$$\scriptstyle{\omega_{P_{W}F}}$$\scriptstyle{\eta_{P_{W}F}}$$\scriptstyle{\eta_{P_{W}P_{W}F}}$$\scriptstyle{\omega_{T_{n}P_{W}F}}$$\scriptstyle{\eta_{P_{W}T_{n}P_{W}F}}$$\scriptstyle{P_{W}\eta_{P_{W}F}}$$\scriptstyle{T_{n}P_{W}\omega_{F}}$$\scriptstyle{T_{n}P_{W}\eta_{P_{W}F}}$

in which, the required map is given by the lower horizontal composite. Since
$P_{W}$ is a homotopically idempotent functor, $P_{W}\omega_{F}$ is a
levelwise weak equivalence. It follows that the bottom horizontal map

$T_{n}P_{W}\omega_{F}\colon T_{n}P_{W}F\longrightarrow T_{n}P_{W}P_{W}F,$

of $(3)$ is a weak equivalence sine $T_{n}$ preserves weak equivalences.

Moreover, $P_{W}$ being homotopically idempotent yields that the vertical map

$\omega_{P_{W}F}\colon P_{W}F\longrightarrow P_{W}P_{W}F$

in $(2)$ is a levelwise weak equivalence. The right-hand vertical map in this
square is also an equivalence by Lemma 5.3.1. By [We95, Theorem 6.3], the top
right hand horizontal map

$\eta_{P_{W}F}\colon P_{W}F\longrightarrow T_{n}P_{W}F,$

is an approximation of order $n$ in the sense of [We95, Definition 5.16]. By
commutativity of $(2)$, the lower horizontal map

$P_{W}\eta_{P_{W}F}\colon P_{W}P_{W}F\longrightarrow P_{W}T_{n}P_{W}F,$

is an approximation of order $n$. The proof of [We95, Theorem 6.3] also
demonstrates that the vertical maps in $(4)$ are approximations of order $n$,
and since three out of the four maps in the lower right square are
approximations of order $n$, so too is the lower right hand horizontal map

$T_{n}P_{W}\eta_{P_{W}F}\colon T_{n}P_{W}P_{W}F\longrightarrow
T_{n}P_{W}T_{n}P_{W}F.$

An application of [We95, Theorem 5.15] yields that this map is a levelwise
weak equivalence as both source and target are polynomial of degree less than
or equal $n$. This concludes the proof that the map

$T_{n}P_{W}(\eta)\colon T_{n}P_{W}F\longrightarrow T_{n}P_{W}T_{n}P_{W}F,$

is a levelwise weak equivalence, and verifying [Bo01, Theorem 9.3(A2)].

Finally we verify [Bo01, Theorem 9.3(A3)]. Let

${A}$${B}$${C}$${D}$$\scriptstyle{k}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{h}$

be a pullback square with $f$ a levelwise fibration between $W$-local
$n$-polynomial functors, and $T_{n}P_{W}h\colon T_{n}P_{W}C\to T_{n}P_{W}D$ a
levelwise weak equivalence. By [Bo01, Theorem 9.9], we see that the fibre of
$k$ is $P_{W}$-acyclic, i.e. $P_{W}(\mathsf{fib}(k))$ is levelwise weakly
contractible. Since $T_{n}$ preserves levelwise weak equivalences, we see that
$T_{n}P_{W}(\mathsf{fib}(k))$ is levelwise weakly contractible, and hence $k$
is a $T_{n}P_{W}$-equivalence.

The fact that the resulting model structure is topological follows from [Bo01,
Theorem 9.1]. ∎

This Bousfield-Friedlander localization results in an identical model
structure to the $W$-local $n$-polynomial model structure of Proposition 5.2.3

######  Proposition 5.3.3.

For a finite cell complex $W$ there is an equality of model structures

${\mathsf{Poly}^{\leq
n}}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})=\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}),$

that is, the $W$-local $n$-polynomial model structure and the $W$-periodic
$n$-polynomial model structure agree. In particular, these model structures
are cellular, proper and topological.

###### Proof.

Both model structures have the same cofibrations, namely the projective
cofibrations. It suffices to show that they share the same fibrant objects.
Working through the definition of a fibrant object in the Bousfield-
Friedlander localization we see that an orthogonal functor $F$ is fibrant if
and only if the canonical map $F\to T_{n}P_{W}F$ is a levelwise weak
equivalence. It follows that $F$ must be $W$-local and $n$-polynomial, hence
fibrant in the $W$-local $n$-polynomial model structure. Conversely, if $F$ is
fibrant in the $W$-local $n$-polynomial model structure, then the map $F\to
P_{W}F$ is a levelwise weak equivalence and there is a commutative diagram

${F}$${{P_{W}F}}$${{T_{n}F}}$${{T_{n}P_{W}F}}$

in which three out of the four arrows are levelwise weak equivalences, hence
so to is the right-hand vertical arrow. It follows that $F$ is fibrant in the
Bousfield-Friedlander localization. ∎

######  Remark 5.3.4.

The nullification condition here is necessary. The above lemma does not hold
in general. To see this, consider the (smashing) localization at the spectrum
$E=H\mathbb{Q}$. The $H\mathbb{Q}$-local model structure is not right proper,
(see Remark 2.3.1) yet if this were expressible as a Bousfield-Friedlander
localization it would necessarily be right proper, [Bo01, Theorem 9.3].

######  Corollary 5.3.5.

For a finite cell complex $W$ a map $f\colon X\to Y$ is a fibration is the
$W$-local $n$-polynomial model structure if and only if $f$ is a fibration in
the projective model structure and the square

${X}$${{T_{n}P_{W}X}}$${{Y}}$${{T_{n}P_{W}Y}}$

is a homotopy pullback square in the projective model structure on
$\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$.

######  Remark 5.3.6.

It is highly unlikely that this result holds in more general localizations
than nullifications. Let $\mathcal{C}$ be a model category and $S$ a set of
maps in $\mathcal{C}$ such that the left Bousfield localization of
$\mathcal{C}$ at $S$ exists. By [Hi03, Proposition 3.4.8(1)] right properness
of $\mathcal{C}$ and $L_{S}\mathcal{C}$ is sufficient for a map $f\colon X\to
Y$ being a fibration in $L_{S}\mathcal{C}$ if and only if $f$ is a fibration
in $\mathcal{C}$ and the square

${X}$${{\hat{X}}}$${Y}$${{\hat{Y}}}$$\scriptstyle{j_{X}}$$\scriptstyle{j_{Y}}$$\scriptstyle{f}$$\scriptstyle{\hat{f}}$

is a homotopy pullback square, where $\hat{f}\colon\hat{X}\to\hat{Y}$ is a
$S$-localization of $f$ in the sense of [Hi03, Definition 3.2.16]. In our
situation, Proposition 2.3.2 guarantees that a homological localization is
right proper if and only if it is a nullification. However, it is not clear in
general if right properness of the base model category and the localized model
category is a necessary condition for the above description of the fibrations
in $L_{S}\mathcal{C}$.

###  6\. Homogeneous functors

####  6.1. Homogeneous functors

The layers of the Weiss tower associated to an orthogonal functor $F$ are the
homotopy fibres of maps $T_{n}F\to T_{n-1}F$ and have two interesting
properties: firstly, they are polynomial of degree less than or equal to $n$
and secondly, their $(n-1)$-polynomial approximation is trivial. We denote the
$n$-th layer of the Weiss tower of $F$ by $D_{n}F$.

######  Definition 6.1.1.

For $n\geq 0$, an orthogonal functor $F$ is said to be _$n$ -reduced_ if its
$(n-1)$-polynomial approximation is levelwise weakly contractible. An
orthogonal functor $F$ is said to be _homogeneous of degree $n$_ if it is both
polynomial of degree less than or equal $n$ and $n$-reduced. We will sometimes
refer to a functor which is homogeneous of degree $n$ as being _$n$
-homogeneous_.

There is a model structure on the category of orthogonal functors which
contains the $n$-homogeneous functors as the bifibrant objects.

######  Proposition 6.1.2 ([BO13, Proposition 6.9]).

There is a model category structure on the category of orthogonal functors
with weak equivalences the $D_{n}$-equivalences and fibrations the fibrations
of the $n$-polynomial model structure. The cofibrant objects are the
$n$-reduced projectively cofibrant objects and the fibrant objects are the
$n$-polynomial functors. In particular, cofibrant-fibrant objects of this
model structure are the projectively cofibrant $n$-homogeneous functors. This
model structure is cellular, proper, stable and topological. We call this the
$n$-homogeneous model structure and denote it by
$\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$.

######  Remark 6.1.3.

The model structure of [BO13, Proposition 6.9] has as weak equivalences those
maps which induce levelwise weak equivalences on the $n$-th derivatives of
their $n$-polynomial approximations. We showed in [TaggartUnitary, Proposition
8.2], that the class of such equivalences is precisely the class of
$D_{n}$-equivalences.

The $n$-homogeneous model structure is (zigzag) Quillen equivalent to spectra
with an action of $O(n)$.

######  Proposition 6.1.4 ([BO13, Proposition 8.3 $\&$ Theorem 10.1]).

Let $n\geq 0$. There is a zigzag of Quillen equivalences

$\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\simeq_{Q}{{\sf{Sp}}}[O(n)].$

On the homotopy category level the Barnes-Oman zigzag of Quillen equivalences
recovers Weiss’ characterisation of homogeneous functors of degree $n$.

######  Proposition 6.1.5 ([We95, Theorem 7.3]).

Let $n\geq 1$. An $n$-homogeneous functor $F$ is determined by and determines
a spectrum $\partial_{n}F$ with an $O(n)$-action. In particular, an
$n$-homogeneous functor $F$ is levelwise weak homotopy equivalent to the
functor

$V\longmapsto\Omega^{\infty}[(S^{\mathbb{R}^{n}\otimes
V}\wedge\partial_{n}F)_{hO(n)}],$

and any functor of the above form is homogeneous of degree $n$.

####  6.2. Local homogeneous functors

######  Definition 6.2.1.

Let $S$ be a set of maps of based spaces. An orthogonal functor $F$ is _$S$
-locally homogeneous of degree $n$_ if it is levelwise $S$-local and
$n$-homogeneous.

Our prototypical example of an $S$-local $n$-homogeneous functor is the
homotopy fibre of the map $T_{n}(L_{S}X)\to T_{n-1}(L_{S}X)$, when
localization with respect to $S$ is “nice”, as made precise below.

######  Lemma 6.2.2.

Let $S$ be a set of maps of based spaces, and $F$ and orthogonal functor. For
$n\geq 1$, there is a homotopy fibre sequence

$D_{n}^{S}F\longrightarrow T_{n}(L_{S}F)\longrightarrow T_{n-1}(L_{S}F),$

in which $D_{n}^{S}(F)$ is

  1. (1)

homogeneous of degree $n$; and,

  2. (2)

$S$-locally $n$-homogeneous if, in addition, the class of $S$-local spaces is
closed under sequential homotopy colimits.

###### Proof.

By [We95, Lemma 5.5] the homotopy fibre of a map between two $n$-polynomial
functors is $n$-polynomial, hence $D_{n}^{S}F$ is $n$-polynomial. Applying
$T_{n-1}$ to the homotopy fibre sequence, yields that the $(n-1)$-polynomial
approximation of $D_{n}^{S}F$ is levelwise weakly contractible, proving $(1)$.

For $(2)$, observe that the homotopy fibre of a map between $S$-local objects
is $S$-local and when the class of $S$-local spaces is closed under sequential
homotopy colimits, $T_{n}L_{S}F$ is $S$-local for all $n$. ∎

######  Examples 6.2.3.

  1. (1)

For homological localization at the Eilenberg-Maclane spectrum associated to a
subring $R$ of the rationals, $D_{n}^{HR}F$ is $HR$-locally $n$-homogeneous.

  2. (2)

For nullification at a finite cell complex $W$, $D_{n}^{W}F$ is $W$-locally
$n$-homogeneous.

In most versions of functor calculus the $n$-homogeneous model structure is a
right Bousfield localization of the $n$-polynomial model structure, see e.g.,
[BR14, Theorem 6.4], [BO13, Proposition 6.9] or [TaggartReality, Proposition
3.2]. In the local picture, the $n$-polynomial model structure need not be
right proper, hence one must find an alternative way to construct an
$n$-homogeneous model structure. We will return to the search for a local
$n$-homogeneous model structure once we have discussed the derivatives in
orthogonal calculus.

###  7\. The derivatives

####  7.1. The derivatives

For each $n\geq 0$, sitting over the space of linear isometries
$\mathcal{J}(U,V)$ is a vector bundle $\gamma_{n}(U,V)$ with fibre over a
linear isometry $f\colon U\to V$ given by $\mathbb{R}^{n}\otimes
f(U)^{\perp}$, where $f(U)^{\perp}$ denotes the orthogonal complement of
$f(U)$ in $V$.

######  Definition 7.1.1.

For $n\geq 0$ define $\mathcal{J}_{n}$ to be the category with the same
objects as $\mathcal{J}$ and morphism space $\mathcal{J}_{n}(U,V)$ given as
the Thom space of $\gamma_{n}(U,V)$.

For $n=0$ this recovers the category $\mathcal{J}_{0}$ of Definition 4.1.1.

######  Remark 7.1.2.

The standard action of $O(n)$ on $\mathbb{R}^{n}$ via the regular
representation induces an action on the vector bundles that is compatible with
the composition, hence $\mathcal{J}_{n}$ is naturally enriched over based
spaces with an $O(n)$-action.

######  Definition 7.1.3.

For $n\geq 0$ define the _$n$ -th intermediate category_ to be the category of
$O(n)\operatorname{\sf{Top}_{\ast}}$-enriched functors from $\mathcal{J}_{n}$
to $O(n)\operatorname{\sf{Top}_{\ast}}$. We will denote this category by
${\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$.

Let $0\leq m\leq n$. The inclusion
$i_{m}^{n}\colon\mathbb{R}^{m}\to\mathbb{R}^{n}$ induces a functor
$i_{m}^{n}\colon\mathcal{J}_{m}\to\mathcal{J}_{n}$. Postcomposition with
$i_{m}^{n}$ induced a topological functor

$\operatorname{\mathsf{res}}_{m}^{n}\colon\sf{Fun}(\mathcal{J}_{n},\operatorname{\sf{Top}_{\ast}})\longrightarrow\sf{Fun}(\mathcal{J}_{m},\operatorname{\sf{Top}_{\ast}}),$

which by [We95, Proposition 2.1] has a right adjoint

$\operatorname{\mathsf{ind}}_{m}^{n}\colon\sf{Fun}(\mathcal{J}_{m},\operatorname{\sf{Top}_{\ast}})\longrightarrow\sf{Fun}(\mathcal{J}_{n},\operatorname{\sf{Top}_{\ast}}),$

given by

$\operatorname{\mathsf{ind}}_{m}^{n}F(U)=\operatorname{\mathsf{nat}}_{m}(\mathcal{J}_{n}(U,-),F),$

where $\operatorname{\mathsf{nat}}_{m}(-,-)$ denotes the (based) space of
natural transformations in
$\sf{Fun}(\mathcal{J}_{m},\operatorname{\sf{Top}_{\ast}})$, and
$\mathcal{J}_{n}(U,-)$ is considered as an object of
$\sf{Fun}(\mathcal{J}_{m},\operatorname{\sf{Top}_{\ast}})$ by restriction.
Combining the restriction and induction functors with change of group
adjunctions from [MM02], we obtain an adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{m}^{n}/O(n-m):{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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106.6729pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Fun}}_{O(m)}(\mathcal{J}_{m},O(m)\operatorname{\sf{Top}_{\ast}}):{\operatorname{\mathsf{ind}}}_{m}^{n}\sf{CI}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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see [BO13, §4].

######  Definition 7.1.4.

Let $F$ be an orthogonal functor. For $n\geq 0$, the _$n$ -th derivative of
$F$_ is given by $\operatorname{\mathsf{ind}}_{0}^{n}{\sf{CI}}F$. In which
case, we write $\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}F$ or
$F^{(n)}$.

The $n$-th intermediate category is equivalent to the category of modules over
the monoid

$\displaystyle n\mathbb{S}\colon\mathcal{I}$ $\displaystyle\longrightarrow
O(n)\operatorname{\sf{Top}_{\ast}},$ $\displaystyle\ V$
$\displaystyle\longmapsto S^{\mathbb{R}^{n}\otimes V},$

in the category of $O(n)$-equivariant $\mathcal{I}$-spaces, where
$\mathcal{I}$ is the category of finite-dimensional inner product subspaces of
$\mathbb{R}^{\infty}$ and linear isometric isomorphisms, see [BO13,
Proposition 7.4]. As such it may be equipped with a stable model structure
reminiscent of the stable model structure on spectra. Given an object $F$ of
the $n$-th intermediate category, we define the $n$-homotopy groups of $F$ as

$n\pi_{k}F=\underset{q}{\operatorname{\mathrm{colim}}}~{}\pi_{nq+k}(F(\mathbb{R}^{q})),$

for $k\in\mathbb{Z}$. The weak equivalences of the $n$-stable model structure
are then the $n\pi_{\ast}$-isomorphisms; those maps $f\colon X\to Y$ such that
for each integer $k$, the induced map

$n\pi_{k}(f)\colon n\pi_{k}(X)\longrightarrow n\pi_{k}(Y),$

is an isomorphism.

######  Proposition 7.1.5 ([BO13, Proposition 7.14]).

There is a model category structure on the $n$-th intermediate category with
weak equivalences the $n\pi_{\ast}$-isomorphisms and fibrations the levelwise
fibrations $X\to Y$ such that the square

${{X(U)}}$${{\Omega^{\mathbb{R}^{n}\otimes V}X(U\oplus
V)}}$${{Y(U)}}$${{\Omega^{\mathbb{R}^{n}\otimes V}Y(U\oplus V)}}$

is a homotopy pullback for all $U,V\in\mathcal{J}_{n}$. The fibrant objects
are the $n\Omega$-spectra, i.e., those objects $F$ such that

$F(U)\to\Omega^{\mathbb{R}^{n}\otimes V}F(U\oplus V),$

is a weak homotopy equivalence for all $U,V\in\mathcal{J}_{n}$. This model
structure is cellular, proper, stable and topological. We call this the
$n$-stable model structure and denote it by
${\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$.

####  7.2. The local $n$-stable model structure

We now equip the $n$-th intermediate category with an $S$-local model
structure which will be intermediate in our classification of $S$-local
$n$-homogeneous functors as levelwise $S$-local spectra with an action of
$O(n)$.

######  Proposition 7.2.1.

Let $S$ be a set of maps of based spaces. There is a model category structure
on the $n$-th intermediate category with cofibrations the cofibrations of the
$n$-stable model structure and fibrant objects the $n\Omega$-spectra which are
levelwise $S$-local. This model structure is cellular, left proper and
topological. We call this the $S$-local $n$-stable model structure and denote
it by
$L_{S}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$.

###### Proof.

This model structure is the left Bousfield localization of the $n$-stable
model structure at the set

$\mathcal{Q}_{n}=\\{O(n)_{+}\wedge\mathcal{J}_{n}(U,-)\wedge j\mid
U\in\mathcal{J},j\in J_{L_{S}\operatorname{\sf{Top}_{\ast}}}\\}.$

∎

We record the following fact which will prove useful later.

######  Lemma 7.2.2.

Let $S$ be a set of maps of based spaces. If $F$ is an $S$-local functor, then
$F^{(n)}=\operatorname{\mathsf{ind}}_{0}^{n}F$ is $S$-local.

###### Proof.

The objectwise smash product

$(-)\wedge(-)\colon{\sf{Fun}}(\mathcal{J}_{n},L_{S}\operatorname{\sf{Top}_{\ast}})\times
L_{S}\operatorname{\sf{Top}_{\ast}}\longrightarrow{\sf{Fun}}(\mathcal{J}_{n},L_{S}\operatorname{\sf{Top}_{\ast}}),$

is a Quillen bifunctor, and the result follows from the definition of
$\operatorname{\mathsf{ind}}_{0}^{n}F$. ∎

####  7.3. The derivatives as spectra

The $n$-th derivative of an orthogonal functor may be thought of as a
spectrum, in particular, the $n$-th intermediate category and the category of
spectra with an $O(n)$-action are Quillen equivalent via the following
adjunction

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83.48582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{\sf{Sp}}}[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.48584pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
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see e.g., [BO13, §8]. We now prove that this result holds $S$-locally for any
set $S$ of maps of based spaces.

######  Theorem 7.3.1.

Let $S$ be a set of maps of based spaces. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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90.15579pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{\sf{Sp}}}(L_{S}\operatorname{\sf{Top}_{\ast}})[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.1558pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence between the $S$-local model structures.

###### Proof.

For the Quillen adjunction apply [Hi03, Theorem 3.3.20(1)], noting that there
is an isomorphism

$(\alpha_{n})_{!}(O(n)_{+}\wedge\mathcal{J}_{n}(U,-)\wedge j)\cong
O(n)_{+}\wedge\mathcal{J}_{1}(\mathbb{R}^{n}\otimes U,-)\wedge j,$

for $j$ a generating acyclic cofibration for the $S$-local model structure on
based spaces.

By [BO13, Proposition 8.3] the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
59.48582pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}:{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
83.48582pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.48582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{\sf{Sp}}}[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.48584pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence. To show that the adjunction is a Quillen
equivalence, it suffices by [HoveySpectra, Proposition 2.3] to show that if
$Y$ is fibrant in ${{\sf{Sp}}}[O(n)]$ such that $(\alpha_{n})^{*}Y$ is fibrant
in the $S$-local $n$-stable model structure, then $Y$ is fibrant in the
$S$-local model structure on ${{\sf{Sp}}}[O(n)]$. This follows readily from
the definitions of fibrant objects in both model structures. ∎

In the case of homological localizations we obtain the following Quillen
equivalence.

######  Corollary 7.3.2.

Let $E$ be a spectrum. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
66.5058pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}:L_{E}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
90.5058pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
90.5058pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{\sf{Sp}}}(L_{E}\operatorname{\sf{Top}_{\ast}})[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.5058pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence between the $E$-local model structures.

###  8\. The classification of local homogeneous functors

####  8.1. The local $n$-homogeneous model structure

We now return to the question of a suitable model structure on the category of
orthogonal functors which caputures the homotopy theory of local
$n$-homgoeneous functors. The case of the homological localization
$E=H\mathbb{Q}$ was discussed by Barnes in [Ba17, §6]

######  Proposition 8.1.1.

Let $S$ be a set of maps of based spaces. There is model category structure on
the category of orthogonal functors with cofibrations the cofibrations of the
$n$-homogeneous model structure and fibrant objects the $n$-polynomial
functors whose $n$-th derivative is levelwise $S$-local. This model structure
is cellular, left proper and topological. We call this the $S$-local
$n$-homogeneous model structure and denote it by
$\mathsf{Homog}^{n}(\mathcal{J},L_{S}\operatorname{\sf{Top}_{\ast}})$.

###### Proof.

We left Bousfield localize the $n$-homogeneous model structure at the set of
maps

$\mathcal{K}_{n}=\\{\mathcal{J}_{n}(U,-)\wedge j\mid U\in\mathcal{J},j\in
J_{L_{S}\operatorname{\sf{Top}_{\ast}}}\\},$

where $\mathcal{J}_{n}(U,V)$ is the Thom space of the $n$-th complement vector
bundle $\gamma_{n}(U,V)$ sitting over the space of linear isometries
$\mathcal{J}(U,V)$. This left Bousfield localization exists since the
$n$-homogeneous model structure is cellular and left proper by [Ba17, Lemma
6.1]. The description of the cofibrations follows immediately.

The fibrant objects are the $\mathcal{K}_{n}$-local objects which are fibrant
in the $n$-homogeneous model structure, i.e., those $n$-polynomial functors
$Z$ for which the induced map

$[\mathcal{J}_{n}(U,-)\wedge B,Z]\longrightarrow[\mathcal{J}_{n}(U,-)\wedge
A,Z],$

is an isomorphism for all maps $\mathcal{J}_{n}(U,-)\wedge
A\to\mathcal{J}_{n}(U,-)\wedge B$ in $\mathcal{K}_{n}$. A straightforward
adjunction argument, and the definition of the $n$-th derivative of an
orthogonal functor yield the required characterisation of the fibrant objects.
∎

######  Corollary 8.1.2.

Let $S$ be a set of maps of based spaces. The cofibrant objects of the
$S$-local $n$-homogeneous model structure are the projectively cofibrant
functors which are $n$-reduced.

###### Proof.

The $S$-local $n$-homogeneous model structure is a particular left Bousfield
localization of the $n$-homogeneous model structure, hence has the same
cofibrant objects. The result follows by the orthogonal calculus version of
[TaggartUnitary, Corollary 8.6]. ∎

We now relate the $S$-local $n$-homogeneous model structure to the $S$-local
$n$-polynomial model structure, exhibiting that the $S$-local $n$-homogeneous
model structure behaves like a right Bousfield localization of the $S$-local
$n$-polynomial model structure.

######  Lemma 8.1.3.

Let $S$ be a set of maps of based spaces. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
49.22017pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.22017pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.22017pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
49.22018pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction.

###### Proof.

To demonstrate that the left adjoint preserves cofibrations it suffices by
[Ho99, Lemma 2.1.20] to show that the identity functor sends the generating
cofibrations of the $S$-local $n$-homogeneous model structure to cofibrations
of the $S$-local $n$-polynomial model structure.

The cofibrations of the $S$-local $n$-homogeneous model structure are the
cofibrations of the $n$-homogeneous model structure, which are contained in
the cofibrations of the $n$-polynomial model structure, which in turn are
precisely the cofibrations of the $S$-local $n$-polynomial model structure,
hence

$\mathds{1}\colon\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})\longrightarrow\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}}),$

preserves cofibrations.

On the other hand, to show that the right adjoint is right Quillen it suffices
to show that the identity functor sends fibrant objects in the $S$-local
$n$-polynomial model structure to fibrant objects in the $S$-local
$n$-homogeneous model structure. This follows from Lemma 7.2.2 since the
fibrant objects in the $S$-local $n$-polynomial model structure are the
$S$-locally $n$-polynomial functors by Proposition 5.2.3 and the fibrant
objects of the $S$-local $n$-homogeneous model structure are the
$n$-polynomial functors with $S$-local $n$-th derivative by Proposition 8.1.1.
∎

####  8.2. Characterisations for homological localizations

In the case of a homological localization, we obtain a characterisation of the
fibrations. A more general result holds when the localizing set $S$ is stable
in the sense of [BarnesRoitzheimStable, Definition 4.2], i.e., when the class
of $S$-local spaces is closed under suspension.

######  Proposition 8.2.1.

If $S$ is a set of maps of based spaces which is stable, then the fibrations
of the $S$-local $n$-homogeneous model structure are those maps $f\colon X\to
Y$ which are fibrations in the $n$-polynomial model structure such that

$X^{(n)}\longrightarrow Y^{(n)},$

is a levelwise fibration of $S$-local spaces.

###### Proof.

We first given an explicit characterisation of the acyclic cofibrations since
the fibrations are characterised by the right lifting property against these
maps. The maps in $\mathcal{K}_{n}$ are cofibrations between cofibrant objects
since $\mathcal{J}_{n}(U,-)$ is cofibrant in
$\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ and the
maps in $J_{L_{S}\operatorname{\sf{Top}_{\ast}}}$ are cofibrations of
$S$-local spaces. Moreover, since the localizing set $S$ is stable, it follows
the set of generating acyclic cofibrations
$J_{L_{S}\operatorname{\sf{Top}_{\ast}}}$ is stable and in turn that the set
$\mathcal{K}_{n}$ is stable. Hence by [BarnesRoitzheimStable, Theorem 4.11],
the generating acyclic cofibrations are given by the set
$J_{\mathsf{Homog}^{n}}\cup\Lambda(\mathcal{K}_{n})$, where
$J_{\mathsf{Homog}^{n}}$ is the set of the generating acyclic cofibrations of
the $n$-homogeneous model structure and $\Lambda(\mathcal{K}_{n})$ the set of
horns on $\mathcal{K}_{n}$ in the sense of [Hi03, Definition 4.2.1]. As horns
in topological model categories are given by pushouts and $\mathcal{K}_{n}$ is
a set of cofibrations between cofibrant objects it suffices to use the set
$J_{\mathsf{Homog}^{n}}\cup\mathcal{K}_{n}$ as the generating acyclic
cofibrations of the $S$-local $n$-homogeneous model structure.

If $f\colon X\to Y$ is a map with the right lifting property with respect to
$J_{\mathsf{Homog}^{n}}\cup\mathcal{K}_{n}$, then $f$ has the right lifting
property with respect to $J_{\mathsf{Homog}^{n}}$ and the right lifting
property with respect to $\mathcal{K}_{n}$ independently. Having the right
lifting property with respect to $J_{\mathsf{Homog}^{n}}$ is equivalent to
being a fibration in the $n$-polynomial model structure. On the other hand, a
map in $\mathcal{K}_{n}$ is of the form $\mathcal{J}_{n}(U,-)\wedge
A\to\mathcal{J}_{n}(U,-)\wedge B$ for $A\to B$ a generating acyclic
cofibration of the $S$-local model structure on based spaces. A lift in the
diagram

${{\mathcal{J}_{n}(U,-)\wedge A}}$${X}$${{\mathcal{J}_{n}(U,-)\wedge
B}}$${Y}$$\scriptstyle{f}$

(indicated by the dotted arrow) exists if and only if the lift in the diagram

${A}$${{\operatorname{\mathsf{nat}}_{0}(\mathcal{J}_{n}(U,-),X)}}$${{B}}$${{\operatorname{\mathsf{nat}}_{0}(\mathcal{J}_{n}(U,-),Y)}}$$\scriptstyle{f_{*}}$

exists, which is equivalent to the statement that $X^{(n)}\to Y^{(n)}$ is a
levelwise fibration of $S$-local spaces, see §7.1. ∎

This specialises to homological localizations.

######  Corollary 8.2.2.

Let $E$ be a spectrum. The fibrations of the $E$-local $n$-homogeneous model
structure are those maps $f\colon X\to Y$ which are fibrations in the
$n$-polynomial model structure such that

$X^{(n)}\longrightarrow Y^{(n)},$

is a levelwise fibration of $E$-local spaces.

###### Proof.

Combine Proposition 8.2.1 with [BarnesRoitzheimStable, Example 4.3]. ∎

######  Remark 8.2.3.

For the case $E=H\mathbb{Q}$, Barnes [Ba17, Theorem 6.2], states that the
fibrant objects of this model structure are the $n$-homogenous functors $F$
with $F^{(n)}$ levelwise $H\mathbb{Q}$-local. This is a typographical error;
it is the _bifibrant_ objects which admit this characterisation.

######  Corollary 8.2.4.

Let $E$ be a spectrum. An orthogonal functor $F$ is fibrant in the $E$-local
$n$-homogeneous model structure if and only if $F$ is $n$-polynomial and
$F^{(n)}$ is levelwise $E$-local. In particular, $F$ is bifibrant if $F$ is
projectively cofibrant, $n$-homogeneous and $X^{(n)}$ is levelwise $E$-local.

###### Proof.

Apply Corollary 8.2.2 to the map $F\to\ast$. ∎

####  8.3. Characterisations for nullifications

In the case of a nullification with respect to a based space $W$ Proposition
8.1.1 is not the only way of constructing a model structure which deserves the
title of the $W$-local $n$-homogeneous model structure. Since the $W$-local
model structure on based spaces is right proper, so too is the $W$-local
$n$-polynomial model structure and hence we can also follow the classical
procedure and preform a right Bousfield localization at the set

$\mathcal{K}_{n}^{\prime}=\\{\mathcal{J}_{n}(U,-)\mid U\in\mathcal{J}\\},$

to obtain a local $n$-homogeneous model category structure.

######  Proposition 8.3.1.

For a finite cell complex $W$ there exists a model structure on the category
of orthogonal functors with weak equivalences those maps $X\to Y$ such that

$(T_{n}P_{W}X)^{(n)}\longrightarrow(T_{n}P_{W}X)^{(n)},$

is a levelwise weak equivalence and with fibrations the fibrations of the
$W$-local $n$-polynomial model structure. This model structure cellular,
proper, stable and topological. We call this the $W$-periodic $n$-homogeneous
model structure and denote it
$\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$.

###### Proof.

This is the right Bousfield localization of the $W$-local $n$-polynomial model
structure. The proof of which follows exactly as in [BO13, Proposition 6.9].
Note that this right Bousfield localization exists since the $W$-local
$n$-polynomial model structure in right proper and cellular when the
localization is a nullification, see Proposition 5.3.3. ∎

This right Bousfield localization behaves like a left Bousfield localization
of the $n$-homogeneous model structure in the following sense.

######  Lemma 8.3.2.

For a finite cell complex $W$ the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
43.10574pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.10574pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.10574pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
43.10576pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction.

###### Proof.

Since the acyclic cofibrations of the $n$-homogeneous model structure are
precisely the acyclic cofibrations of the $n$-polynomial model structure and
similarly, the acyclic cofibrations of $W$-periodic $n$-homogeneous model
structure are precisely the acyclic cofibrations of the $W$-local
$n$-polynomial model structure, the identity functor preserves acyclic
cofibrations by Lemma 5.2.4.

On the other hand, by [Hi03, Proposition 3.3.16(2)], cofibrations between
cofibrant objects in a right Bousfield localization are cofibrations in the
underlying model structure, hence Lemma 5.2.4 shows that the identity functor
preserves cofibrations between cofibrant objects. The result follows by
[DuggerReplacement, Corollary A.2]. ∎

######  Remark 8.3.3.

We will see in Corollary 8.5.2 that although the $W$-local $n$-homogeneous
model structure and the $W$-periodic $n$-homogeneous model structure are not
identical, the identity functor yields a Quillen equivalence between them and
hence they have canonically equivalent homotopy categories.

####  8.4. Differentiation as a Quillen functor

The $n$-th derivative of a functor $F$ is a right Quillen functor as part of a
Quillen equivalence between the $n$-homogeneous model structure and the
intermediate category; the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
73.33556pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n):{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
97.33556pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
97.33556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}):{\operatorname{\mathsf{ind}}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.33557pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence, [BO13, Theorem 10.1]. We now show that this extends
to the $S$-local situation with respect to a set of maps of based spaces.

######  Theorem 8.4.1.

Let $S$ be a set of maps of based spaces. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
80.00552pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n):L_{S}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
104.00552pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
104.00552pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}}):{\operatorname{\mathsf{ind}}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
80.00554pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence between the $S$-local model structures.

###### Proof.

The left adjoint applied to the localizing set of the $S$-local $n$-stable
model structure is precisely the localization set of the $S$-local
$n$-homogeneous model structure, hence the result follows from [Hi03, Theorem
3.3.20(1)]. ∎

######  Corollary 8.4.2.

Let $S$ be a set of maps of based spaces, and $n\geq 0$. There is a zigzag of
Quillen equivalences

$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})\simeq_{Q}{{\sf{Sp}}}(L_{S}\operatorname{\sf{Top}_{\ast}})[O(n)].$

######  Example 8.4.3.

Let $R$ be a subring of the rationals. Then there is a zigzag of Quillen
equivalences

$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{HR}\operatorname{\sf{Top}_{\ast}})\simeq_{Q}{{\sf{Sp}}}_{HR}[O(n)],$

between $HR$-local $n$-homogeneous functors and $HR$-local spectra with an
action of $O(n)$.

####  8.5. Characterisations for nullifications

An analogous Quillen equivalence is obtained between the $W$-local
intermediate category and the $W$-periodic $n$-homogeneous model structure of
Proposition 8.3.1 which recall is obtained as a right Bousfield localization
of the $W$-local $n$-polynomial model structure.

######  Theorem 8.5.1.

For a finite cell complex $W$ the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
81.16054pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n):L_{W}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
105.16054pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.16054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}):{\operatorname{\mathsf{ind}}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.16055pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence.

###### Proof.

By [BO13, Lemma 9.2], the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
73.33556pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n):{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
97.33556pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
97.33556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.33557pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction, hence the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
73.33556pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n):{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
97.33556pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
97.33556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.33557pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is also a Quillen adjunction. This Quillen adjunction extends to give a
Quillen adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
81.16054pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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105.16054pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.16054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.16055pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

by [Hi03, Theorem 3.1.6(1) $\&$ Proposition 3.3.18(1)], since the $n$-th
derivative of a $W$-local $n$-polynomial functor is $W$-local by Lemma 7.2.2,
and an $n\Omega$-spectrum by [We95, Corollary 5.12]. An application of [Hi03,
Theorem 3.1.6(2) $\&$ Proposition 3.3.18(2)] extends the above Quillen
adjunction to a Quillen adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
81.16054pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n):L_{W}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
105.16054pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.16054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.16055pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

since a map $f$ is a weak equivalence between fibrant objects in the
$W$-periodic $n$-homogeneous model structure if and only if
$\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}(f)$is a levelwise weak
equivalence.

We now move on to showing that the adjunction is a Quillen equivalence. First
note that the right adjoint reflects weak equivalences, since if $f\colon X\to
Y$ is a map between fibrant orthogonal functors in the $W$-local
$n$-homogeneous model structure, such that

$\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}T_{n}P_{W}(f)\colon\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}T_{n}P_{W}(X)\longrightarrow\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}T_{n}P_{W}(Y)$

is a $W$-local equivalence in
$L_{W}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$,
then $\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}T_{n}P_{W}(f)$ is a
levelwise weak equivalence by the $W$-local Whitehead’s Theorem and [BO13,
Theorem 10.1]. It follows that $f$ is a weak equivalence in
$\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$.

To prove the Quillen equivalence, by [Ho99, Corollary 1.3.16], it is left to
show that the derived unit is a weak equivalence. Let $F$ be a cofibrant
object of the $W$-local $n$-stable model structure. As in [BO13, Theorem
10.1], there is a commutative diagram

${F}$${{\widehat{c}((\alpha_{n})^{*}\Psi)}}$${{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}T_{n}P_{W}\operatorname{\mathsf{res}}_{0}^{n}F/O(n)}}$${{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}T_{n}P_{W}\operatorname{\mathsf{res}}_{0}^{n}(\widehat{c}((\alpha_{n})^{*}\Psi))}}$$\scriptstyle{1}$$\scriptstyle{3}$$\scriptstyle{2}$$\scriptstyle{4}$

in which, $\Psi$ is a fibrant replacement of $(\alpha_{n})_{!}F$ in the Borel
stablisation of $W$-local spaces with an $O(n)$-action and $\widehat{c}$
denotes cofibrant replacement in the $W$-local $n$-stable model structure. By
_loc. cit._ the maps labelled “$1$” and “$2$” are $n$-stable equivalences,
hence $W$-local $n$-stable equivalences, thus it suffices to show that the map
labelled “$3$” is a $W$-local $n$-stable equivalence. This follows from the
proof of [BO13, Theorem 10.1], and the observation that a $T_{n}$-equivalence
is a $T_{n}P_{W}$-equivalence by the commutativity of the diagram

${{T_{n}A}}$${{T_{n}B}}$${{T_{n}P_{W}A}}$${{T_{n}P_{W}B.}}$$\scriptstyle{T_{n}(f)}$$\scriptstyle{T_{n}\omega_{A}}$$\scriptstyle{T_{n}\omega_{B}}$$\scriptstyle{T_{n}P_{W}(f)}$

∎

Proposition 8.1.1 and Proposition 8.3.1 provide two different model structures
which both capture the homotopy theory of $W$-locally $n$-homogeneous
functors. However, these model structures are not identical. For instance, the
$W$-local model structure of Proposition 8.1.1 has fibrant objects the
$n$-polynomial functors which have $W$-local $n$-th derivative, whereas the
fibrant objects of the $W$-periodic $n$-homogeneous model structure
(Proposition 8.3.1) are the $W$-local $n$-polynomial functors. These model
structures are Quillen equivalent via the identity functor. Recall that the
$W$-local $n$-homogeneous model structure, which is a left Bousfield
localization of the $n$-homogeneous model structure, see Proposition 8.1.1, is
denoted by
$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})$, and
the $W$-periodic $n$-homogeneous model structure is denoted by
$\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$.

######  Corollary 8.5.2.

For a finite cell complex $W$ the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
50.37518pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
74.37518pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
74.37518pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
50.3752pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence.

###### Proof.

Since cofibrations between cofibrant objects in
$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})$ are
projective cofibrations which are $T_{n}$-equivalences, and the cofibrations
between cofibrant objects of
$\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$ are
the projective cofibrations, it follows that the identity functor

$\mathds{1}\colon\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})\longrightarrow\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}}),$

necessarily preserves cofibrations between cofibrant objects. On the other
hand, the identity functor

$\mathds{1}\colon\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})\longrightarrow\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}}),$

preserves fibrant objects since if $X$ is levelwise $W$-local,
$\operatorname{\mathsf{ind}}_{0}^{n}X$ is levelwise $W$-local, by Lemma 7.2.2.
It follows that the adjunction is a Quillen adjunction. To see that the
adjunction is a Quillen equivalence, there is a commutative square

${{L_{W}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})}}$${{L_{W}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$

of Quillen adjunctions, in which three-out-of-four are Quillen equivalences by
Theorem 8.4.1 and Theorem 8.5.1. Hence the remaining Quillen adjunction must
also be a Quillen equivalence. ∎

It follows that there is a zigzag of Quillen equivalences

$\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})\simeq_{Q}{\sf{Sp}}(L_{W}\operatorname{\sf{Top}_{\ast}})[O(n)],$

whenever both model structures exist.

####  8.6. The classification

As in the classical theory, any $S$-locally $n$-homogeneous functor may be
expressed concretely in terms of a levelwise $S$-local spectrum with an action
of $O(n)$. The proof of which follows as in the classical setting, [We95,
Theorem 7.3] and can be realised through the derived equivalence of homotopy
categories provided by our zigzag of Quillen equivalences.

######  Theorem 8.6.1.

Let $S$ be a set of maps of based spaces and $n\geq 1$. An $S$-local
$n$-homogeneous functor $F$ is determined by and determines a levelwise
$S$-local spectrum with an $O(n)$-action, denoted $\partial_{n}^{S}F$. In
particular, an $S$-local $n$-homogeneous functor $F$ is levelwise weak
homotopy equivalent to the functor

$V\longmapsto\Omega^{\infty}[(S^{\mathbb{R}^{n}\otimes
V}\wedge\partial_{n}^{S}F)_{hO(n)}],$

and any functor of the above form is levelwise $S$-local and $n$-homogeneous.

####  8.7. The classification for homology theories

This let’s us characterise the $n$-th layer of the $E$-local Weiss tower for a
“nice” localizations since by Lemma 6.2.2 the $n$-th layer is $E$-locally
$n$-homogeneous.

######  Corollary 8.7.1.

Let $E$ be a spectrum such that the class of local spaces is closed under
sequential homotopy colimits. If $F$ is an orthogonal functor, then for $n\geq
1$, $D_{n}^{E}F$ is levelwise weakly equivalent to the functor given by

$V\longmapsto\Omega^{\infty}[(S^{\mathbb{R}^{n}\otimes
V}\wedge\partial_{n}^{E}F)_{hO(n)}].$

##  Part III Applications

Before moving on to the initial applications of our theory of local orthogonal
calculus we briefly recall the various constructions from Part II,
particularly the various model structures and how they relate to each other.

Let $S$ be a set of maps of based spaces and let
$L_{S}\operatorname{\sf{Top}_{\ast}}$ denote the $S$-local model structure on
based spaces. This model structure may be transferred (Proposition 4.2.1) to
the category of orthogonal functors to produce a model structure
${\sf{Fun}}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$ in which the
weak equivalences and fibrations are the levelwise $S$-local weak equivalences
and fibrations respectively. A left Bousfield localization (Proposition 5.2.3)
resulted in the $S$-local $n$-polynomial model structure $\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$, in which the fibrant
objects were the $S$-local $n$-polynomial functors. In nice cases such as
nullification at a finite cell complex, we demonstrated (Proposition 5.3.3)
that a fibrant replacement in this model structure is given by the composite
of the localization functor $L_{S}$ with the $n$-polynomial approximation
functor $T_{n}$.

To understand the layers of the $S$-local Weiss tower better we began with the
$n$-homogenous model structure of Barnes and Oman [BO13], which contains the
homogeneous of degree $n$ functors as the bifibrant objects and preformed a
left Bousfield localization (Proposition 8.1.1) to obtain the $S$-local
$n$-homogeneous model structure
$\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})$
which contains the $n$-homogeneous functors with $S$-local $n$-th derivative
as the bifibrant objects.

In the case of a nullification at a finite cell complex $W$, the $W$-local
$n$-homogeneous model structure can chararcterised (up to Quillen equivalence
via the identity functor, see Corollary 8.5.2) as the $W$-periodic
$n$-homogeneous model structure
$\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})$ in
which the bifibrant objects are the $W$-local $n$-homogeneous functors.

For a general set of maps of based spaces $S$ we showed (Theorem 8.4.1) that
taking the $n$-th derivative realises a Quillen equivalence between the
$S$-local $n$-homogenous model structure and the $S$-local $n$-th intermediate
category and that this latter category is canonically Quillen equivalent
(Theorem 7.3.1) to the category of spectra in $S$-local spaces with an
$O(n)$-action. This produces the following zigzag of Quillen equivalences

${{\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})}}$${{L_{S}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{{\sf{Sp}}(L_{S}\operatorname{\sf{Top}_{\ast}})[O(n)].}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$$\scriptstyle{(\alpha_{n})_{\ast}}$$\scriptstyle{(\alpha_{n})_{!}}$

For the case of a nullification with respect to a finite cell complex $W$, we
summarise our model structures in Figure 1. Note that the Quillen adjunctions
forming the lower “$\big{\ulcorner}$” shape are Quillen equivalences.

${{{\sf{Fun}}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{W}\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{W}\operatorname{\sf{Top}_{\ast}})}}$${{L_{W}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{{\sf{Sp}}(L_{W}\operatorname{\sf{Top}_{\ast}})[O(n)]}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{(\alpha_{n})_{!}}$$\scriptstyle{(\ref{prop:
n-poly for
nullification})}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$$\scriptstyle{(\alpha_{n})^{*}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{(\ref{prop:
S-local n-poly model structure})}$$\scriptstyle{(\ref{prop: E local n-homog
model})}$$\scriptstyle{(\ref{cor: QE for W-local
n-homog})}$$\scriptstyle{(\ref{thm: E-local diff as
Quillen})}$$\scriptstyle{(\ref{thm: E-local spectra and intermediate})}$
Figure 1. Diagram of Quillen adjunctions for $W$-periodic orthogonal calculus

In the case of a homological localization with respect to a homology theory
$E_{\ast}$, we summarise our model structures in Figure 2. In this case, the
lower two Quillen adjunctions are Quillen equivalences.

${{{\sf{Fun}}(\mathcal{J}_{0},L_{E}\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{E}\operatorname{\sf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0},L_{E}\operatorname{\sf{Top}_{\ast}})}}$${{L_{E}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{{\sf{Sp}}(L_{E}\operatorname{\sf{Top}_{\ast}})[O(n)]}}$${{{\sf{Sp}}_{HR}[O(n)]}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{(\alpha_{n})_{!}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$$\scriptstyle{(\alpha_{n})^{*}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{(\ref{prop:
S-local n-poly model structure})}$$\scriptstyle{(\ref{prop: E local n-homog
model})}$$\scriptstyle{(\ref{thm: E-local spectra and
intermediate})}$$\scriptstyle{(\ref{thm: E-local diff as
Quillen})}$$\scriptstyle{(\ref{lem: R-local and stablization of
R-local})}$$\scriptstyle{(E=HR,R\subseteq\mathbb{Q})}$ Figure 2. Diagram of
Quillen adjunctions for $E$-local orthogonal calculus

###  9\. Bousfield classes

####  9.1. Bousfield classes

The concept of Bousfield classes were introduced (in the stable setting) by
Bousfield in [BousfieldLocalSpectra], and extensively studied by Ravenel in
[RavenelLocalizations]. For a spectrum $E$, the _Bousfield class_ of $E$,
denoted $\langle E\rangle$, is the equivalence class of of all spectra
$E^{\prime}$ such that the class of $E^{\prime}$-acyclic spectra is the class
of $E$-acyclic spectra. If $\langle E\rangle=\langle E^{\prime}\rangle$, then
the classes of $E_{\ast}$-isomorphisms and $E_{\ast}^{\prime}$-isomorphisms
agree and hence the localization functors (on spaces or spectra) agree. The
collection of all Bousfield classes forms a lattice, with partial ordering
$\langle E\rangle\leq\langle E^{\prime}\rangle$ given by reverse containment,
i.e., if and only if the class of $E^{\prime}$-acyclic spectra is contained in
the class of $E$-acyclic spectra.

A similar story remains true unstably. Given a based space $W$, Bousfield,
[BousfieldPeriodicity, §9] and Farjoun [FarjounCellular], introduced the
_unstable Bousfield class_ of $W$, or the _nullity class_ of $W$. This is the
equivalence class $\langle W\rangle$ of all spaces $W^{\prime}$ such that the
class of $W$-local spaces agrees with the class of $W^{\prime}$-local spaces.
There is a partial ordering $\langle W\rangle\leq\langle W^{\prime}\rangle$
given by reverse containment, i.e., if and only if every $W^{\prime}$-periodic
space is $W$-periodic. In particular, the relation $\langle
W\rangle\leq\langle W^{\prime}\rangle$ implies that every $W$-local
equivalence is a $W^{\prime}$-local equivalence and there is a natural
transformation $P_{W}\to P_{W^{\prime}}$, which is a
$W^{\prime}$-localization.

######  Theorem 9.1.1.

Let $S$ and $S^{\prime}$ be sets of maps of based spaces. The class of
$S$-local spaces agrees with the class of $S^{\prime}$ local spaces if and
only if for every orthogonal functor $F$, the $S$-local Weiss tower of $F$ is
levelwise weakly equivalent to the $S^{\prime}$-local Weiss tower of $F$.

###### Proof.

If the class of $S$-local spaces agrees with the class of $S^{\prime}$-local
spaces, then the localization functors $L_{S}$ and $L_{S^{\prime}}$ agree on
the level of spaces and hence on the level of orthogonal functors. In
particular, for every orthogonal functor $F$, the canonical map555This map is
induced from the $S$-local objects being contained in the $S^{\prime}$-local
objects. We could also use the canonical $L_{S^{\prime}}F\to L_{S}F$ since the
$S$-local objects also contained the ${S^{\prime}}$-local objects. $L_{S}F\to
L_{S^{\prime}}F$ is a levelwise weak equivalence. Now, consider the
commutative diagram

${{D_{n}^{S}F}}$${{T_{n}^{S}F}}$${{T_{n-1}^{S}F}}$${{D_{n}^{S^{\prime}}F}}$${{T_{n}^{S^{\prime}}F}}$${{T_{n-1}^{S^{\prime}}F}}$

in which the rows are homotopy fibre sequences. For each $n\geq 0$, the map
$T_{n}^{S}F\to T_{n}^{S^{\prime}}F$ is a levelwise weak equivalence since
polynomial approximation preserves levelwise weak equivalences. It follows
that the left-most vertical arrow is also a levelwise weak equivalence and
that the $S$-local Weiss tower is levelwise weakly equivalent to the
$S^{\prime}$-local Weiss tower.

The converse is immediate from specialising for every based space $A$, to the
constant functor at $A$. ∎

As corollaries, we obtain a relationship between Bousfield classes and the
local Weiss towers.

######  Corollary 9.1.2.

Let $W$ and $W^{\prime}$ be based spaces. For every orthogonal functor $F$ the
$W$-local Weiss tower of $F$ and the $W^{\prime}$-local Weiss tower of $F$
agree if and only if $\langle W\rangle=\langle W^{\prime}\rangle$.

######  Corollary 9.1.3.

Let $E$ and $E^{\prime}$ be spectra. For every orthogonal functor $F$ the
$E$-local Weiss tower of $F$ and the $E^{\prime}$-local Weiss tower of $F$
agree if and only if $\langle E\rangle=\langle E^{\prime}\rangle$.

On the model category level, we have the following.

######  Theorem 9.1.4.

Let $S$ and $S^{\prime}$ be sets of maps of based spaces. If the class of
$S$-local spaces agrees with the class of $S^{\prime}$-local spaces, then the
following are equalities of model structures

${\sf{Fun}}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})={\sf{Fun}}(\mathcal{J}_{0},L_{S^{\prime}}\operatorname{\sf{Top}_{\ast}}),$
${\mathsf{Poly}^{\leq
n}}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})={\mathsf{Poly}^{\leq
n}}(\mathcal{J}_{0},L_{S^{\prime}}\operatorname{\sf{Top}_{\ast}}),$
${\mathsf{Homog}^{n}}(\mathcal{J}_{0},L_{S}\operatorname{\sf{Top}_{\ast}})={\mathsf{Homog}^{n}}(\mathcal{J}_{0},L_{S^{\prime}}\operatorname{\sf{Top}_{\ast}}).$

###### Proof.

First note that the $S$-local model structure on based spaces agrees with the
$S^{\prime}$-local model structure on based spaces since these model
structures have the same cofibrations and the same fibrant objects, since a
space is $S$-local if and only if it is $S^{\prime}$-local. This equality
lifts to the local projective model structures on the category of orthogonal
functors since a functor is levelwise $S$-local if and only if it is levelwise
$S^{\prime}$-local under our assumption.

As left Bousfield localization does not alter the cofibrations, the
cofibrations of the $S$-local $n$-polynomial model structure agree with the
cofibrations of the $S^{\prime}$-local $n$-polynomial model structure. These
model structures also have the same fibrant objects since a functor is
$S$-locally $n$-polynomial if and only if it is $S^{\prime}$-local
$n$-polynomial under our assumption.

For the local $n$-homogeneous model structures, recall that these are certain
left Bousfield localizations of the $n$-homogeneous model structure (see
Proposition 8.1.1), hence have the same cofibrations. As before, these model
structures have the same fibrant objects since our assumption together with
Lemma 7.2.2 implies that the $n$-th derivative of a functor is $S$-local if
and only if it is $S^{\prime}$-local, and the fibrant objects are the
$n$-polynomial functors with local derivatives, see Proposition 8.1.1. ∎

With respect to the partial ordering on Bousfield classes, we obtain the
following more general result. The proof of which relies on the fact that if
the class of $S^{\prime}$-local spaces is contained in the class of $S$-local
spaces, then the induced map $L_{S}\to L_{S^{\prime}}$ is an
$S^{\prime}$-local equivalence.

######  Lemma 9.1.5.

Let $S$ and $S^{\prime}$ be sets of maps of based spaces and $F$ an orthogonal
functor. If the class of $S^{\prime}$-local spaces is contained in the class
of $S$-local spaces then,

  1. (1)

there is an $S^{\prime}$-local equivalence $D_{n}^{S}F\to
D_{n}^{S^{\prime}}F$; and,

  2. (2)

if $F$ is reduced, then the $S$-local Weiss tower of $F$ is
$S^{\prime}$-locally equivalent to the $S^{\prime}$-local Weiss tower of $F$.

###### Proof.

For (1), note that the map on derivatives
$\partial_{n}^{S}F\to\partial_{n}^{S^{\prime}}F$ induced by the natural
transformation $L_{S}\to L_{S^{\prime}}$ is an $S^{\prime}$-local equivalence,
hence the $n$-homogeneous functors which correspond to these spectra are