# Comparing orthogonal calculus and calculus with reality

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

We show that there exists a suitable $C_{2}$-fixed points functor from
calculus with Reality to the orthogonal calculus of Weiss which recovers
orthogonal calculus “up to a shift” in an analogous way with the recovery of
real topological $K$-theory from Atiyah’s $K$-theory with Reality via
appropriate $C_{2}$-fixed points.

###### Contents

  1. 1 Introduction
  2. 2 A cofree model for calculus with Reality
  3. 3 Polynomial functors
  4. 4 The Weiss tower
  5. 5 Higher categorical analysis

##  1\. Introduction

### Background

Orthogonal calculus [Wei95] is a homotopy theoretic tool developed to study
geometric topology. Weiss’ key observation, stemming from work of Weiss and
Williams [WW88], is that many objects of interest to geometric topologists may
be packaged as functors from the category of Euclidean spaces and linear
isometries to the category of (based) spaces. Key examples include the functor
which assigns to a Euclidean space $V$ the classifying space
$\mathsf{BDiff}(V)$ of the group of diffeomorphisms of $V$, or the functor
which assigns to a Euclidean space $V$ the space of (smooth) embeddings
$\mathsf{Emb}(M\times V,N\times V)$ for $M$ and $N$ fixed smooth manifolds,
possibly with boundary conditions.

To a functor $F$ from the category of Euclidean spaces to (based) spaces, the
calculus assigns a tower of functors

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

under $F$, which we refer to as _the Weiss tower_. The $n$-th layer of the
Weiss tower is the homotopy fibre $D_{n}F$ of the map $T_{n}F\to T_{n-1}F$
induced by the inclusion $\mathbb{R}^{n}\hookrightarrow\mathbb{R}^{n+1}$.
Weiss [Wei95] classified the homotopy type of the $n$-th layers of this tower
in terms of spectra with an action of the $n$-th orthogonal group $O(n)$, and
the orthogonal calculus program aims to understand the homotopy type of the
input functor $F$ in terms of the homotopy type of the spectra classifying the
layers. Informally, one may think of this as a calculus in which total
Stiefel-Whitney classes appear as first-order approximations and total
Pontryagin classes as second-order approximations.

Orthogonal calculus fits into a larger framework of functor calculus, a
categorification of Taylor’s theorem from differential topology, pioneered by
Goodwillie [Goo90, Goo92, Goo03]. The aim of any version of functor calculus
is to split a functor into “polynomial” parts, and reconstruct the homotopy
type of the functor from these polynomial pieces. Depending on the natural
structure of the functor you wish to study, one may want a calculus which
captures this structure. For example a version of orthgonal calculus based on
complex inner product spaces was of fundamental importance to Arone’s study of
Mitchell’s finite spectra with $\mathcal{A}_{k}$-free cohomology [Aro98] which
arise naturally as stablisations of Weiss’ _cross-effects_ or _derivatives_ ,
and to Hahn and Yuan’s [HY19] study of multiplicative structures in the stable
splittings of $\Omega\mathsf{SL}_{n}(\mathbb{C})$.

Since the original version of functor calculus there has been a small industry
focused on the development of other versions of calculus that capture
different structures, and in investigating the relationships between these
various versions of functor calculus. Examples include the discrete calculus
of Bauer, Johnson and McCarthy [BJM15] together with its comparison to
Goodwillie calculus, and the manifold calculus of Goodwillie and Weiss [Wei96,
Wei99, GW99] and its the relationship with orthogonal calculus as exploited by
Arone, Lambrechts and Volić [ALV07] in their study of the stable rational
homotopy type of spaces of embeddings.

Motivated by the analogy between orthogonal calculus and real topological
$K$-theory, the author developed a series of other calculi: unitary calculus
[Tag19], which sits in analogy with complex topological $K$-theory, and
calculus with Reality [Tag22], which is the calculus version of Atiyah’s
$K$-theory with Reality. These calculi are related in precisely the same
manner as their analogous $K$-theories. The complexification-realification
adjunction on the level of vector spaces induces a relationship between
orthogonal and unitary calculus [Tag21] which is well-behaved if the functors
have convergent Weiss towers. This is the calculus version of the existence of
maps (of spectra)

$\displaystyle r:KU\to KO,$ $\displaystyle c:KO\to KU,$

relating real and complex topological $K$-theory. Moreover, we have a calculus
version of the statement that the underlying non-equviariant spectrum of
Aityah’s $K$-theory with Reality spectrum $KR$ is the complex topological
$K$-theory spectrum $KU$; forgetting the “complex conjugation” action on
calculus with Reality recovers unitary calculus [Tag20].

### Statement of main results

Figure 1 depicts the known relations between the various calculi.

---
$\textstyle{\begin{array}[]{c}\text{Calculus with}\\\
\text{Reality}\end{array}}$[Tag20]$\scriptstyle{\begin{array}[]{c}\text{Forget}\\\
C_{2}-\text{action}\\\
\end{array}}$$\scriptstyle{\begin{array}[]{c}C_{2}-\text{fixed}\\\
\text{points}\\\ \end{array}}$$\textstyle{\begin{array}[]{c}\text{Orthogonal
calculus}\\\
\end{array}}$[Tag21]$\scriptstyle{\begin{array}[]{c}\text{Complexification/}\\\
\text{Realification}\\\
\end{array}}$$\textstyle{\begin{array}[]{c}\text{Unitary calculus}\\\
\end{array}}$

Figure 1. Relationship between various versions of Weiss calculus

In this work we complete these comparisons by completing the dotted arrow in
the diagram, i.e., by demonstrating the calculus version of the fact that the
spectrum $KO$ of real topological $K$-theory is the (homotopy) $C_{2}$-fixed
points of $KR$. We construct an appropriate fixed points functor from calculus
with Reality to orthogonal calculus and show that this functor interacts well
with the Weiss towers.

The major technical difficulty to overcome is the non-existence of a good
fixed points functor. Calculus with Reality is constructed with respect to the
underlying non-equivariant equivalences of $C_{2}$-spaces, which are preserved
by homotopy $C_{2}$-fixed points, but _not_ $C_{2}$-fixed points, in general.
On the other hand, the polynomial approximations are constructed as a
(sequential) homotopy colimit of a (compact) homotopy limit. These are
preserved by $C_{2}$-fixed points but _not_ by homotopy $C_{2}$-fixed points,
in general. In some sense the problem is that our model for $C_{2}$-spaces is
based on _free_ $C_{2}$-spaces, which interact poorly with the various
versions of fixed points. We choose to model $C_{2}$-spaces by _cofree_
$C_{2}$-spaces, on which homotopy $C_{2}$-fixed points and $C_{2}$-fixed
points agree, and hence we can exploit all of the nice properties of both
functors. We then apply this to the equivalence between free and cofree
$C_{2}$-spaces, see Proposition 2.1.3, to produce a cofree model for calculus
with Reality.

######  Remark 1.0.1.

Although we do not state it, our methods for showing that calculus with
Reality based on cofree $C_{2}$-spaces agrees with calculus with Reality based
on free $C_{2}$-spaces extend without much work to show that if $\mathcal{C}$
is a topological model category on which there exists a suitable notion of
Weiss calculus, and $\mathcal{D}$ is a Quillen equivalent topological model
category with left Quillen functor $L:\mathcal{C}\to\mathcal{D}$ which
preserves all weak equivalences then there exists a suitable notion of Weiss
calculus on $\mathcal{D}$ which is equivalent to the Weiss calculus on
$\mathcal{C}$. This is simmilar to the analysis carried out by Walter [Wal06]
in Goodwillie calculus.

We exhibit an equivalence of homotopy theories between the polynomial functors
in the free model for calculus with Reality constructed in [Tag22] and the
homotopy theory of these functors based on cofree $C_{2}$-spaces.

######  Theorem (Proposition 2.3.6).

Let $n$ be a non-negative integer. The identity functor induces an equivalence
of $\infty$-categories between the $\infty$-category of $n$-polynomial
functors based on free $C_{2}$-spaces and the $\infty$-category of
$n$-polynomial functors based on cofree $C_{2}$-spaces.

We preform a similar analysis for homogeneous functors in the free model for
calculus with Reality constructed in [Tag22] and the homotopy theory of these
functors based on cofree $C_{2}$-spaces.

######  Theorem (Proposition 2.5.3).

Let $n$ be a non-negative integer. The identity functor induces an equivalence
of $\infty$-categories between the $\infty$-category of $n$-homogeneous
functors based on free $C_{2}$-spaces and the $\infty$-category of
$n$-homogeneous functors based on cofree $C_{2}$-spaces.

We further show that calculus with Reality may be modelled on cofree
$C_{2}$-spaces by demonstrating that the equivalence of $\infty$-categories
between the $\infty$-category of $n$-homogeneous functors and the
$\infty$-category of spectra with an action of $C_{2}\ltimes U(n)$ coming from
the classication of $n$-homogeneous functors naturally lifts to a suitable
euqivalence between the cofree models. The following result is a combination
of the results in $\S$2.6.

######  Theorem.

There is a commutative diagram

${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})}}$${{\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n)\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n)\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{free}}}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{cofree}}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]^{\mathsf{free}}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]^{\mathsf{cofree}}}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{\psi}$$\scriptstyle{L_{\psi}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{L_{\psi}}$$\scriptstyle{\psi}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{ind_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$

of adjoint pairs in which every adjoint pair induces an equivalence of
$\infty$-categories.

Using the cofree model and exploiting the equivalence between homotopy fixed
points and fixed points, we can directly compare the polynomial
approximations. To distinguish between constructions made in orthogonal
calculus and constructions made in calculus with Reality we invoke a
superscript $\mathbf{O}$ and $\mathbf{R}$ to denote orthogonal calculus and
calculus with Reality constructions, respectively.

######  Theorem (Theorem 3.3.2).

Let $n$ be a non-negative integer. If $F$ is an fibrant functor in the cofree
model for calculus with Reality, then there is a levelwise weak equivalence of
orthogonal functors

$c^{\ast}(T_{n}^{\mathbf{R}}F)^{C_{2}}\longrightarrow
T_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}}).$

Our comparison is complete by examining the layers of the respective Weiss
towers.

######  Theorem (Corollary 4.1.3).

Let $n$ be a non-negative integer. If $F$ is an fibrant functor in the cofree
model for calculus with Reality, then there is a levelwise weak equivalence of
orthogonal functors

$c^{\ast}(D_{n}^{\mathbf{R}}F)^{C_{2}}\longrightarrow
D_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}}).$

To complete our study of the relationship between calculus with Reality and
orthogonal calculus we provide a complete relationship between the higher
categorical classifications for both calculi. This results in a diagram of
$\infty$-categories (induced by a diagram of model categories), see Figure 2,
which commutes up to natural equivalence, see Section 5. In order to construct
such a commuative diagram we construct “multiplicity $n$” versions of
Schwede’s comparison functors between orthognoal spectra and real spectra, see
e.g., [Sch19, Example 7.11].

### Conventions

We abuse language and refer to based compactly generated weak Hausdorff spaces
as spaces, and similarly for based spaces with an action of a compact Lie
group $G$, which we will refer to as $G$-spaces.

We model $\infty$-categories by quasi-categories as introduced by Boardman and
Vogt [BV] and developed extensively by Joyal [Joy02] and Lurie [Lur09]. Given
a topological model category $\mathcal{M}$, the category
$\mathcal{M}^{\mathsf{fc}}$ of fibrant-cofibrant objects defines a topological
category, the nerve of which defines a quasi-category. We refer to the quasi-
category $N(\mathcal{M}^{\mathsf{fc}})$ as the _underlying_ $\infty$-category
of $\mathcal{M}$, and denote it by $\mathcal{M}_{\infty}$. Given a Quillen
adjunction (_resp._ Quillen equivalence) of model categories
$\textstyle{F:\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{N}:G}$,
there is a well-defined adjunction (_resp._ equivalence) of underlying
$\infty$-categories,
$\textstyle{\mathds{L}F:\mathcal{M}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{N}_{\infty}:\mathds{R}G}$.

### Acknowledgements

The author wishes to thank L. Pol for numerous helpful conversations about
this work. A portion of this work was completed while the author was in
residence at the Institut Mittag-Leffler in Djursholm, Sweden in 2022 as part
of the program ‘Higher algebraic structures in algebra, topology and geometry’
supported by the Swedish Research Council under grant no. 2016-06596.

##  2\. A cofree model for calculus with Reality

###  2.1. Models for equivariant spaces

In the construction of calculus with Reality [Tag22], we considered the
_underlying_ or _coarse_ model structure on based $C_{2}$-spaces in which a
map $f:X\to Y$ is weak equivalence or fibration if and only if $f$ is a weak
equivalence or fibration in the Quillen model structure on based spaces. The
cofibrant objects were the _free_ $C_{2}$-spaces, i.e., those spaces which are
(up to underlying weak equivalence) of the form $(EC_{2})_{+}\wedge X$ for $X$
a based space. This model structure is the transfer of the Quillen model
structure on based spaces along the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
37.0479pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(C_{2})_{+}\wedge(-):\operatorname{\mathsf{Top}_{\ast}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
61.0479pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
61.0479pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C_{2}\operatorname{\mathsf{Top}_{\ast}}:i^{\ast}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
37.04791pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

and we denote this model structure by
$C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}$. In this section we
present an alternative model for this homotopy theory which is presented by
_cofree_ $C_{2}$-spaces, i.e., we give a model structure on
$C_{2}\operatorname{\mathsf{Top}_{\ast}}$ in which the weak equivalences are
still underlying equivalences but the fibrant objects are $C_{2}$-spaces of
the form $\operatorname{\mathsf{Map}}_{\ast}((EC_{2})_{+},X)$ for $X$ a based
space. Much of what we say in this section is true for arbitrary groups $G$,
but we specialise to $G=C_{2}$ as it is the only case of interest for us.

We start by recalling the _fine_ (or “genuine”) model structure on
$C_{2}$-spaces of which our cofree model structure will be a localization.

######  Lemma 2.1.1 ([MM02, Theorem III.1.8]).

There is a model structure on the category of $C_{2}$-spaces such that a map
$f:X\to Y$ is a weak equivalence or fibration if and only if the induced map
on $H$-fixed points $f^{H}:X^{H}\to Y^{H}$ is a weak equivalence or fibration
of based spaces for all subgroups $H$ of $C_{2}$. We denote this model
structure by $C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{fine}}$. This
model structure is cellular, proper and topological.

We now state the _cofree_ model structure as a localization of the fine model
structure, see also [Law20, Example 7.9]. In particular homotopy $C_{2}$-fixed
points are precisely the right derived functor of $C_{2}$-fixed points in this
model structure.

######  Proposition 2.1.2.

There is a model structure on the category of $C_{2}$-spaces such that a map
$f:X\to Y$ is a weak equivalence or cofibration if it is an underlying weak
equivalence or cofibration in the fine model structure, respectively. The
fibrant objects are the cofree $C_{2}$-spaces. We denote this model structure
by $C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}$. This model
structure is cellular, proper and topological.

###### Proof.

This model structure is the left Bousfield localization of the fine model
structure at the map $(EC_{2})_{+}\to\ast$, which exists by [Hir03, Theorem
4.1.1]. A space $X$ is $(EC_{2})_{+}$-local if and only if the induced map

$X^{C_{2}}\longrightarrow X^{hC_{2}},$

is an underlying weak equivalence, i.e., if and only if $X$ is a cofree
$C_{2}$-space. In particular, a localization functor is given by

$X\longmapsto\operatorname{\mathsf{Map}}_{\ast}((EC_{2})_{+},X).$

It follows that map $f:X\to Y$ is a weak equivalence if and only if

$\operatorname{\mathsf{Map}}_{\ast}((EC_{2})_{+},X)^{H}\longrightarrow\operatorname{\mathsf{Map}}_{\ast}((EC_{2})_{+},Y)^{H}$

is a weak equivalence of based spaces for all $H\leq C_{2}$. This last is
equivalent to $f:X\to Y$ and $f^{hC_{2}}:~{}X^{hC_{2}}\to Y^{hC_{2}}$ begin
underlying weak equivalences, which since homotopy fixed points preserve
underlying weak equivalences, is equivalent to $f:X\to Y$ being an underlying
weak equivalence. ∎

######  Proposition 2.1.3.

The identity functor

$\mathds{1}:C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}\longrightarrow
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}},$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

We prove that the identity functor is a left Quillen equivalence. The identity
functor

$\mathds{1}:C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}\to
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{fine}},$

is a left Quillen functor from the free model structure to the fine model
structure, and since the cofree model structure is a left Bousfield
localization of the fine model structure, we obtain a left Quillen functor

$\mathds{1}:C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}\longrightarrow
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}.$

The weak equivalences in the free and cofree model structure agree, hence it
suffices to show that either of the (co)units is a weak equivalence. This
follows readily from the fact that for any $C_{2}$-space $X$, the map

$(EC_{2})_{+}\wedge\operatorname{\mathsf{Map}}_{\ast}((EC_{2})_{+},X)\longrightarrow
X,$

is an underlying weak equivalence. ∎

###  2.2. A cofree model for input functors

Let $\mathds{k}$ denote either $\mathbb{R}$ or $\mathbb{C}$ , and let
$\mathcal{U}$ denote the infinite-dimensional $\mathds{k}$ inner product
spaces $\mathbb{R}^{\infty}$, $\mathbb{C}^{\infty}$ or
$\mathbb{C}\otimes\mathbb{R}^{\infty}$. The infinite-dimensional inner product
space $\mathcal{U}$ is the indexing universe for the various versions of
orthogonal calculus, e.g., $\mathcal{U}=\mathbb{R}^{\infty}$ indexes
orthogonal calculus and $\mathcal{U}=\mathbb{C}\otimes\mathbb{R}^{\infty}$
indexes calculus with Reality. Define $\mathcal{J}$ to be the category of
finite-dimensional inner product subspaces of $\mathcal{U}$ with morphism the
linear isometries. The category $\mathcal{J}$ is topologically enriched as
$\mathcal{J}(V,W)$ is the Stiefel manifold of $\dim(V)$-frames in $W$. By
adding a disjoint basepoint to the morphism space produces a
$\operatorname{\mathsf{Top}_{\ast}}$-enriched category, which we continue to
denote by $\mathcal{J}$. In the case of calculus with Reality this category is
$C_{2}\operatorname{\mathsf{Top}_{\ast}}$-enriched with $C_{2}$ acting by
conjugation by complex conjugation, see [Tag22, §1.2].

The input category for calculus with Reality is the category of
$C_{2}$-equivariant continuous functors
$\mathsf{Fun}(\mathcal{J}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})$
with $C_{2}$ acting by conjugation, while the input category for orthogonal
calculus is the category of continuous functors
$\mathsf{Fun}(\mathcal{J}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})$.
When discussing both calculi at once the phrase “continuous functors” should
be interpretted in the appropriate sense. For calculus with Reality, the
adjunction between free and cofree $C_{2}$-spaces induces an equivalence of
$\infty$-categories between the $\infty$-categories of functors.

######  Proposition 2.2.1.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
45.67796pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Fun}(\mathcal{J}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.67796pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
69.67796pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}(\mathcal{J}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
45.67798pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

We prove that the adjunction is a Quillen adjunction between projective model
structures. Since (acyclic) fibrations are defined levelwise in
$\mathsf{Fun}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})$,
and the identity functor

$\mathds{1}:C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}\longrightarrow
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}},$

is right Quillen, it follows that the induced adjunction on projective model
structures is a Quillen adjunction.

To show that the adjunction is a Quillen equivalence it suffices to show that
the derived (co)units are equivalences. Unravelling the definition of the
(co)unit, one observes that the Quillen equivalence between
$C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}$ and
$C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}$ forces the
(co)units to be equivalences on the level of functor categories. ∎

###  2.3. A cofree model for polynomial functors

For the remainder of this section we let $\mathcal{C}$ denote either
$\operatorname{\mathsf{Top}_{\ast}}$ or
$C_{2}\operatorname{\mathsf{Top}_{\ast}}$.

######  Definition 2.3.1.

Let $n$ be a non-negative integer. A functor $F:\mathcal{J}\to\mathcal{C}$ is
said to be _polynomial of degree less than or equal $n$_ or equivalently _$n$
-polynomial_ if for all $V\in\mathcal{J}$, the canonical map

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

is a weak equivalence in $\mathcal{C}$.

######  Remark 2.3.2.

The poset of subspaces of $\mathds{k}^{n+1}$ is a topological poset in that it
has a _space_ of object and _space_ of morphisms, with continuous sorce,
target, composition and insertion of identity maps. The space of objects is a
disjoint union of Grassmannian manifolds, while the space of morphisms is a
disjoint union of flag manifolds.

There is a model strcuture which captures the homotopy theory of
$n$-polynomial functors, and hence in which fibrant replacement is
$n$-polynomial approximation.

######  Lemma 2.3.3 ([BO13, Proposition 6.5 $\&$ Proposition 6.6],[Tag22,
Proposition 2.15]).

Let $n$ be a non-negative integer. There is a cellular, proper and topological
model category structure on the category of continuous functors
$\mathsf{Fun}(\mathcal{J},\mathcal{C})$ with cofibrations the projective
cofibrations and fibrant objects are the levelwise fibrant $n$-polynomial
functors. We denote this model structure by $\mathsf{Poly}^{\leq
n}(\mathcal{J},\mathcal{C})$.

######  Remark 2.3.4.

The $n$-polynomial model structure is the left Bousfield localization of the
projective model structure at the set

$\left\\{\ \underset{0\neq
U\subset\mathds{k}^{n+1}}{\operatorname{\mathrm{hocolim}}}~{}\mathcal{J}(V\oplus
U,-)\longrightarrow\mathcal{J}(V,-)\ \mid\ V\in\mathcal{J}\ \right\\}.$

By the relevant version of [Wei95, Proposition 4.2] to the known version of
Weiss calculus under consideration, there is an identification

$S\gamma_{n+1}(V,W)_{+}\cong\underset{0\neq
U\subset\mathds{k}^{n}}{\operatorname{\mathrm{hocolim}}}~{}\mathcal{J}_{0}(V\oplus
U,W),$

of the sphere bundle of $\gamma_{n+1}(V,W)$ with a topological homotopy
colimit of Stiefel manifolds, hence the set of maps is equivalently described
via the projection maps of the bundle $S\gamma_{n+1}(V,-)$.

######  Remark 2.3.5.

The underlying $\infty$-category of the $n$-polynomial model structure is the
$\infty$-category of $n$-polynomial functors. This should be equivalently
defined as the full $\infty$-subcategory of the $\infty$-category of functors
$\mathsf{Fun}(\mathcal{J},\mathcal{C})$ spanned by the $n$-polynomial
functors, if one suitably interprets Definition 2.3.1 in an appropriate
$\infty$-categorical sense.

For calculus with Reality, the adjunction between free and cofree
$C_{2}$-spaces induces an equivalence of $\infty$-categories between the
underlying $\infty$-categories of the $n$-polynomial model structures.

######  Proposition 2.3.6.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
48.8642pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Poly}^{\leq
n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
72.8642pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.8642pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
48.8642pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

We prove that the adjunction is a Quillen adjunction between $n$-polynomial
model structures. Apply [Hir03, Theorem 3.3.20], noticing that the localizing
set of the cofree $n$-polynomial model structure is the derived image of the
localizing set of the free $n$-polynomial model structure, noting that the
representable functors are necessarily cofibrant in any projective model
structure. ∎

We now define the $n$-polynomial approximation functor. The idea being that
iterations $(\tau_{n})^{k}$ of $\tau_{n}$ become close and closer to being
$n$-polynomial as $k$ increases.

######  Definition 2.3.7.

Let $n$ be a non-negative integer. For any continuous functor
$F:\mathcal{J}\to\mathcal{C}$, define the _$n$ -polynomial approximation_
$T_{n}F$ of $F$ to be the sequential homotopy colimit

$\operatorname{\mathrm{hocolim}}~{}(F\longrightarrow\tau_{n}(F)\longrightarrow(\tau_{n})^{2}(F)\longrightarrow\cdots).$

######  Remark 2.3.8.

In all known versions of Weiss calculus the above homotopy colimit produces an
$n$-polynomial functor, see, e.g., [Wei98]. This is a sophisticated result
which is the heart of much of the work in constructing new variants of Weiss
calculus. For example, the only known proof that $T_{n}F$ is $n$-polynomial
for orthogonal calculus requires: a calculation of the connectivity of the map
$S\gamma_{n+1}(V,W)\to\mathcal{J}(V,W)$; showing that the functor $\tau_{n}$
increases connectivity using an elaborate calculation of the connectivity of
topological homotopy limits; and, that

$T_{n}(S\gamma_{n+1}(V,W))\to T_{n}(\mathcal{J}(V,W)),$

is a weak equivalence, where $S\gamma_{n+1}(V,W)$ is the sphere bundle of the
$(n+1)$-fold orthogonal complement bundle $\gamma_{n+1}(V,W)$ over
$\mathcal{J}(V,W)$ with fibre over $f$ given by $\mathds{k}^{n+1}\otimes
f(V)^{\perp}$. Weiss uses this fact to show that for every continuous functor
$F:\mathcal{J}\to\operatorname{\mathsf{Top}_{\ast}}$ and every non-negative
integer $i$, the square

${{(\tau_{n})^{i}F}}$${{T_{n}F}}$${{(\tau_{n})^{i+1}F}}$${{\tau_{n}T_{n}F}}$

can always be factored through a weak equivalence, and hence the resulting
right-hand side is a weak equivalence. In all known versions of Weiss calculus
$T_{n}F$ is always $n$-polynomial, see e.g., [Wei98], [Tag19, Lemma 3.8] and
[Tag22, Lemma 2.13].

######  Remark 2.3.9.

In all known versions of Weiss calculus the $n$-polynomial model structure may
be equivalently described as the Bousfield-Friedlander localization of the
projective model structure at the endofunctor

$T_{n}:\mathsf{Fun}(\mathcal{J},\mathcal{C})\longrightarrow\mathsf{Fun}(\mathcal{J},\mathcal{C}),$

see e.g. [BO13, Proposition 6.6], [Tag19, Proposition 3.9], and [Tag22,
Proposition 2.15].

###  2.4. A cofree model for the intermediate categories

In Weiss calculus the layers of the Weiss towe are classified by spectra with
an appropriate group action. This classification is through a zigzag of
Quillen equivalences, and passes through what is referred to as the $n$-th
intermediate category. Heuristically the $n$-th intermediate category is a
model for coordinate free spectra of multiplicity $n$, and is a natural home
for the $n$-th derivative of a functor.

Sitting over the space of linear isometries $\mathcal{J}(V,W)$ is the $n$-th
orthogonal complement vector bundle $\gamma_{n}(V,W)$, with fibre over a
linear isometry $f$ given by $\mathds{k}^{n}\otimes f(V)^{\perp}$. Define the
_$n$ -th jet category_ $\mathcal{J}_{n}$ to be the category with the same
objects as $\mathcal{J}$ and morphism space given by the Thom space of the
$n$-th orthogonal complement vector bundle. For the remainder of this section
let $G_{n}$ denote the group such that spectra with a $G_{n}$-action
classifies $n$-homogeneous functors in Weiss calculus. For example, for
orthogonal calculus $G_{n}=O(n)$, see [Wei95], and for calculus with Reality,
$G_{n}=C_{2}\ltimes U(n)$, see [Tag22]. We denote by $\mathcal{C}[G_{n}]$ the
category of $G_{n}$-objects in $\mathcal{C}$. For
$\mathcal{C}=\operatorname{\mathsf{Top}_{\ast}}$ and $G_{n}=O(n)$ this is the
category of $O(n)$-spaces, and for
$\mathcal{C}=C_{2}\operatorname{\mathsf{Top}_{\ast}}$ and $G_{n}=C_{2}\ltimes
U(n)$ this is the category of $C_{2}\ltimes U(n)$-spaces.

######  Definition 2.4.1.

Let $n$ be a non-negative integer. Define the _$n$ -th intermediate category_
to be the category of $G_{n}$-equivariant continuous functors from
$\mathcal{J}_{n}$ to $\mathcal{C}[G_{n}]$. We denote this category by
$\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}])$.

The $n$-th intermediate category may be equipped with stable model structure,
called the $n$-stable model structure, which is a left Bousfield localization
of the projective model structure. This is reminiscent of the stable model
structure on spectra since the $n$-th intermediate category is equivalent to a
category of modules over the commutative monoid

$n\mathbb{S}:V\longmapsto S^{\mathds{k}^{n}\otimes V},$

in the category of $G_{n}$-equivariant $\mathcal{I}$-spaces, where
$\mathcal{I}$ is the category with the same objects as $\mathcal{J}$ but only
linear isometric isomorphisms, see e.g., [BO13, Proposition 7.4] or [Tag22,
Proposition 4.13].

######  Lemma 2.4.2 ([BO13, Proposition 7.14],[Tag22, Theorem 4.18]).

Let $n$ be a non-negative integer. There is a model category structure on the
$n$-th intermediate category such that the cofibrations are the projective
cofibrations and the fibrant objects are the $n\Omega$-spectra, i.e., those
levelwise fibrant functors $X$ for which the adjoint structure maps

$X(V)\to\Omega^{\mathds{k}^{n}\otimes W}X(V\oplus W),$

are weak equivalences in $\mathcal{C}$ for all $V,W\in\mathcal{J}_{n}$. We
call this the $n$-stable model structure and denote it by
$\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}])$.

######  Remark 2.4.3.

The weak equivalences of the $n$-stable model structure are the “stable
equivalences”, i.e., the local equivalences. In all known versions of Weiss
calculus, the stable equivalences are precisely the
$n\pi_{\ast}$-isomorphisms, where the $n$-homotopy groups of a functor
$X:\mathcal{J}_{n}^{\mathcal{C}}\to\mathcal{C}$ are given by

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

see e.g., [BO13, Proposition 7.14], [Tag19, Proposition 5.6], and [Tag22,
Proposition 5.18].

For calculus with Reality, the adjunction between free and cofree
$C_{2}$-spaces induces an equivalence of $\infty$-categories between the
underlying $\infty$-categories of the $n$-stable model structures.

######  Proposition 2.4.4.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
68.19736pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
92.19736pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
92.19736pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\times
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.19737pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

An analogous argument to that of Proposition 2.2.1 yields a Quillen
equivalence between projective model structures on the $n$-th intermediate
categories. To lift to a Quillen equivalence between the $n$-stable model
structures, it suffices by [Hir03, Theorem 3.3.20] to show that the derived
image of the localizing set of the free $n$-stable model structure under the
left adjoint is precisely the localising set for the cofree $n$-stable model
structure. This follows as the localising set in both cases is

$\\{\mathcal{J}_{n}(V\oplus W,-)\wedge S^{\mathds{k}^{n}\otimes
W}\longrightarrow\mathcal{J}_{n}(V,-)\mid V,W\in\mathcal{J}_{n}\\},$

and the source and target are cofibrant. ∎

Let ${\mathsf{Sp}}^{\mathcal{U}}[G_{n}]$ denote the category of
$\mathcal{U}$-spectra with a $G_{n}$-action. For orthogonal calculus this is
precisely the category ${\mathsf{Sp}}^{\mathbf{O}}[O(n)]$ of orthogonal
spectra with a $O(n)$-action and for calculus with Reality this is the
category ${\mathsf{Sp}}^{\mathbf{R}}[U(n)]$ of _real spectra_ with a
$U(n)$-action, see e.g., [Tag22, §5]. For calculus with Reality, the
adjunction between free and cofree $C_{2}$-spaces induces an equivalence of
$\infty$-categories between the underlying $\infty$-categories of the
categories of _real spectra_ with a $U(n)$-action.

######  Proposition 2.4.5.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.57358pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{free}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.57358pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
56.57358pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{cofree}}:\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.57358pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

By [Tag22, Proposition 5.4], we may identify the underlying $(1-)$category of
the stable model structures as the category of $U(n)$-objects in the category
of $C_{2}$-equivariant continuous functors
$\mathsf{Fun}(\mathcal{J}_{1},C_{2}\operatorname{\mathsf{Top}_{\ast}})$. The
statement and proof are then the $n=1$ case of Proposition 2.4.4. ∎

There is a well-defined continuous functor

$\alpha_{n}:\mathcal{J}_{n}\longrightarrow\mathcal{J}_{1},V\longmapsto\mathds{k}^{n}\otimes
V,$

and given an object $X\in{\mathsf{Sp}}^{\mathcal{U}}[G_{n}]$, we let $g\in
G_{n}$ act on $(\alpha_{n})^{\ast}X$ by

$X(g\otimes V)\circ g_{X(\mathds{k}^{n}\otimes
V)}:(\alpha_{n})^{\ast}X(V)\longrightarrow(\alpha_{n})^{\ast}X(V),$

where $X(g\otimes V)$ is the image under the functor $X$ of the _internal_
action of $g$ on $\mathds{k}^{n}\otimes V$, and $g_{X(\mathds{k}^{n}\otimes
V)}$ is the _external_ action of $g$ on the $G_{n}$-object
$X(\mathds{k}^{n}\otimes V)$. This action defines an equivariant continuous
functor

$(\alpha_{n})^{\ast}:{\mathsf{Sp}}^{\mathcal{U}}[G_{n}]\longrightarrow\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}]),$

with left adjoint

$(\alpha_{n})_{!}:\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}])\to{\mathsf{Sp}}^{\mathcal{U}}[G_{n}],$

given by (enriched) left Kan extension along
$\alpha_{n}:\mathcal{J}_{n}\to\mathcal{J}_{1}$.

######  Proposition 2.4.6 ([BO13, Proposition 8.3],[Tag22, Theorem 5.2]).

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
47.0281pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}:\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.0281pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.0281pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathsf{Sp}}^{\mathcal{U}}[G_{n}]:(\alpha_{n})^{\ast}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.02812pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###  2.5. A cofree model for homogeneous functors

In the known versions of Weiss calculus the layers of the tower, i.e., the
homotopy fibres

$D_{n}F=\operatorname{\mathrm{hofib}}(T_{n}F\longrightarrow T_{n-1}F),$

have two useful properties: firstly they are polynomial of degree less than or
equal $n$, and secondly, their $(n-1)$-polynomial approximation is trivial.

######  Definition 2.5.1.

Let $n$ be a non-negative integer. A continuous functor
$F:\mathcal{J}\to\mathcal{C}$ is _homogeneous of degree $n$_ or equivalently
_$n$ -homogeneous_ if it is $n$-polynomial and $T_{n-1}F$ is levelwise
equivalent to the terminal object of $\mathcal{C}$. This last property is
referred to as being _$n$ -reduced_.

There is a model structure which captures the homotopy theory of
$n$-homogeneous functors.

######  Lemma 2.5.2 ([BO13, Proposition 6.9],[Tag22, Proposition 3.2]).

Let $n$ be a non-negative integer. There is a model category structure on the
category of continuous functors $\mathsf{Fun}(\mathcal{J},\mathcal{C})$ such
that the fibrations are the fibrations of the $n$-polynomial model structure,
and the weak equivalences are the $D_{n}$-equivalences. The fibrant objects
are the $n$-polynomial functors, and the cofibrant objects are the
projectively cofibrant $n$-reduced functors. We call this the $n$-homogeneous
model structure and denote it by
$\mathsf{Homog}^{n}(\mathcal{J},\mathcal{C})$.

For calculus with Reality, the adjunction between free and cofree
$C_{2}$-spaces induces an equivalence of $\infty$-categories between the
underlying $\infty$-categories of the $n$-homogeneous model structure.

######  Proposition 2.5.3.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
52.31142pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Homog}^{n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.31142pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.31142pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
52.31143pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

By [Hir03, Theorem 3.3.20(2)], it suffices to show that the class of colocal
equivalences for the cofree model agrees with the class of colocal
equivalences for the free model. Let $X\to Y$ be a $D_{n}$-equivalence in the
free model for caluclus with Reality, i.e., $D_{n}X\to D_{n}Y$, is a levelwise
weak equivalence in $C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}$.
Since the identity functor

$\mathds{1}:C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}}\to
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}},$

preserves all weak equivalences, it follows that $D_{n}X\to D_{n}Y$ is a
levelwise weak equivalence in
$C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}$.

For the converse, let $X\to Y$ be a $D_{n}$-equivalence in the cofree model
for calculus with Reality. It suffices to show that given a functor $F$, the
projective fibrant replacement functor in
$\mathsf{Fun}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})$
is a fibrant replacment in
$\mathsf{Fun}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})$
since then we will have a commutative diagram

${{D_{n}X}}$${{D_{n}Y}}$${{\mathsf{R}_{2}D_{n}X}}$${{\mathsf{R}_{2}D_{n}Y}}$

in which two-out-of-three maps are weak equivalences in free projective model
structure, hence the top horiztonal map must also be a weak equivalence in the
free projective model structure. Let $F:\mathcal{J}\to\mathcal{C}$ be any
functor, and denote by $F\to\mathsf{R}_{2}(F)$ the fibrant replacement of $F$
in the cofree projective model structure. We can cofibrantly replace the map
$F\to\mathsf{R}_{2}(F)$ in the free model structure, obtaining a diagram

${{\mathsf{Q}_{1}(F)}}$${X}$${{\mathsf{Q}_{1}\mathsf{R}_{2}(F)}}$${{\mathsf{R}_{2}(X)}}$

in which the lower horizontal map is the derived counit of the Quillen
equivalence between the free and cofree projective model structures, and
$\mathsf{Q}_{1}$ denotes cofibrant replacement in the free model structure. It
follows that three-out-of-four of the maps in the above square are weak
equivalences in free projective model structure, hence so to is the right-hand
vertical map, and the result follows. ∎

###  2.6. A cofree model for differentiation

For $m\leq n$, the inclusion $\mathds{k}^{m}\to\mathds{k}^{n}$ onto the first
$m$-coordinates induces an inclusion of categories,
$i_{m}^{n}\colon\mathcal{J}_{m}\to\mathcal{J}_{n}$. Define the restriction
functor

$\operatorname{\mathsf{res}}_{0}^{n}:\mathsf{Fun}(\mathcal{J}_{n},\mathcal{C})\longrightarrow\mathsf{Fun}(\mathcal{J}_{m},\mathcal{C}),$

to be precomposition with $i_{m}^{n}$, and define the induction functor

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

to be the enriched right Kan extension along $i_{m}^{n}$. In the case $m=0$,
the induction functor $\operatorname{\mathsf{ind}}_{0}^{n}$ is called the
$n$-th derivative. Combining this adjunction with the change of group functors
from [MM02, V.2] provides an adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
65.60461pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/\mathrm{Aut}(\mathds{k}^{n}):\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.60461pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
89.60461pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}(\mathcal{J},\mathcal{C}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.60461pt\raise-2.15277pt\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>
######  Theorem 2.6.1 ([BO13, Theorem 10.1], [Tag22, Theorem 6.8]).

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
65.60461pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/\mathrm{Aut}(\mathds{k}^{n}):\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},\mathcal{C}[G_{n}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.60461pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
89.60461pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J},\mathcal{C}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.60461pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###  2.7. The equivalence of calculi

Consider the commutative diagram

(1)
${{\mathsf{Homog}^{n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{\mathsf{Homog}^{n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})}}$${{\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$

In Weiss calculus the largest task in setting up the theory is demonstrating
that the restriction-induction adjunction is an equivalence between the
underlying $\infty$-categories of the $n$-homogeneous model structure and the
$n$-stable model structure. The diagram (1), implies this result for cofree
calculus with Reality if it is true for free calculus with Reality.

######  Theorem 2.7.1.

Let $n$ be a non-negative integer. The adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
75.85768pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n):\mathsf{Homog}^{n}(\mathcal{J},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
99.85768pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
99.85768pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
75.85768pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

Diagram (1) is a diagram in which three-out-of-four adjoint pairs are Quillen
equivalences. By the two-out-of-three property for Quillen equivalences, the
result will follow if the lower horizontal adjunction is a Quillen
adjunction111Note that Quillen adjunctions do not satisfy a two-out-of-three
property, see e.g., [Bal21, Example 4.1.10] hence we cannot use the same
tactic to prove that the lower horizontal adjunction is a Quillen adjunction.
The required Quillen adjunction follows readily from the standard techniques,
see e.g., [BO13, Lemmas 9.1, 9.2 and 9.4]. In particular, it suffices to show
that if $F$ is fibrant in the $n$-homogeneous model structure then
$\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}(F)$ is an
$n\Omega$-spectrum in the cofree $n$-th intermediate category, and that the
right adjoint applied to a weak equivalence between levelwise fibrant
$n$-polynomial functors is a weak equivalence in the $n$-stable model
structure. The latter is a direct consequence of the formation of the
$n$-homogeneous model structure as a right Bousfield localization of the
$n$-polynomial model structure, see Lemma 2.5.2. The former follows from
[Wei95, Corollary 5.12]. ∎

Now consider the diagram

(2) ${{\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{free}}}}$${{\mathsf{Fun}_{G_{n}}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{cofree}}}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$

######  Theorem 2.7.2.

Let $n$ be a non-negative integer. The adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
80.80125pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n}^{\mathbf{R}})_{!}:\mathsf{Fun}_{U(n)}(\mathcal{J}_{n},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
104.80125pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
104.80125pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{cofree}}:(\alpha_{n}^{\mathbf{R}})^{\ast}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
80.80127pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

iunduces an adjunction of undelrying $\infty$-categories.

###### Proof.

It suffices to show that the lower horizontal adjunction in Diagram (2) is a
Quillen adjunction. The required Quillen adjunction follows readily from the
standard techniques, see e.g., [BO13, Proposition 8.3]. ∎

###  2.8. Calculus with Reality on free and cofree $C_{2}$-spaces

In calculus with Reality, the classification of $n$-homogeneous functors has
one extra step, and is given by the zigzag

(3)
${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n)\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{free}})}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{free}}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]^{\mathsf{free}}}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{\psi}$$\scriptstyle{L_{\psi}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$

where ${\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes U(n)]$ is the category of
orthogonal spectra with an action of $C_{2}\ltimes U(n)$. The adjoint pair
$(L_{\psi},\psi)$ has right adjoint given by

$\psi(X)(V)=\operatorname{\mathsf{Map}}_{\ast}(S^{iV},X(V_{\mathbb{C}})),$

and left adjoint given by an appropriate enriched coend formula, see e.g.
[Tag22, Proposition 5.5]. Note that this last adjunction induces an
equivalence of underlying $\infty$-categories by [Tag22, Theorem 5.8]. To
replace the zigzag of free model structures by a zigzag of cofree model
structures it suffices to show that the diagram

(4)
${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{free}}}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{cofree}}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]^{\mathsf{free}}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]^{\mathsf{cofree}}}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{L_{\psi}}$$\scriptstyle{\psi}$$\scriptstyle{\psi}$$\scriptstyle{L_{\psi}}$

is a commutative diagram of Quillen equivalences. This will follow from the
following result.

######  Proposition 2.8.1.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
48.3926pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{\psi}:{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]^{\mathsf{cofree}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
72.3926pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.3926pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathsf{Sp}}^{\mathbf{R}}[U(n)]^{\mathsf{cofree}}:\psi}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
48.39261pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an equivalence of underlying $\infty$-categories.

###### Proof.

Since Diagram (4) clearly commutes on the level of $(1-)$categories, it
suffices to exhibit that the adjunction in question is a Quillen adjunction,
then the diagram will be a diagram of Quillen adjunctions in which three-out-
of-four are Quillen equivalences, hence the fourth adjoint pair must also be a
Quillen equivalence. To see that the adjunction is a Quillen adjunction note
that by construction the right adjoint preserves acyclic fibrations as acyclic
fibrations are precisely the levelwise acyclic fibrations. It hence suffices
to show that the right ajdoint preserves fibrant objects. The proof of [Tag22,
Proposition 5.6] demonstrates that the right adjoint sends real
$\Omega$-spectra to orthogonal $\Omega$-spectra, hence it suffices to show
that if $Xin{\mathsf{Sp}}^{\mathbf{R}}[U(n)]$ is levelwise cofree that $\psi
X$ is levelwise cofree. This follows by definition and the interaction of
loops and mapping spaces. The result on underlying $\infty$-categories
follows. ∎

##  3\. Polynomial functors

###  3.1. Recovering input functors

Using the invariance of Weiss calculus under Quillen equivalence, we shall
base our comparison between calculus with Reality and orthogonal calculus on
the cofree model for calculus with Reality. To ease exposition we will neglect
the superscripts for model categories when it will not cause confusion.

The complexification of real vector spaces induces a functor
$c:\mathcal{J}^{\mathbf{O}}\to\mathcal{J}^{\mathbf{R}}$, and hence a Quillen
adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.63051pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{!}:\mathsf{Fun}(\mathcal{J}^{\mathbf{O}}_{0},\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.63051pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.63051pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}(\mathcal{J}^{\mathbf{R}}_{0},\operatorname{\mathsf{Top}_{\ast}}):c^{\ast}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
35.63052pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

on the level of enriched functor categories, see e.g., [Tag21, Lemma 3.2]. On
the other hand, the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.3837pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}:\operatorname{\mathsf{Top}_{\ast}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
41.3837pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.3837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{fine}}:(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
17.3837pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

extends to an adjunction between spaces and cofree $C_{2}$-spaces

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.3837pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}:\operatorname{\mathsf{Top}_{\ast}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
41.3837pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.3837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}:(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
17.3837pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

since cofree $C_{2}$-spaces are a left Bousfield localization of the fine
model structure. This lifts to a Quillen adjunction on the level of functor
categories

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.69485pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}:\mathsf{Fun}(\mathcal{J}^{\mathbf{R}}_{0},\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.69485pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.69485pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}(\mathcal{J}^{\mathbf{R}}_{0},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
35.69485pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

with both model structures equipped with the projective model structure. It
should be noted that on the left-hand side we have a category of
$\operatorname{\mathsf{Top}_{\ast}}$-enriched functors, while on the right-
hand side we have a category of
$C_{2}\operatorname{\mathsf{Top}_{\ast}}$-enriched functors. This different in
enrichment does not interfere with the existence of the adjunction. The result
is a composite Quillen adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
37.9753pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}c_{!}:\mathsf{Fun}(\mathcal{J}_{0}^{\mathbf{0}},\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
61.9753pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
61.9753pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):c^{*}(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
37.97531pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

and hence an adjunction of underlying $\infty$-categories through which the
right adjoint sends a functor with Reality to an orthogonal functor. Given a
functor $F$ in orthogonal calculus, the image of $F$ under the unit of the
adjunction recovers $F$ up to a shift in degree of the vector spaces.

###  3.2. Polynomial functors

We begin our comparison with the polynomial functors. Unlike many other
comparisons between calculi, see e.g., [BE16, Tag21] we do not need to invoke
the classification theorem for homogeneous functors in order to compare
polynomial functors. In both versions of Weiss calculus the $n$-polynomial
functors are defined via homotopy limits indexed on topological categories,
see Definition 2.3.1. In the orthogonal calculus the indexing poset
$\mathbf{O}_{\leq n+1}$ is the poset of non-zero inner product subspaces of
$\mathbb{R}^{n+1}$, and in calculus with Reality the indexing poset
$\mathbf{R}_{\leq n+1}$ is the poset of non-zero inner product subspaces of
$\mathbb{C}\otimes\mathbb{R}^{n+1}$ of the form $\mathbb{C}\otimes
U=:U_{\mathbb{C}}$, for $U\in\mathbf{O}_{\leq n+1}$. The comparison of
$n$-polynomial functors boils down to a (co)finality statement of homotopy
(co)limits indexed on such topological categories, see e.g., [Lin13,
Appendix].

######  Theorem 3.2.1.

Let $n$ be a non-negative integer. If $F$ is a fibrant $n$-polynomial functor
in the cofree model for calculus with Reality, then the orthogonal functor
$c^{\ast}(F)^{C_{2}}$ is $n$-polynomial in orthogonal calculus.

###### Proof.

For $(1)$, since $F$ is levelwise fibrant in the cofree model structure, we
can replace $C_{2}$-fixed points by homotopy $C_{2}$-fixed points. By [Wei95,
Definition 5.1], we have to show that for each $V\in\mathcal{J}^{\mathbf{O}}$,
the canonical map

$\rho_{\mathbf{O}}:F(V_{\mathbb{C}})^{hC_{2}}\longrightarrow\underset{U\in\mathbf{O}_{\leq
n+1}}{\operatorname{\mathrm{holim}}}~{}F(V_{\mathbb{C}}\oplus
U_{\mathbb{C}})^{hC_{2}},$

is a weak equivalence. This map factors as the composite

${{F(V_{\mathbb{C}})^{hC_{2}}}}$${{\underset{U\in\mathbf{O}_{\leq
n+1}}{\operatorname{\mathrm{holim}}}~{}F(V_{\mathbb{C}}\oplus
U_{\mathbb{C}})^{hC_{2}}}}$${{\left[\underset{U_{\mathbb{C}}\in\mathbf{R}_{\leq
n+1}}{\operatorname{\mathrm{holim}}}~{}F(V_{\mathbb{C}}\oplus
U_{\mathbb{C}})\right]^{hC_{2}}}}$${{\left[\underset{U\in\mathbf{O}_{\leq
n+1}}{\operatorname{\mathrm{holim}}}~{}F(V_{\mathbb{C}}\oplus
U_{\mathbb{C}})\right]^{hC_{2}}}}$$\scriptstyle{(\rho_{\mathbf{R}})^{hC_{2}}}$$\scriptstyle{c^{\ast}}$$\scriptstyle{\rho_{\mathbf{O}}}$$\scriptstyle{\simeq}$

where the right-hand vertical map follows from the fact that right derived
functors commute with homotopy limits, and the lower horizontal map is the
induced map from the map of topological posets $c:\mathbf{O}_{\leq
n+1}\to\mathbf{R}_{\leq n+1}$. It suffices to show that this lower horizontal
map is a weak equivalence, since by assumption the left-hand vertical map is a
weak equivalence. By the dual of [Lin13, Lemma A.5], it suffices to prove that
for every $V_{\mathbb{C}}\in\mathbf{R}_{\leq n+1}$ the comma category
$(c\downarrow V_{\mathbb{C}})$ has a topologically terminal object222Let
$\mathcal{C}$ be a topological category, i.e., a category internal to spaces.
An object $\ast$ of $\mathcal{C}$ is _topologically terminal_ if it is
terminal in the underlying category, and the unique map of spaces
$!:\mathsf{Ob}(\mathcal{C})\to\mathsf{Mor}(\mathcal{C})$ sending every object
$c\in\mathcal{C}$ to the unique map $c\to\ast$ is continuous, see e.g.,
[Lin13, Remark A.6].

The comma category $(c\downarrow V_{\mathbb{C}})$ has objects pairs
$(U,f_{U})$ with $U\in\mathbf{O}_{\leq n+1}$ and $f_{U}:U_{\mathbb{C}}\to
V_{\mathbb{C}}$ in $\mathbf{R}_{\leq n+1}$. A morphism
$h:(U,f_{U})\to(W,f_{W})$ in $(c\downarrow V_{\mathbb{C}})$ is the data of a
linear isometry $h:U\to W$ in $\mathbf{O}_{\leq n+1}$ such that the triangle

${{U_{\mathbb{C}}}}$${{W_{\mathbb{C}}}}$${{V_{\mathbb{C}}}}$$\scriptstyle{f_{U}}$$\scriptstyle{f_{U}}$$\scriptstyle{c(h)}$

commutes in $\mathbf{R}_{\leq n+1}$. Since $\mathbf{R}_{\leq n+1}$ is the
poset of subspaces of $\mathbb{C}\otimes\mathbb{R}^{n+1}$ of the form
$\mathbb{C}\otimes U$ for $U$ a subspace of $\mathbb{R}^{n+1}$ ordered by
inclusion, it follows that every map $U_{\mathbb{C}}\to V_{\mathbb{C}}$ in
$\mathbf{R}_{\leq n+1}$ must be either the identity on $U_{\mathbb{C}}$ or the
inclusion of a complex subspace. One can readily check that the inclusion of a
complex subspace is fixed under the $C_{2}$-action by complex conjugation
hence is of the form $c(f)$ for $g$ a real linear map. It follows that every
map $f:U_{\mathbb{C}}\to V_{\mathbb{C}}$ must be of the form $c(g)$ for
$g:U\to V$ in $\mathbf{O}_{\leq n+1}$.

The comma category $(c\downarrow V_{\mathbb{C}})$ is a category object in
spaces with topology inherited from the product topology on
$\mathsf{Ob}(\mathbf{O}_{\leq n+1})\times\mathsf{Mor}(\mathbf{R}_{\leq n+1})$.
On the underlying non-topological category, the object
$(V,\operatorname{\mathrm{Id}}_{V_{\mathbb{C}}})$ is terminal in $(c\downarrow
V_{\mathbb{C}})$, with the unique map
$(U,f_{U})\to(V,\operatorname{\mathrm{Id}}_{V_{\mathbb{C}}})$ given by
$f_{U}$, using the above fact that $f_{U}=c(g)$ for some $g:U\to V$ in
$\mathbf{O}_{\leq n+1}$. It is left to show that the unique map

$!:\mathsf{Ob}(c\downarrow
V_{\mathbb{C}})\longrightarrow\mathsf{Mor}(c\downarrow V_{\mathbb{C}}),$

is continuous. To see this, note that the unique map is the restriction of
domain and codomain of a the projection map

$\pi_{2}:\mathsf{Ob}(\mathbf{O}_{\leq n+1})\times\mathsf{Mor}(\mathbf{R}_{\leq
n+1})\longrightarrow\mathsf{Mor}(\mathbf{R}_{\leq n+1}),$

hence continuous. ∎

######  Corollary 3.2.2.

Let $n$ be a non-negative integer. If $F$ is an fibrant functor in the cofree
model for calculus with Reality, then there is a levelwise equivalence of
orthogonal functors

$c^{\ast}(\tau_{n}^{\mathbf{R}}F)^{C_{2}}\longrightarrow\tau_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}}).$

###### Proof.

The above proof provides an equivalence on homotopy $C_{2}$-fixed points, and
the cofree assumption on $F$ concludes that all maps from $C_{2}$-fixed points
to homotopy $C_{2}$-fixed points are weak equivalences. ∎

###  3.3. Polynomial approximation

To give an equivalence on polynomial approximations, we first provide an
equivalence on iterations of $\tau_{n}$, used in constructing the polynomial
approximation functor.

######  Lemma 3.3.1.

Let $k$ and $n$ be non-negative integers. If $F$ is an fibrant functor in the
cofree model for calculus with Reality, then there is a levelwise equivalence
of orthogonal functors

$c^{\ast}((\tau_{n}^{\mathbf{R}})^{k}F)^{C_{2}}\longrightarrow(\tau_{n}^{\mathbf{O}})^{k}(c^{\ast}F^{C_{2}}).$

###### Proof.

This follows as in Corollary 3.2.2 by writing the iterated homotopy limit
$(\tau_{n})^{k}$ as a single homotopy limit, and noting that our fibrant
assumption insures homotopy invariance. ∎

Taking the homotopy colimit of Lemma 3.3.1 over $k$ provides an equivalence on
polynomial approximations.

######  Theorem 3.3.2.

Let $n$ be a non-negative integer. If $F$ is an fibrant functor in the cofree
model for calculus with Reality, then there is a levelwise equivalence of
orthogonal functors

$c^{\ast}(T_{n}^{\mathbf{R}}F)^{C_{2}}\longrightarrow
T_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}}).$

###### Proof.

Inductively use Lemma 3.3.1 with the definition of $T_{n}$, and the fact that
$C_{2}$-fixed points commute with sequential homotopy colimits. ∎

With this we can show that on the level of homotopy theories (or
$\infty$-categories) the adjunction between functor categories lifts to an
equivalence between the $\infty$-categories of $n$-polynomial functors in the
calculi.

######  Proposition 3.3.3.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
44.00041pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}c_{!}:\mathsf{Poly}^{\leq
n}(\mathcal{J}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.00041pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.00041pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}):c^{\ast}(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.00043pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an adjunction on the underlying $\infty$-categories.

###### Proof.

We prove that the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
44.00041pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}c_{!}:\mathsf{Poly}^{\leq
n}(\mathcal{J}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.00041pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.00041pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}):c^{\ast}(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.00043pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction. The acyclic fibrations of the $n$-polynomial model
structure from calculus with Reality are the levelwise acyclic fibrations
which the right adjoint preserves by the discussion in Subsection 3.1.

Moreover, A map $f:E\to F$ is a fibration in the $n$-polynomial model
structure from calculus with Reality if and only if $f:E\to F$ is a underlying
fibration (which are preserved by the right adjoint), and the canonical square

${E}$${F}$${{T_{n}^{\mathbf{R}}(E)}}$${{T_{n}^{\mathbf{R}}(F)}}$$\scriptstyle{f}$$\scriptstyle{T_{n}^{\mathbf{R}}(f)}$

is a homotopy pullback in the projective model structure. Since the right
adjoint preserves homotopy pullbacks, we see that we have a diagram

${{c^{\ast}E^{C_{2}}}}$${{c^{\ast}F^{C_{2}}}}$${{c^{\ast}(T_{n}^{\mathbf{R}}(E))^{C_{2}}}}$${{c^{\ast}(T_{n}^{\mathbf{R}}(F)^{C_{2}}}}$${{T_{n}^{\mathbf{O}}(c^{\ast}E^{C_{2}})}}$${{T_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}})}}$$\scriptstyle{\simeq}$$\scriptstyle{c^{\ast}f^{C_{2}}}$$\scriptstyle{c^{\ast}(T_{n}^{\mathbf{R}}(f))^{C_{2}}}$$\scriptstyle{T_{n}^{\mathbf{O}}(c^{*}f^{C_{2}})}$$\scriptstyle{\simeq}$

in which the top square is a homotopy pullback as the lower vertical arrows
are weak equivalences, and hence the outer square must also be a homotopy
pullback. It follows that the right adjoint preserves fibrations. ∎

##  4\. The Weiss tower

###  4.1. Homogeneous functors

The next step in understanding the relationship between Weiss towers is
understanding how the layers of the towers interact. More generally, we obtain
the following relationship between homogeneous functors.

######  Theorem 4.1.1.

Let $n$ be a non-negative integer. If $F$ is a homogeneous functor of degree
$n$ in the cofree model for calculus with Reality, then $c^{\ast}F^{C_{2}}$ is
homogeneous functor of degree $n$ in orthogonal calculus.

###### Proof.

An homogeneous functor of degree $n$ is precisely a polynomial functor of
degree less than or equal $n$, such that the $(n-1)$-polynomial vanishes. In
light of Theorem 3.2.1, it suffices to show that triviality of
$(n-1)$-polynomial approximations is preserved, which follows from Theorem
3.3.2. ∎

As with the $\infty$-categories of polynomial functors, the adjoint pair
between functor categories induces an adjoint pair between $\infty$-categories
of homogeneous functors.

######  Proposition 4.1.2.

Let $n$ be a non-negative integer. The adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
47.44765pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{i^{\ast}c_{!}:\mathsf{Homog}^{n}(\mathcal{J}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.44765pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.44765pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}}^{\mathsf{cofree}}):c^{\ast}(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.44765pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

induces an adjunction on the underlying $\infty$-categories.

###### Proof.

We show that the adjunction on the level of model categories is a Quillen
adjunction. The right adjoint preserves fibrations since the fibrations of the
$n$-homogeneous model structure coincide with the fibrations of the
$n$-polynomial model structure, which are preserved by Proposition 3.3.3. By
(the calculus with Reality version of) [Tag19, Proposition 8.3] the acyclic
fibrations of the $n$-homogeneous model structure are the fibrations of the
$(n-1)$-polynomial model structure which are also $D_{n}$-equivalences. Since
fibrations of the $(n-1)$-polynomial model structure are preserved, to
complete the proof it suffices to prove that the right adjoint is homotopical
and hence that the weak equivalences are reflected. This follows from (the
calculus of Reality version of) [Tag19, Proposition 8.2], Theorem 3.3.2, and
that right adjoints commute with right Kan extensions. ∎

Using that $D_{n}F$ is the universal approximation to $F$ in the
$\infty$-category of $n$-homogeneous functors we obtain the following
relationship between $n$-homogeneous approximations, or $n$-th layers of Weiss
towers.

######  Corollary 4.1.3.

Let $n$ be a non-negative integer. If $F$ is an fibrant functor in the cofree
model for calculus with Reality, then there is a levelwise equivalence of
orthogonal functors

$c^{\ast}(D_{n}^{\mathbf{R}}F)^{C_{2}}\longrightarrow
D_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}}).$

###  4.2. Weiss towers

We gather together the results obtained so far to give the relationship
between Weiss towers in orthogonal calculus and Weiss towers in calculus with
Reality. Since the Weiss tower of a functor $F$ is a cofiltered object in the
category of input functors, one can view the Weiss tower as a functor

$\mathsf{Tow}:\mathsf{Fun}(\mathcal{J},\mathcal{C})\longrightarrow\mathsf{CoFil}(\mathsf{Fun}(\mathcal{J},\mathcal{C})),$

where $\mathcal{C}$ is either $\operatorname{\mathsf{Top}_{\ast}}$ or
$C_{2}\operatorname{\mathsf{Top}_{\ast}}$ for our purposes.

######  Theorem 4.2.1.

If $F$ is a fibrant functor in the cofree model for calculus with Reality,
then there is a levelwise weak equivalence between the image of the Weiss
tower of $F$ in calculus with Reality under the functor $c^{\ast}(-)^{C_{2}}$
and the Weiss tower of the functor $c^{\ast}F^{C_{2}}$ in orthogonal calculus,
i.e.,

$c^{\ast}\mathsf{Tow}^{\mathbf{R}}(F)^{C_{2}}\longrightarrow\mathsf{Tow}^{\mathbf{O}}(c^{\ast}F^{C_{2}}).$

###### Proof.

By Theorem 3.3.2, the $n$-th term in the Weiss towers agree up to levelwise
weak equivalence and by Corollary 4.1.3, the homotopy fibres of the maps
between the $n$-th approximations and $(n-1)$-st approximations argree up to
levelwise weak equivalence, hence the towers agree up to levelwise weak
equivalence. In particular, we obtain a commutative diagram

${{c^{\ast}(D_{n}^{\mathbf{R}}F)^{C_{2}}}}$${{D_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}})}}$${{c^{\ast}(T_{n}^{\mathbf{R}}F)^{C_{2}}}}$${{T_{n}^{\mathbf{O}}(c^{\ast}F^{C_{2}})}}$${{c^{\ast}(T_{n-1}^{\mathbf{R}}F)^{C_{2}}}}$${{T_{n-1}^{\mathbf{O}}(c^{\ast}F^{C_{2}})}}$

in which the columns are homotopy fibre sequences and the horizontal arrows
are levelwise weak equivalences. ∎

##  5\. Higher categorical analysis

Model categories for functor calculus have proved a useful tool in
computations. We have seen that on the point-set level orthogonal calculus is
recoverable from calculus with Reality. In this section, we complete the
comparison by considering the model structures which appear in the model
categorical classification of homogeneous functors, or more precisely the
underlying $\infty$-categories of these model categories. Figure 2 displays
the relationship between the model categories for orthogonal calculus and the
model categories for calculus with Reality. In this section we explore in
which sense this diagram is commutative333We say that a diagram of adjoint
pairs is commutative if the respective diagram of right adjoints commutes. One
can easily show that this is equivalent to the respective diagram of left
adjoints commuting. on the level of underlying $\infty$-categories.

${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]}}$${{{\mathsf{Sp}}^{\mathbf{O}}[O(n)]}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]}}$${{\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n-1}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Poly}^{\leq
n-1}(\mathcal{J}_{0}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Fun}_{C_{2}}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})}}$${{\mathsf{Fun}(\mathcal{J}_{0}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})}}$$\scriptstyle{c^{\ast}(-)^{C_{2}}}$$\scriptstyle{c^{\ast}(-)^{C_{2}}}$$\scriptstyle{c^{\ast}(-)^{C_{2}}}$$\scriptstyle{c^{\ast}(-)^{C_{2}}}$$\scriptstyle{\psi_{n}(-)^{C_{2}}}$$\scriptstyle{(-)^{C_{2}}}$$\scriptstyle{(C_{2}\ltimes
U(n))_{+}\wedge_{C_{2}\times O(n)}(-)}$$\scriptstyle{(C_{2}\ltimes
U(n)_{+}\wedge_{C_{2}\times
O(n)}(L_{\psi_{n}}(-))}$$\scriptstyle{i^{\ast}c_{!}}$$\scriptstyle{i^{\ast}c_{!}}$$\scriptstyle{i^{\ast}c_{!}}$$\scriptstyle{i^{\ast}c_{!}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\mathds{1}}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{\psi}$$\scriptstyle{L_{\psi}}$$\scriptstyle{(\alpha_{n}^{\mathbf{O}})^{\ast}}$$\scriptstyle{(\alpha_{n}^{\mathbf{O}})_{!}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{3}$$\scriptstyle{4}$$\scriptstyle{5}$
Figure 2. Model categories for calculus with Reality and orthogonal calculus.

Observe that by the model categorical work of Barnes and Oman [BO13], the
right-hand column consists of Quillen adjunctions, and by [Tag22] the left-
hand column also consists of Quillen adjunctions, hence both columns may be
suitably derived to give adjunctions of $\infty$-categories. By Section 3.1,
Proposition 3.3.3, and Proposition 4.1.2, we have shown that the lower three
horizontal adjunctions induce adjunctions of $\infty$-categories, and the
lower three squares ($\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$) are clearly
commutative. To complete our comparisons it suffices to exhibit adjunctions
between the respective $\infty$-categories of spectra and the intermediate
categories, and show that the remaining square (4) and pentagon (5) are
commutative on the $\infty$-categorical level.

###  5.1. The adjunctions

For the categories of spectra, we construct an adjunction of
$\infty$-categories using classical stable homotopy theory.

######  Lemma 5.1.1.

Let $n$ be a non-negative integer. There is an adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
74.5189pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(C_{2}\ltimes U(n))_{+}\wedge_{C_{2}\times
O(n)}(-):{\mathsf{Sp}}^{\mathbf{O}}[O(n)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
98.5189pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
98.5189pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]:(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
74.51892pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

which induces an adjunction between underlying $\infty$-categories.

###### Proof.

We work on the model category level and provide a Quillen adjunction between
model categories. By a standard equivariant stable homotopy theory argument,
for a subgroup $H$ of a compact Lie group $G$, there is a Quillen adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
55.90213pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{+}\wedge_{N_{G}(H)}(-):{\mathsf{Sp}}^{\mathbf{O}}[W_{G}(H)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
79.90213pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
79.90213pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathsf{Sp}}^{\mathbf{O}}[G]:(-)^{H}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
55.90213pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

between stable model structures. It hence suffices to show that the normaliser
of $C_{2}$ in $C_{2}\ltimes U(n)$ is $C_{2}\times O(n)$ and that the
associated Weyl group is $O(n)$. By examining the normaliser condition one
sees that the normaliser of $C_{2}$ is $C_{2}\ltimes U(n)$ is

$C_{2}\ltimes\\{M\in U(n)\mid MM^{T}=I_{n}\\},$

We claim that this is precisely $C_{2}\times O(n)$. To see this, note that If
$M$ is as above, then $M\in U(n)$ implies $g\cdot M^{T}=M^{\ast}=M^{-1}$ for
$g\in C_{2}$ the non-identity element. In particular, $g\cdot M^{T}=M^{T}$,
i.e., $M^{T}$ and hence $M$ must be a real matrix. It follows that $M\in
O(n)$. An analogous converse argument yields the identification

$C_{2}\ltimes\\{M\in U(n)\mid MM^{T}=I_{n}\\}=C_{2}\times O(n).$

The condition on the Weyl group follows as $C_{2}\ltimes O(n)/C_{2}\cong
O(n)$. ∎

Similarly, we obtain the following adjunction between the intermediate
categories.

######  Lemma 5.1.2.

Let $n$ be a non-negative integer. There is an adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
72.79495pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm{I}}_{C_{2}\times O(n)}^{C_{2}\ltimes U(n)}\circ
L_{\psi_{n}}:\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
96.79495pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
96.79495pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}):\psi_{n}(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
72.79497pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

which induces an adjunction of underlying stable $\infty$-categories, where
${\rm{I}}_{C_{2}\times O(n)}^{C_{2}\ltimes U(n)}$ is short-hand notation for
the functor

$(C_{2}\ltimes U(n))_{+}\wedge_{C_{2}\times
O(n)}(-):\mathsf{Fun}_{O(n)}(\iota^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}})\to\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}).$

###### Proof.

The adjunction is the composite of the adjunction

(5) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
56.12704pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{\psi_{n}}:\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
80.12704pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
80.12704pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}_{O(n)}(\iota^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}}):\psi_{n}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.12704pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

with the adjunction

(6) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
66.17pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm{I}}^{C_{2}\ltimes U(n)}_{C_{2}\times
O(n)}:\mathsf{Fun}_{O(n)}(\iota^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
90.17pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
90.17pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}}):(-)^{C_{2}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.17pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

where
$\mathsf{Fun}_{O(n)}(\iota^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}})$
is the category of $O(n)$-equivariant functors from
$\mathcal{J}_{n}^{\mathbf{R}}$ to $O(n)\operatorname{\mathsf{Top}_{\ast}}$,
with $O(n)$ acting on $\mathcal{J}_{n}^{\mathbf{R}}$ via restriction along the
subgroup inclusion

$\iota:O(n)\hookrightarrow U(n)\hookrightarrow C_{2}\ltimes U(n).$

It suffices to demonstrate that each term in the composite is a Quillen
adjunction, where the category
$\mathsf{Fun}_{O(n)}(\iota^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}})$
may be equipped with an $n$-stable model structure analogous to the $n$-stable
model structures in any version of Weiss calculus. The right adjoint of the
adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
55.16667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{\psi}:\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
79.16667pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
79.16667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Fun}_{O(n)}(i^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}}):\psi_{n}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
55.16667pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is given by
$\psi_{n}(X)(V)=\operatorname{\mathsf{Map}}_{\ast}(S^{i\mathbb{R}^{n}\otimes
V},X(V_{\mathbb{C}}))$, while the left adjoint is given by a suitable coend
formula analogous to the left adjoint $L_{\psi}$ of the functor $\psi$ from
calculus with Reality, see e.g., [Tag22, Proposition 5.5].

The adjoint pair (6) is clearly a Quillen adjunction, hence it is left to show
that the adjoint pair (5) is also. It is clear by definition that $\psi_{n}$
preserves levelwise (acyclic) fibrations, hence preserves acyclic fibrations
of the $n$-stable model structures. It suffices to show that $\psi_{n}$
preserves fibrant objects as then it will preserve fibrations between fibrant
objects. To see that $\psi_{n}$ preserves $n\Omega$-spectra, let
$X\in\mathsf{Fun}_{O(n)}(\iota^{\ast}\mathcal{J}_{n}^{\mathbf{R}},O(n)\operatorname{\mathsf{Top}_{\ast}})$
be an $n\Omega$-spectrum, i.e., for every
$V_{\mathbb{C}},W_{\mathbb{C}}\in\mathcal{J}_{n}^{\mathbf{R}}$ the adjoint
structure maps

$X(V_{\mathbb{C}})\longrightarrow\Omega^{\mathbb{C}^{n}\otimes
W_{\mathbb{C}}}X(V_{\mathbb{C}}\oplus W_{\mathbb{C}}),$

are weak equivalences of based spaces. We want to show that the adjoint
structure maps

$(\psi_{n}X)(V)\longrightarrow\Omega^{\mathbb{R}^{n}\otimes
W}(\psi_{n}X)(V\oplus W),$

are weak homotopy equivalences for all $V,W\in\mathcal{J}_{n}^{\mathbf{O}}$.
This follows from writing these structure maps as

$\Omega^{i(\mathbb{R}^{n}\otimes
V)}X(V_{\mathbb{C}})\longrightarrow\Omega^{\mathbb{R}^{n}\otimes
W}\Omega^{i(\mathbb{R}^{n}\otimes(V\oplus W))}X(V_{\mathbb{C}}\oplus
W_{\mathbb{C}}),$

and using that $X$ is an $n\Omega$-spectrum to write the left-hand side as

$\Omega^{i(\mathbb{R}^{n}\otimes V)}\Omega^{\mathbb{C}^{n}\otimes
W_{\mathbb{C}}}X(V_{\mathbb{C}}\oplus W_{\mathbb{C}}),$

and comparing the results by identifying $\mathbb{C}^{n}$ with
$\mathbb{R}^{n}\otimes i\mathbb{R}^{n}$. ∎

###  5.2. The commutativity of Figure 2

We begin with the commutativity of the top pentagon (5) of Figure 2.

######  Proposition 5.2.1.

Let $n$ be a non-negative integer. The diagram

${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]_{\infty}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[O(n)]_{\infty}}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]_{\infty}}}$${{\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$$\scriptstyle{\mathds{R}\psi}$$\scriptstyle{\mathds{L}L_{\psi}}$$\scriptstyle{\mathds{R}(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{\mathds{L}(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{\mathds{R}(\alpha_{n}^{\mathbf{O}})^{\ast}}$$\scriptstyle{\mathds{L}(\alpha_{n}^{\mathbf{O}})_{!}}$$\scriptstyle{\mathds{R}(-)^{C_{2}}}$$\scriptstyle{(C_{2}\ltimes
U(n))_{+}\wedge^{\mathds{L}}_{C_{2}\times
O(n)}(-)}$$\scriptstyle{\mathds{R}(\psi_{n}(-)^{C_{2}})}$$\scriptstyle{(C_{2}\ltimes
U(n))_{+}\wedge^{\mathds{L}}_{C_{2}\times O(n)}(L_{\psi_{n}}(-))}$

is a commutative diagram of $\infty$-categories up to natural equivalence.

###### Proof.

Consider the corresponding diagram of model categories, and let $X$ be an
object of ${\mathsf{Sp}}^{\mathbf{R}}[U(n)]$. For $V$ in
$\mathcal{J}_{n}^{\mathbf{O}}$, we have that

$((\alpha_{n}^{\mathbf{O}})^{\ast}\circ(-)^{C_{2}}\circ\psi)X(V)=\operatorname{\mathsf{Map}}_{\ast}(S^{i(\mathbb{R}^{n}\otimes
V)},X(\mathbb{C}^{n}\otimes V_{\mathbb{C}}))^{C_{2}},$

and

$(\psi_{n}\circ(-)^{C_{2}}\circ(\alpha_{n}^{\mathbf{R}})^{\ast})X(V)=\operatorname{\mathsf{Map}}_{\ast}(S^{i(\mathbb{R}^{n}\otimes
V)},X(\mathbb{C}^{n}\otimes V_{\mathbb{C}})^{C_{2}}).$

If $X(\mathbb{C}^{n}\otimes V_{\mathbb{C}})$ is fibrant in the cofree model
structure, then the above two mapping spaces are naturally isomorphic, hence
it suffices to exhibit that the derived image of $X$ in
$\mathsf{Fun}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})_{\infty}$
is cofree. This follows readily from the fact that the image of $X$ in
$\mathsf{Fun}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})_{\infty}$
is the image of $X$ under a series of right derived functors. ∎

We conclude by demonstating that the square (4) commmutes on the level of
underlying $\infty$-categories.

######  Proposition 5.2.2.

Let $n$ be a non-negative integer. The diagram

${{\mathsf{Fun}_{C_{2}\ltimes U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$$\scriptstyle{\mathds{R}\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\mathds{L}\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{\mathds{L}\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\mathds{R}\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{(C_{2}\ltimes
U(n))_{+}\wedge^{\mathds{L}}_{C_{2}\times
O(n)}(L_{\psi_{n}}(-))}$$\scriptstyle{\mathds{R}(\psi_{n}(-)^{C_{2}})}$$\scriptstyle{\mathds{L}(i^{\ast}c_{!})}$$\scriptstyle{\mathds{R}(c^{\ast}(-)^{C_{2}})}$

is a commutative diagram of $\infty$-categories, up to natural equivalence.

###### Proof.

It suffices to show that that the diagram

${{{\mathsf{Sp}}^{\mathbf{O}}[C_{2}\ltimes
U(n)]_{\infty}}}$${{{\mathsf{Sp}}^{\mathbf{O}}[O(n)]_{\infty}}}$${{{\mathsf{Sp}}^{\mathbf{R}}[U(n)]_{\infty}}}$${{\mathsf{Fun}_{C_{2}\ltimes
U(n)}(\mathcal{J}_{n}^{\mathbf{R}},(C_{2}\ltimes
U(n))\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Fun}_{O(n)}(\mathcal{J}_{n}^{\mathbf{O}},O(n)\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{R}},C_{2}\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$${{\mathsf{Homog}^{n}(\mathcal{J}_{0}^{\mathbf{O}},\operatorname{\mathsf{Top}_{\ast}})_{\infty}}}$$\scriptstyle{\mathds{R}\psi}$$\scriptstyle{\mathds{L}L_{\psi}}$$\scriptstyle{\mathds{R}\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$$\scriptstyle{\mathds{L}\operatorname{\mathsf{res}}_{0}^{n}/U(n)}$$\scriptstyle{\mathds{R}(\alpha_{n}^{\mathbf{R}})^{\ast}}$$\scriptstyle{\mathds{L}(\alpha_{n}^{\mathbf{R}})_{!}}$$\scriptstyle{\mathds{R}(c^{\ast}(-)^{C_{2}})}$$\scriptstyle{\mathds{L}(i^{\ast}c_{!})}$$\scriptstyle{\mathds{R}(-)^{C_{2}}}$$\scriptstyle{(C_{2}\ltimes
U(n))_{+}\wedge^{\mathds{L}}_{C_{2}\times
O(n)}(-)}$$\scriptstyle{\mathds{L}(\alpha_{n}^{\mathbf{O}})_{!}}$$\scriptstyle{\mathds{R}(\alpha_{n}^{\mathbf{O}})^{\ast}}$$\scriptstyle{\mathds{L}\operatorname{\mathsf{res}}_{0}^{n}/O(n)}$$\scriptstyle{\mathds{R}\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}}$

commutes, by which we mean that there is a natural equivalence of composites

$\mathds{L}(\alpha_{n})_{!}\circ\mathds{R}\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}\circ\mathds{R}(c^{\ast}(-)^{C_{2}})\simeq\mathds{R}(-)^{C_{2}}\circ\mathds{R}\psi\circ\mathds{L}(\xi_{n})_{!}\circ\mathds{R}(\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{\ast}).$

By an adjoint argument using that the vertical adjunctions are equivalences of
$\infty$-categories, it suffices to exhibit a natural equivalence

$\mathds{R}(c^{\ast}(-)^{C_{2}})\circ\mathds{L}\operatorname{\mathsf{res}}_{0}^{n}/U(n)\circ\mathds{R}(\xi_{n})^{\ast}\circ\mathds{L}L_{\psi}\simeq\mathds{L}\operatorname{\mathsf{res}}_{0}^{n}/O(n)\circ\mathds{R}(\alpha_{n})^{\ast}\circ\mathds{R}(-)^{C_{2}}.$

Let $\Theta$ be an object in the $\infty$-category of spectra with an action
of $C_{2}\ltimes U(n)$, or equivalently a bifibrant object in the model
category. By the classification theorem for calculus with Reality, see [Tag22,
Theorem 7.1], the left-hand side applied to $\Theta$ is naturally levelwise
weakly equivalent to the functor

$V\longmapsto(\Omega^{\infty}[(S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}}\wedge\Theta)_{hU(n)}])^{C_{2}},$

whereas by the classification theorem for orthogonal calculus, see [Wei95,
Theorem 7.3] and [BO13, Theorem 10.3], the right-hand side is naturally
levelwise weakly equivalent to the functors

$V\longmapsto\Omega^{\infty}[(S^{\mathbb{R}^{n}\otimes_{\mathbb{R}}V}\wedge\Theta^{C_{2}})_{hO(n)}],$

hence it suffices to exhibit a natural isomorphism between these functors.
First note that $C_{2}$-fixed points commutes with $\Omega^{\infty}$, hence
there is a natural equivalence

$(\Omega^{\infty}[(S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}}\wedge\Theta)_{hU(n)}])^{C_{2}}\simeq\Omega^{\infty}[(S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}}\wedge\Theta)_{hU(n)}]^{C_{2}},$

which by definition of the homotopy orbits we may rewrite as

$(\Omega^{\infty}[(S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}}\wedge\Theta)\wedge_{U(n)}EU(n)_{+}])^{C_{2}}\simeq\Omega^{\infty}[(S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}}\wedge\Theta)\wedge_{U(n)}EU(n)_{+}]^{C_{2}}.$

Since $C_{2}$ acts diagonally on the right-hand side, there is a natural
isomorphism

$[(S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}}\wedge\Theta)\wedge_{U(n)}EU(n)_{+}]^{C_{2}}\simeq((S^{\mathbb{C}^{n}\otimes_{\mathbb{C}}V_{\mathbb{C}}})^{C_{2}}\wedge\Theta^{C_{2}})\wedge_{U(n)^{C_{2}}}(EU(n)_{+})^{C_{2}},$

which by calculation is naturally isomorphic to

$(S^{\mathbb{R}^{n}\otimes_{\mathbb{R}}V}\wedge\Theta^{C_{2}})\wedge_{O(n)}EO(n)_{+}=(S^{\mathbb{R}^{n}\otimes_{\mathbb{R}}V}\wedge\Theta^{C_{2}})_{hO(n)},$

hence the result follows. ∎

## References

  * [ALV07] G. Arone, P. Lambrechts, and I. Volić.  Calculus of functors, operad formality, and rational homology of embedding spaces.  Acta Math., 199(2):153–198, 2007.
  * [Aro98] G. Arone.  Iterates of the suspension map and Mitchell’s finite spectra with $A_{k}$-free cohomology.  Math. Res. Lett., 5(4):485–496, 1998.
  * [Bal21] S. Balchin.  A handbook of model categories, volume 27 of Algebra and Applications.  Springer, Cham, 2021.
  * [BE16] D. Barnes and R. Eldred.  Comparing the orthogonal and homotopy functor calculi.  J. Pure Appl. Algebra, 220(11):3650–3675, 2016.
  * [BJM15] K. Bauer, B. Johnson, and R. McCarthy.  Cross effects and calculus in an unbased setting.  Trans. Amer. Math. Soc., 367(9):6671–6718, 2015.  With an appendix by R. Eldred.
  * [BO13] D. Barnes and P. Oman.  Model categories for orthogonal calculus.  Algebr. Geom. Topol., 13(2):959–999, 2013.
  * [BV] J. M. Boardman and R. M. Vogt.  Homotopy invariant algebraic structures on topological spaces.
  * [Goo90] T. G. Goodwillie.  Calculus. I. The first derivative of pseudoisotopy theory.  $K$-Theory, 4(1):1–27, 1990.
  * [Goo03] T. G. Goodwillie.  Calculus. III. Taylor series.  Geom. Topol., 7:645–711, 2003.
  * [Goo92] T. G. Goodwillie.  Calculus. II. Analytic functors.  $K$-Theory, 5(4):295–332, 1991/92.
  * [GW99] T. G. Goodwillie and M. Weiss.  Embeddings from the point of view of immersion theory. II.  Geom. Topol., 3:103–118, 1999.
  * [Hir03] P. S. Hirschhorn.  Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2003.
  * [HY19] J. Hahn and A. Yuan.  Multiplicative structure in the stable splitting of $\Omega SL_{n}(\mathbb{C})$.  Adv. Math., 348:412–455, 2019.
  * [Joy02] A. Joyal.  Quasi-categories and Kan complexes.  volume 175, pages 207–222. 2002.  Special volume celebrating the 70th birthday of Professor Max Kelly.
  * [Law20] T. Lawson.  An introduction to Bousfield localization.  arXiv e-prints, page arXiv:2002.03888, February 2020.
  * [Lin13] J. A. Lind.  Diagram spaces, diagram spectra and spectra of units.  Algebr. Geom. Topol., 13(4):1857–1935, 2013.
  * [Lur09] J. Lurie.  Higher topos theory, volume 170 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2009.
  * [MM02] M. A. Mandell and J. P. May.  Equivariant orthogonal spectra and $S$-modules.  Mem. Amer. Math. Soc., 159(755):x+108, 2002.
  * [Sch19] S. Schwede.  Lectures on equivariant stable homotopy theory.  http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf, 2019\.
  * [Tag19] N. Taggart.  Unitary calculus: model categories and convergence.  arXiv e-prints, page arXiv:1911.08575, Nov 2019.
  * [Tag20] N. Taggart.  Recovering unitary calculus from calculus with reality.  arXiv e-prints, page arXiv:2010.05948, October 2020.
  * [Tag21] N. Taggart.  Comparing the orthogonal and unitary functor calculi.  Homology Homotopy Appl., 23(2):227–256, 2021.
  * [Tag22] N. Taggart.  Unitary functor calculus with reality.  Glasg. Math. J., 64(1):197–230, 2022.
  * [Wal06] B. C. Walter.  Rational homotopy calculus of functors.  ProQuest LLC, Ann Arbor, MI, 2006.  Thesis (Ph.D.)–Brown University.
  * [Wei95] M. Weiss.  Orthogonal calculus.  Trans. Amer. Math. Soc., 347(10):3743–3796, 1995.
  * [Wei96] M. Weiss.  Calculus of embeddings.  Bull. Amer. Math. Soc. (N.S.), 33(2):177–187, 1996.
  * [Wei98] M. Weiss.  Erratum: “Orthogonal calculus”.  Trans. Amer. Math. Soc., 350(2):851–855, 1998.
  * [Wei99] M. Weiss.  Embeddings from the point of view of immersion theory. I.  Geom. Topol., 3:67–101, 1999.
  * [WW88] M. Weiss and B. Williams.  Automorphisms of manifolds and algebraic $K$-theory. I.  $K$-Theory, 1(6):575–626, 1988.