$\textup{C}^{*}$-algebra $A$ by Hilbert bimodules. If $A$ is separable or of
Type I, then the following are equivalent:

  1. (1)

the dual groupoid $\widehat{A}\rtimes S$ is residually topologically free;

  2. (2)

the action $\mathcal{E}$ is residually aperiodic;

  3. (3)

for any $I\in\mathbb{I}^{\mathcal{E}}(A)$, the full crossed product for the
restricted action $\mathcal{E}|_{A/I}$ has a unique pseudo-expectation namely,
the canonical $\mathcal{M}_{\mathrm{loc}}$-expectation;

  4. (4)

for any $I\in\mathbb{I}^{\mathcal{E}}(A)$, $(A/I)^{+}$ supports $C$ for each
intermediate $\textup{C}^{*}$-subalgebra $A/I\subseteq C\subseteq
A/I\rtimes_{\mathrm{ess}}S$ for the restricted action $\mathcal{E}|_{A/I}$;

  5. (5)

for any $I\in\mathbb{I}^{\mathcal{E}}(A)$, $A/I$ detects ideals in each
intermediate $\textup{C}^{*}$-subalgebra $A/I\subseteq C\subseteq
A/I\rtimes_{\mathrm{ess}}S$ for the restricted action $\mathcal{E}|_{A/I}$.

For general $A$, (1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(5) and
(2)$\Rightarrow$(4)$\Rightarrow$(5).

###### Proof.

A subset of $\widehat{A}$ is closed and invariant if and only if it is of the
form $X=\widehat{A/I}$ for an $\mathcal{E}$-invariant ideal $I$. The dual
groupoid of the induced action on $A/I$ is the restriction $X\rtimes S$ of the
dual groupoid to $X$. Hence the implications
(1)$\Rightarrow$(2)$\Rightarrow$(3), (4) follow from [Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations]*Corollary 4.8 and Theorem 3.6, applied
to the quotients $A/I$ and intermediate $\textup{C}^{*}$-algebras. Lemma 2.22
shows that (4) implies (5), and [Pitts-
Zarikian:Unique_pseudoexpectation]*Theorem 3.5 shows that (3) implies (5). If
$A$ is separable or of Type I, so are its quotients $A/I$, and then
[Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]*Proposition 6.1 shows
that (5) implies (1). ∎

The following result justifies introducing the notion of essential exactness –
it is an instance of condition (1) in Theorem 2.33.

######  Theorem 5.9.

Let $B$ be an $S$-graded $\textup{C}^{*}$-algebra with a grading
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ that forms a residually aperiodic
action of $S$ on $A\mathrel{\vcentcolon=}\mathcal{E}_{1}$ (this holds if the
dual groupoid $\widehat{A}\rtimes S$ is residually topologically free). The
following are equivalent:

  1. (1)

$B\cong A\rtimes_{\mathrm{ess}}S$ and $\mathcal{E}$ is essentially exact;

  2. (2)

$A$ separates ideals in $B$, so
$\mathbb{I}(B)\cong\mathbb{I}^{\mathcal{E}}(A)$;

  3. (3)

$A^{+}$ residually supports $B$;

  4. (4)

$A^{+}$ fills $B$.

If the above equivalent conditions hold and the primitive ideal space
$\check{A}$ is second countable, then the quasi-orbit map induces a
homeomorphism $\check{B}\cong\check{A}/{\sim}$, where $\check{A}/{\sim}$ is
the quasi-orbit space of the dual groupoid $\check{A}\rtimes S$, that is,
$\mathfrak{p}_{1},\mathfrak{p}_{2}\in\check{A}$ satisfy
$\mathfrak{p}_{1}\sim\mathfrak{p}_{2}$ if and only if
$\overline{(\check{A}\rtimes
S)\cdot\mathfrak{p}_{1}}=\overline{(\check{A}\rtimes
S)\cdot\mathfrak{p}_{2}}$.

###### Proof.

The $\textup{C}^{*}$-inclusion $A\subseteq B$ is symmetric and residually
aperiodic by Propositions 2.12 and 5.2. Hence by Theorem 2.33 conditions
(2)–(4) are equivalent to the condition that for each
$I\in\mathbb{I}^{B}(A)=\mathbb{I}^{\mathcal{E}}(A)$ the unique pseudo-
expectation $E^{I}\colon B/BIB\to A/I$ is almost faithful. Let us assume this.
Then there is a commutative diagram

${A\rtimes S/I\rtimes S\cong A/I\rtimes
S}$${B/BIB}$${\mathcal{M}_{\mathrm{loc}}(A/I)}$${\mathrm{I}(A/I),}$$\scriptstyle{\Psi}$$\scriptstyle{EL^{I}}$$\scriptstyle{E^{I}}$$\scriptstyle{\hookrightarrow}$

where $\Psi$ is the homomorphism that exists by universality of $A/I\rtimes S$
because $B/BIB$ is graded by $\mathcal{E}|_{A/I}$ by Lemma 2.13, $EL^{I}$ is
the canonical essential expectation for $A/I\rtimes S$, and
$\mathcal{M}_{\mathrm{loc}}(A/I)\hookrightarrow\mathrm{I}(A/I)$ is the
canonical embedding. The diagram commutes because the inclusion $A/I\subseteq
A/I\rtimes S$ is aperiodic and hence there is a unique pseudo-expectation by
[Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]*Theorem 3.6. As a
consequence,
$\ker\Psi={\Psi}^{-1}(0)={\Psi}^{-1}(\mathcal{N}_{E^{I}})=\mathcal{N}_{EL}$
and thus $\Psi$ factors through an isomorphism
$A/I\rtimes_{\mathrm{ess}}S\cong B/BIB$. Since this holds for every
$I\in\mathbb{I}^{\mathcal{E}}(A)$ we get that $B\cong
A\rtimes_{\mathrm{ess}}S$ and that $\mathcal{E}$ is essentially exact
($A\rtimes_{\mathrm{ess}}S/I\rtimes_{\mathrm{ess}}S\cong
B/BIB\cong(A/I)\rtimes_{\mathrm{ess}}S$ for
$I\in\mathbb{I}^{\mathcal{E}}(A)$).

Theorem 2.33 implies easily that (1) implies (2). This finishes the proof that
the four conditions are equivalent. The remaining claims follow mostly from
the last part of Theorem 2.33. That the quasi-orbit space has the asserted
form follows from [Kwasniewski-Meyer:Stone_duality]*Theorem 6.22. ∎

######  Corollary 5.10.

Let $\mathcal{A}=(A_{\gamma})_{\gamma\in G}$ be a Fell bundle over an étale
groupoid $G$ with locally compact, Hausdorff unit space $X$, and put
$A=\textup{C}_{0}(\mathcal{A}|_{X})$. Define the dual groupoid
$\widehat{A}\rtimes G$ as in Example 3.6. Assume that it is residually
topologically free (this holds, for instance, if $G$ is residually
topologically free and the base map for the $\textup{C}^{*}$-bundle
$\mathcal{A}$ is open and closed). Assume also one of the following

  1. (1)

$B\mathrel{\vcentcolon=}\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A})$ and
$\mathcal{A}$ is essentially exact;

  2. (2)

$B\mathrel{\vcentcolon=}\textup{C}^{*}_{\mathrm{r}}(\mathcal{A})$,
$\mathcal{A}$ is exact, and the unit space in $\check{A}\rtimes G$ is closed
(the latter is automatic if $G$ is Hausdorff);

  3. (3)

$B\mathrel{\vcentcolon=}\textup{C}^{*}(\mathcal{A})$, $\mathcal{A}$ is
separable, and $G$ is amenable and Hausdorff.

Then $A$ separates ideals in $B$ and, even more, $A^{+}$ fills $B$. The
lattice $\mathbb{I}(B)$ is naturally isomorphic to the lattice of
$G$-invariant ideals in $A$. If, in addition, $\check{A}$ is second countable,
then $\check{B}\cong\check{A}/{\sim}$, where $\check{A}/{\sim}$ is the quasi-
orbit space of the dual groupoid $\check{A}\rtimes G$, that is,
$\mathfrak{p}_{1},\mathfrak{p}_{2}\in\check{A}$ satisfy
$\mathfrak{p}_{1}\sim\mathfrak{p}_{2}$ if and only if
$\overline{(\check{A}\rtimes
G)\cdot\mathfrak{p}_{1}}=\overline{(\check{A}\rtimes
G)\cdot\mathfrak{p}_{2}}$.

###### Proof.

The claims in brackets follow from Lemma 5.7. The claim in case (1) follows
from Theorem 5.9 (see also Example 4.25). Case (2) follows from (1) and
Remarks 3.16 and 4.24. Example 4.32 explains why (3) is a special case of (2).
∎

Theorem 5.9 allows us to describe the ideal structure of $B$ in terms of $A$
under the following assumptions:

(5.11) $\mathcal{E}\text{ is an essentially exact, residually aperiodic action
and }B=A\rtimes_{\mathrm{ess}}S.$

We are going to study whether $B$ is purely infinite using the same
assumption. Before we do this, we simplify (5.11) in the presence of a
conditional expectation.

######  Proposition 5.12.

If there is a genuine conditional expectation $E\colon B\to A\subseteq B$,
then (5.11) is equivalent to

(5.13) $\mathcal{E}$ is closed, exact, residually aperiodic and
$B=A\rtimes_{\mathrm{r}}S$.

For any $\textup{C}^{*}$-inclusion $A\subseteq B$, an action $\mathcal{E}$ as
in (5.13) exists if and only if $A\subseteq B$ is regular, residually
aperiodic and there is a conditional expectation $E\colon B\to A$ which is
residually faithful in the sense that $E$ descends to a faithful conditional
expectation $E^{I}\colon B/BIB\to A/I$ for any $I\in\mathbb{I}^{B}(A)$ (see
Lemma 2.19).

###### Proof.

(5.13) implies (5.11) by Remark 4.24. Conversely, assume (5.11). Then $E\colon
B\to A$ is the unique pseudo-expectation for $A\subseteq B$ by [Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations]*Theorem 3.6. Hence $E=EL$,
$\mathcal{E}$ is closed and $A\rtimes_{\mathrm{r}}S=A\rtimes_{\mathrm{ess}}S$.
The same argument works for all the restrictions $\mathcal{E}^{I}$ and
quotient inclusions $A/I\subseteq B/BIB$ for
$I\in\mathbb{I}^{\mathcal{E}}(A)=\mathbb{I}^{B}(A)$. This gives (5.13).
Combining this reasoning with [Kwasniewski-Meyer:Cartan]*Theorem 6.3 also
gives the second part of the assertion. ∎

######  Corollary 5.14.

If $A$ is type $I$, then an action $\mathcal{E}$ as in (5.13) exists if and
only if $A\subseteq B$ is residually Cartan, that is, for each
$I\in\mathbb{I}^{B}(A)$, $A/I\subseteq B/BIB$ is a noncommutative Cartan
subalgebra in the sense of Exel [Exel:noncomm.cartan].

###### Proof.

Combine the second part of Proposition 5.12 and [Kwasniewski-
Meyer:Cartan]*Theorem 6.3. ∎

###  5.2. Pure infiniteness criteria

In this section, we assume that $B$ is an $S$-graded $\textup{C}^{*}$-algebra
with a grading $\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ and $A$ is the unit
fibre of the grading. We give pure infiniteness criteria for $B$ under the
assumption (5.11). This covers (5.13) and the assumptions in Corollary 5.10 as
special cases. In view of Theorem 5.9, the following two theorems are
immediate corollaries of Theorems 2.35 and 2.36:

######  Theorem 5.15.

Assume (5.11). Let $\mathcal{F}\subseteq A^{+}$ fill $A$ and be invariant
under $\varepsilon$-cut-downs. Then $B$ is strongly purely infinite if and
only if each pair of elements $a,b\in\mathcal{F}$ has the matrix
diagonalisation property in $B$.

######  Theorem 5.16.

Assume (5.11). Let $\mathcal{F}\subseteq A^{+}$ residually support $A$.
Suppose that $\mathbb{I}^{\mathcal{E}}(A)$ is finite or the projections in
$\mathcal{F}$ separate the ideals in $\mathbb{I}^{\mathcal{E}}(A)$. Then $B$
is strongly purely infinite (with the ideal property) if and only if every
element in $\mathcal{F}\setminus\\{0\\}$ is properly infinite in $B$.

In order to use these results, we need conditions that suffice for $a,b\in
A^{+}$ to have the matrix diagonalisation property in $B$ or for $a\in
A^{+}\setminus\\{0\\}$ to be properly infinite in $B$. Checking the matrix
diagonalisation property is usually difficult. Nevertheless, the following
lemma may be useful (see [Kirchberg-Sierakowski:Strong_pure, Kwasniewski-
Meyer:Aperiodicity]):

######  Lemma 5.17.

Let $a,b\in A^{+}\setminus\\{0\\}$. Suppose that for each
$c\in\mathcal{E}_{t}$, $t\in S$, and each $\varepsilon>0$ there are
$n,m\in\mathbb{N}$ and $a_{i}\in a\mathcal{E}_{s_{i}}$, $s_{i}\in S$, for
$i=1,\dotsc,n$ and $b_{j}\in b\mathcal{E}_{t_{j}}$, $t_{j}\in S$, for
$j=1,\dotsc,m$ such that

$a\approx_{\varepsilon}\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad
b\approx_{\varepsilon}\sum_{i=1}^{m}b_{i}^{*}b_{i},\quad\sum_{i,j=1,i\neq
j}^{n}a_{i}^{*}a_{j}\approx_{\varepsilon}0,\quad\sum_{i,j=1,i\neq
j}^{n,m}b_{i}^{*}b_{j}\approx_{\varepsilon}0$

and $\sum_{i=1,j=1}^{n,m}a_{i}^{*}cb_{j}\approx_{\varepsilon}0$. Then $a,b\in
A^{+}\setminus\\{0\\}$ have the matrix diagonalisation property in $B$.

###### Proof.

Let $\mathcal{C}\mathrel{\vcentcolon=}\bigcup_{t\in S}\mathcal{E}_{t}$ and
$\mathcal{S}\mathrel{\vcentcolon=}\bigcup_{t\in S}\mathcal{E}_{t}$. We claim
that $a,b\in A^{+}\setminus\\{0\\}$ have the matrix diagonalisation property
with respect to $\mathcal{C}$ and $\mathcal{S}$ as introduced in [Kirchberg-
Sierakowski:Filling_families]*Definition 4.6. Indeed, let
$x\in\mathcal{E}_{t}$ be such that $\left(\begin{smallmatrix}a&x^{*}\\\
x&b\end{smallmatrix}\right)\in M_{2}(B)^{+}$ and let $\varepsilon>0$. Let
$a_{i}\in a\mathcal{E}_{s_{i}}$ and $b_{j}\in b\mathcal{E}_{t_{j}}$ satisfy
the conditions described in the assertion with
$c\mathrel{\vcentcolon=}a^{\nicefrac{{1}}{{2}}}xb^{\nicefrac{{1}}{{2}}}$. We
may write $a_{i}=a^{\nicefrac{{1}}{{2}}}x_{i}$ and
$b_{j}=b^{\nicefrac{{1}}{{2}}}y_{j}$ for some $x_{i}$, $y_{j}$. Let
$d_{1}\mathrel{\vcentcolon=}\sum_{i=1}^{n}x_{i}$ and
$d_{2}\mathrel{\vcentcolon=}\sum_{j=1}^{n}y_{j}$. The assumed estimates imply
that

$d_{1}^{*}ad_{1}\approx_{2\varepsilon}a,\qquad
d_{2}^{*}bd_{2}\approx_{2\varepsilon}b,\qquad
d_{1}^{*}xd_{2}\approx_{\varepsilon}0.$

This proves our claim. Clearly, $\mathcal{S}$ is a multiplicative subsemigroup
of $B$, $\mathcal{S}^{*}\mathfrak{S}\mathcal{S}\subseteq\mathfrak{S}$,
$A\mathcal{S}A\subseteq\mathcal{S}$, and the closed linear span of
$\mathcal{C}$ is $B$. Thus $a,b\in A^{+}\setminus\\{0\\}$ have the matrix
diagonalisation property in $B$ by [Kirchberg-
Sierakowski:Filling_families]*Lemma 5.6. ∎

Now we will focus on ways to check whether $a\in A^{+}\setminus\\{0\\}$ is
properly infinite in $B$. The following definition generalises [Kwasniewski-
Szymanski:Pure_infinite]*Definition 5.1 and [Kwasniewski-
Meyer:Aperiodicity]*Definition 5.5 from groups to inverse semigroups. A
crucial point is that the properties depend only on the Fell bundle
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$, not on the norm in $B$.

######  Definition 5.18.

An element $a\in A^{+}\setminus\\{0\\}$ is called

  1. (1)

_$\mathcal{E}$ -infinite_ if there is $b\in A^{+}\setminus\\{0\\}$ such that
for each $\varepsilon>0$, there are $n,m\in\mathbb{N}$, $t_{i}\in S$ and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$, such that

$a\approx_{\varepsilon}\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad
b\approx_{\varepsilon}\sum_{i=n+1}^{n+m}a_{j}^{*}a_{j},\quad\sum_{\genfrac{}{}{0.0pt}{}{i,j=1}{i\neq
j}}^{n+m}\lVert a_{i}^{*}a_{j}\rVert\leq\varepsilon;$

  2. (2)

_residually $\mathcal{E}$-infinite_ if $a+I$ is $\mathcal{E}|_{A/I}$-infinite
for all $I\in\mathbb{I}^{\mathcal{E}}(A)$ with $a\notin I$;

  3. (3)

_properly $\mathcal{E}$-infinite_ if for all $\varepsilon>0$ there are
$n,m\in\mathbb{N}$, $t_{i}\in S$ and $a_{i}\in a\mathcal{E}_{t_{i}}$ for
$1\leq i\leq n+m$, such that

$a\approx_{\varepsilon}\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad
a\approx_{\varepsilon}\sum_{j=n+1}^{m}a_{j}^{*}a_{j},\quad\sum_{\genfrac{}{}{0.0pt}{}{i,j=1}{i\neq
j}}^{n+m}\lVert a_{i}^{*}a_{j}\rVert\leq\varepsilon;$

  4. (4)

_$\mathcal{E}$ -paradoxical_ if the condition in (3) holds with
$\varepsilon=0$, that is, there are $n,m\in\mathbb{N}$, $t_{i}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$, such that

$a=\sum_{i=1}^{n}a_{i}^{*}a_{i},\qquad
a=\sum_{j=n+1}^{n+m}a_{j}^{*}a_{j},\qquad a_{i}^{*}a_{j}=0\quad\text{for
}i\neq j.$

######  Lemma 5.19.

If $a\in A^{+}\setminus\\{0\\}$ is $\mathcal{E}$-infinite, then it is infinite
in $B$. If $a\in A^{+}\setminus\\{0\\}$ is properly $\mathcal{E}$-infinite,
then it is properly infinite in $B$.

###### Proof.

First let $a\in A^{+}\setminus\\{0\\}$ be $\mathcal{E}$-infinite. Let $b\in
A^{+}\setminus\\{0\\}$ be as in Definition 5.18.(1). For $\varepsilon>0$,
there are $n,m\in\mathbb{N}$ and $t_{i}\in S$ and $a_{i}\in
a\mathcal{E}_{t_{i}}$ for $i=1,\dotsc,n+m$, $n,m\in\mathbb{N}$ as in
Definition 5.18.(1). Let $x\mathrel{\vcentcolon=}\sum_{i=1}^{n}a_{i}$ and
$y\mathrel{\vcentcolon=}\sum_{i=n+1}^{m}a_{i}$. Then $x,y\in aB$. Simple
estimates such as

$\lVert
x^{*}x-a\rVert=\bigg{\lVert}\sum_{j=1}^{n}a_{j}^{*}a_{j}-a+\sum_{i,j=1,i\neq
j}^{n}a_{i}^{*}a_{j}\bigg{\rVert}\leq\varepsilon+\varepsilon$

show that $x^{*}x\approx_{2\varepsilon}a$, $y^{*}y\approx_{2\varepsilon}b$ and
$y^{*}x\approx_{\varepsilon}0$. Hence $a$ is infinite in $B$ (see Definition
2.34). The proof when $a$ is properly $\mathcal{E}$-infinite is the same with
$b=a$. ∎

Let us compare the definitions of infinite and properly infinite elements in
Definition 2.34 to the definitions of $\mathcal{E}$-infinite and properly
$\mathcal{E}$-infinite elements in Definition 5.18. There are two differences.
First, we now choose the elements $x,y\in aB$ in the subalgebra
$A\rtimes_{\mathrm{alg}}S$, so that we may write them as a finite sum $\sum
a_{i}$ with $a_{i}\in a\mathcal{E}_{t_{i}}$. Secondly, we estimate each
product $a_{i}^{*}a_{j}$ for $i\neq j$ separately. The first change does not
achieve much because $A\rtimes_{\mathrm{alg}}S$ is dense in $B$ and we only
aim for approximate equalities anyway. The second change simplify the
estimates a lot because $\lVert a_{i}^{*}a_{j}\rVert$ is computed in the
Hilbert $A$-bimodule $\mathcal{E}_{t_{i}^{*}t_{j}}$, whereas the norm
estimates in Definition 2.34 involve the $\textup{C}^{*}$-norm of $B$. For an
$\mathcal{E}$-paradoxical element, we even assume the products
$a_{i}^{*}a_{j}$ for $i\neq j$ to vanish exactly. This is once again much
easier to check. Paradoxical elements are also important because they are
related to paradoxical decompositions, which were studied already by Banach
and Tarski. In the setting of purely infinite crossed products, their
importance was highlighted by Rørdam and Sierakowski [Rordam-
Sierakowski:Purely_infinite]. The implications among our infiniteness
conditions hinted at above are summarised in the following proposition:

######  Proposition 5.20.

Assume that $A$ separates ideals in $B$. Consider the following conditions
$a\in A^{+}\setminus\\{0\\}$ may satisfy:

  1. (1)

$a$ is properly infinite in $B$;

  2. (2)

for each $\varepsilon>0$ there are $n,m\in\mathbb{N}$, $t_{i}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$, such that

$a\approx_{\varepsilon}\sum_{i,j=1}^{n}a_{i}^{*}a_{j},\qquad
a\approx_{\varepsilon}\sum_{i,j=n+1}^{m}a_{i}^{*}a_{j},\qquad\sum_{i,j=1,i\neq
j}^{n+m}a_{i}^{*}a_{j}\approx_{\varepsilon}0;$

  3. (3)

$a$ is residually $\mathcal{E}$-infinite;

  4. (4)

$a$ is properly $\mathcal{E}$-infinite;

  5. (5)

$a$ is $\mathcal{E}$-paradoxical.

Then (1)$\Leftrightarrow$(2)$\Leftarrow$(3)$\Leftarrow$(4)$\Leftarrow$(5).

###### Proof.

The implications (5)$\Rightarrow$(4)$\Rightarrow$(3) are straightforward. By
[Kirchberg-Rordam:Non-simple_pi]*Proposition 3.14, $a$ is properly infinite if
and only if it is residually infinite. Since $A$ separates ideals in $B$, any
ideal in $B$ comes from an invariant ideal in $A$, as in the definition that
$a$ is residually $\mathcal{E}$-infinite. Together with Lemma 5.19, this shows
that (3) implies (1).

According to Definition 2.34, $a\in A^{+}\setminus\\{0\\}$ is properly
infinite in $B$ if and only if, for all $\varepsilon>0$, there are $x,y\in
a\cdot B$ with $x^{*}x\approx_{\varepsilon}a$, $y^{*}y\approx_{\varepsilon}b$
and $x^{*}y\approx_{\varepsilon}0$. Without loss of generality, we may pick
$x,y\in a\cdot(\sum_{t\in S}\mathcal{E}_{t})$ because $\sum_{t\in
S}\mathcal{E}_{t}$ is dense in $B$. So $x=\sum_{j=1}^{n}a_{j}$ and
$y=\sum_{j=n+1}^{n+m}a_{j}$ for some $n,m\in\mathbb{N}$, $t_{i}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$. The relations
$x^{*}x\approx_{\varepsilon}a$, $y^{*}y\approx_{\varepsilon}b$ and
$x^{*}y\approx_{\varepsilon}0$ translate to those described in (2). This
proves that (1) and (2) are equivalent. ∎

It is unclear whether the implications in Proposition 5.20 may be reversed.

######  Remark 5.21.

The example of graph $\textup{C}^{*}$-algebras shows that it may be much
easier to check that an element is residually $\mathcal{E}$-infinite than that
it is properly $\mathcal{E}$-infinite (see also [Kwasniewski-
Szymanski:Pure_infinite]*Remark 7.10).

######  Corollary 5.22.

Assume (5.11). Let $\mathcal{F}\subseteq A^{+}$ residually support $A$.
Suppose that $\mathbb{I}^{\mathcal{E}}(A)$ is finite or that $\mathcal{F}$
consists of projections, or that the projections in $\mathcal{F}$ separate the
ideals in $\mathbb{I}^{\mathcal{E}}(A)$. If every element in
$\mathcal{F}\setminus\\{0\\}$ is residually $\mathcal{E}$-infinite, then
$A\rtimes_{\mathrm{ess}}S$ is purely infinite and has the ideal property.

###### Proof.

Combine Theorem 5.16 and Proposition 5.20. ∎

We may simplify our conditions further if $A$ is commutative. Then
$A\rtimes_{\mathrm{ess}}S\cong\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ for a
twisted étale groupoid $G$ with object space $\hat{A}$. The twist $\Sigma$ is
always locally trivial. Therefore, the bisections that trivialise the twist
$\Sigma$ form a wide inverse subsemigroup $S^{\prime}$ among all bisections of
$G$ (see [BussExel:Regular.Fell.Bundle]*Theorem 7.2). Then
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)\cong
A\rtimes_{\mathrm{ess}}S^{\prime}$. The action of $S^{\prime}$ on $A$ is
equivalent to a twisted action as in [BussExel:Regular.Fell.Bundle]*Definition
4.1, that is, each $\mathcal{E}_{t}$ for $t\in S$ comes from an isomorphism
between two ideals in $A$. We assume this because it allows us to identify
elements of $\mathcal{E}_{t}$ with $\textup{C}_{0}$-functions on
$s(\mathcal{E}_{t})\subseteq\hat{A}$. This discussion shows how to turn any
inverse semigroup action on a commutative $\textup{C}^{*}$-algebra into a
twisted action by partial automorphisms.

######  Lemma 5.23.

Assume that $A$ is commutative and that $S$ acts on $A$ by a twisted action by
partial automorphisms as in Example 3.2. Equip $\widehat{A}$ with the dual
action of $S$. Let $a\in A^{+}$ and
$V\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\neq 0\\}$.
Consider the following conditions:

  1. (1)

the condition in Definition 5.18.(1) holds with $\varepsilon=0$;

  2. (2)

there are $b\in(aAa)^{+}\setminus\\{0\\}$, $n\in\mathbb{N}$ and $t_{i}\in S$,
$a\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n$ such that

$a=\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad\text{ and }\quad a_{i}^{*}a_{j}=0,\quad
a_{i}^{*}b=0\qquad\text{for all }\,i,j=1,\dotsc,n,i\neq j;$

  3. (3)

there are $n\in\mathbb{N}$, $t_{1},\dotsc,t_{n}\in S$, and open subsets
$V_{1},\dotsc,V_{n}\subseteq V$ such that

    1. (a)

$V_{i}$ is contained in the domain of $t_{i}$ for $1\leq i\leq n$;

    2. (b)

$(t_{i}\cdot V_{i})\cap(t_{j}\cdot V_{j})=\emptyset$ if $1\leq i<j\leq n$;

    3. (c)

$V=\bigcup_{i=1}^{n}V_{i}$ and $\overline{\bigcup_{i=1}^{n}t_{i}\cdot
V_{i}}\subsetneq V$;

  4. (4)

$a$ is $\mathcal{E}$-infinite.

Then (1)$\Leftrightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4). The implications
(1)$\Leftrightarrow$(2)$\Rightarrow$(4) hold in full generality.

###### Proof.

If (2) holds, then taking $m\mathrel{\vcentcolon=}1$,
$t_{n+1}\mathrel{\vcentcolon=}1$ and $a_{n+1}\mathrel{\vcentcolon=}\sqrt{b}$
we get (1). Conversely, if (1) holds, then there are $t_{i}\in S$, $a_{i}\in
a\mathcal{E}_{t_{i}}$ for $i=1,\dotsc,n+m$, such that
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}$, $\sum_{i=n+1}^{n+m}a_{i}^{*}a_{i}\neq 0$ and
$a_{i}^{*}a_{j}=0$ for $i\neq j$. Thus putting
$b\mathrel{\vcentcolon=}\sum_{i=n+1}^{n+m}a_{i}a_{i}^{*}$ gives (2). This
shows that (1) and (2) are equivalent.

Now let $b\in(aAa)^{+}\setminus\\{0\\}$, $n\in\mathbb{N}$, and $t_{i}\in S$,
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $i=1,\dotsc,n$ be as in (2). We identify
the fibres $\mathcal{E}_{t}$ for $t\in S$ with spaces of sections of the
associated line bundle over $G$. Put $U_{i}\mathrel{\vcentcolon=}\\{\gamma\in
G\,{:}\,\mathopen{}\lVert a_{i}(\gamma)\rVert>0\\}$ for $i=1,\dotsc,n$ and
$W\mathrel{\vcentcolon=}\\{x\in X\,{:}\,\mathopen{}\lVert
b(\gamma)\rVert>0\\}$. Then $V_{i}\mathrel{\vcentcolon=}s(U_{i})=\\{x\in
X\,{:}\,\mathopen{}(a_{i}^{*}a_{i})(x)>0\\}\subseteq V$ is contained in the
domain of $t_{i}$ for $1\leq i\leq n$. The equality
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}$ implies that
$V=\bigcup_{i=1}^{n}s(U_{i})=\bigcup_{i=1}^{n}V_{i}$. Similarly,
$a_{i}^{*}a_{j}=0$ holds if and only if $(t_{i}\cdot V_{i})\cap(t_{j}\cdot
V_{j})=r(U_{i})\cap r(U_{j})=\emptyset$ for all $i\neq j$ and $a_{i}^{*}b=0$
holds if and only if $W\subseteq
V\setminus\bigcup_{i=1}^{n}r(U_{i})=V\setminus\bigcup_{i=1}^{n}(t_{i}\cdot
V_{i})$; here we identify functions in $\mathcal{E}_{t}$ with
$\textup{C}_{0}$-functions on bisections. Such a $W$ exists if and only if
$\overline{\bigcup_{i=1}^{n}t_{i}\cdot V_{i}}\subsetneq V$. Hence (2) implies
(3).

Next we show that (3) implies (4). Let $t_{1},\dotsc,t_{n}\in S$, and
$V_{1},\dotsc,V_{n}\subseteq V$ be as in (3). Let $b\in A^{+}\setminus\\{0\\}$
be any function that vanishes outside the open set
$V\bigm{\backslash}\overline{\bigcup_{i=n+1}^{n}t_{i}\cdot V_{i}}$. Fix
$\varepsilon>0$. Let
$K\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\geq\varepsilon\\}$.
Let $w_{1},\dotsc,w_{n}\in A$ be a partition of unity subordinate to the open
covering $K\subseteq\bigcup_{i=1}^{n}V_{i}$. Let
$a_{i}\mathrel{\vcentcolon=}(a-\varepsilon)_{+}^{1/2}\cdot w_{i}^{1/2}$ for
$i=1,\dotsc,n$. These functions vanish outside $K$, and $a_{i}$ belongs to the
domain of $t_{i}$. Since $\mathcal{E}_{t_{i}}$ comes from a partial
automorphism, we may view $a_{i}$ as an element of $\mathcal{E}_{t_{i}}$. It
belongs to $a\cdot\mathcal{E}_{t_{i}}$ because the support of $a_{i}$ is
contained in $V$. The product $a_{i}^{*}a_{j}$ is defined using the Fell
bundle structure. If $i\neq j$, then $a_{i}^{*}a_{j}=0$ because $(t_{i}\cdot
V_{i})\cap(t_{j}\cdot V_{j})=\emptyset$. Similarly, we get $a_{i}^{*}b=0$. And

$\sum_{i=1}^{n}a_{i}^{*}a_{i}=\sum_{i=1}^{n}(a-\varepsilon)_{+}\cdot
w_{i}=(a-\varepsilon)_{+}\approx_{\varepsilon}a.$

Hence $a$ is $\mathcal{E}$-infinite. ∎

######  Remark 5.24.

For strongly boundary group actions (see [Laca-Spielberg:Purely_infinite])
and, more generally, for filling actions (see [Jolissaint-
Robertson:Simple_purely_infinite]) condition (3) in Lemma 5.23 holds for every
nonempty open subset $V$. Thus if $\mathcal{E}$ comes from such an action,
then every element in $A^{+}\setminus\\{0\\}$ is $\mathcal{E}$-infinite. This
also holds when $A$ is noncommutative (see [Kwasniewski-
Szymanski:Pure_infinite]*Lemma 5.12).

######  Remark 5.25.

An étale, Hausdorff, locally compact groupoid $H$ is _locally contracting_ if
for each nonempty open set $U$ in the unit space $H^{0}$ of $H$ there is a
bisection $B\subseteq H$ with $\overline{r(B)}\subsetneq s(B)\subseteq U$ (see
[Anantharaman-Delaroche:Purely_infinite]). Given a wide inverse subsemigroup
$S\subseteq\operatorname{Bis}(H)$, we may strengthen this criterion by
requiring $B\subseteq t$ for some $t\in S$. Then we may rewrite
$\overline{r(B)}\subsetneq s(B)\subseteq U$ as follows: there is $t\in S$ and
$V\subseteq U$ contained in the domain of $t$ with $\overline{t\cdot
V}\subsetneq V$. This is the case $n=1$ of condition (3) in Lemma 5.23. As a
result, if the dual groupoid $\widehat{A}\rtimes S$ is locally contracting,
then for any $a\in A^{+}\setminus\\{0\\}$ there is $0\neq a_{2}\leq a$ that is
$\mathcal{E}$-infinite; namely, choose $U=\operatorname{supp}a$ and then
$a_{2}$ with $\operatorname{supp}a_{2}=V$ and $a_{2}\leq a$ for $V$ as above.

Condition (3) in Lemma 5.23 could be relaxed so that it still implies
$\mathcal{E}$-infiniteness, by using compact subsets of $V$. We formulate the
relevant condition implying $\mathcal{E}$-proper infiniteness:

######  Lemma 5.26.

Retain the assumptions of Lemma 5.23. In particular, let $a\in A^{+}$ and
$V\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\neq 0\\}$. If
for each compact subset $K\subseteq V$ there are $n,m\in\mathbb{N}$,
$t_{1},\dotsc,t_{n+m}\in S$, and open subsets $V_{1},\dotsc,V_{n+m}\subseteq
V$ such that $(t_{i}\cdot V_{i})\cap(t_{j}\cdot V_{j})=\emptyset$ if $1\leq
i<j\leq n+m$, $K\subseteq\bigcup_{i=1}^{n}V_{i}$ and
$K\subseteq\bigcup_{i=n+1}^{n+m}V_{i}$, then $a$ is $\mathcal{E}$-properly
infinite.

###### Proof.

Fix $\varepsilon>0$. Let
$K\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\geq\varepsilon\\}$.
Choose $n,m,t_{i},V_{i}$ as in the assumption of the lemma. Let
$w_{1},\dotsc,w_{n}\in A$ and $w_{n+1},\dotsc,w_{n+m}\in A$ be partitions of
unity subordinate to the open coverings $K\subseteq\bigcup_{i=1}^{n}V_{i}$ and
$K\subseteq\bigcup_{i=n+1}^{n+m}V_{i}$, respectively. Let
$a_{i}\mathrel{\vcentcolon=}(a-\varepsilon)_{+}^{1/2}\cdot w_{i}^{1/2}$ for
$i=1,\dotsc,n+m$. As in the proof of the implication (3)$\Rightarrow$(4) in
Lemma 5.23 one sees that treating $a_{i}$ as an element of
$\mathcal{E}_{t_{i}}$, the elements $a_{i}$ satisfy the relations in
Definition 5.18.(3). ∎

Now we assume, in addition, that the spectrum $\widehat{A}$ is totally
disconnected. This implies that the compact open bisections form a basis for
the topology and that $A$ is spanned by projections. We are going to see that
a projection is $\mathcal{E}$-paradoxical if and only if its support is
$(2,1)$-paradoxical as defined in [Boenicke-Li:Ideal]. Such open subsets give
purely infinite elements in the type semigroup considered in [Boenicke-
Li:Ideal, Rainone-Sims:Dichotomy, Ma:Purely_infinite_groupoids].

######  Definition 5.27 ([Boenicke-Li:Ideal]).

Let $G$ be an ample groupoid. We say that a compact open set $V\subseteq
G^{0}$ is _$(2,1)$ -paradoxical_ if there are $n,m\in\mathbb{N}$ and compact
open bisections $U_{i}\subseteq G$ for $1\leq i\leq n+m$ such that
$r(U_{i})\subseteq V$ for $1\leq i\leq n+m$ and

$V=\bigsqcup_{i=1}^{n}s(U_{i}),\qquad V=\bigsqcup_{i=n+1}^{n+m}s(U_{i}),\qquad
r(U_{i})\cap r(U_{j})=\emptyset\quad\text{for }i\neq j.$

######  Proposition 5.28.

Let $\mathcal{E}$ be an action of an inverse semigroup $S$ by Hilbert
bimodules on a commutative $\textup{C}^{*}$-algebra $A$ with totally
disconected spectrum $\widehat{A}$; equivalently, the dual groupoid
$G\mathrel{\vcentcolon=}\widehat{A}\rtimes S$ is ample. A projection $a\in
A^{+}$ is $\mathcal{E}$-paradoxical if and only if its support
$V\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\neq 0\\}$ is
$(2,1)$-paradoxical.

###### Proof.

Suppose first that $a\in A^{+}\setminus\\{0\\}$ is $\mathcal{E}$-paradoxical.
That is, there are $n,m\in\mathbb{N}$, $t_{1},\dotsc,t_{n+m}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ such that
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}=\sum_{i=n+1}^{n+m}a_{i}^{*}a_{i}$ and
$a_{i}^{*}a_{j}=0$ for $i\neq j$. Let $1\leq i\leq n+m$. Recall that we may
treat $\mathcal{E}_{t_{i}}$ as spaces of sections $A_{U_{i}}$ of a line bundle
over $G=\widehat{A}\rtimes S$ that are supported on open bisections
$U_{i}\in\operatorname{Bis}(G)$. Thus $U_{i}\mathrel{\vcentcolon=}\\{\gamma\in
G\,{:}\,\mathopen{}\lVert a_{i}(\gamma)\rVert>0\\}$ is an open bisection of
$G$ contained in $U_{i}$. Since $a_{i}\in aA_{U_{i}}$, we have
$r(U_{i})=\\{x\in\widehat{A}\,{:}\,\mathopen{}(a_{i}a_{i}^{*})(x)>0\\}\subseteq
V$. And $a_{i}^{*}a_{j}=0$ implies that $r(U_{i})\cap r(U_{j})=\emptyset$ for
all $i\neq j$. Since $\\{x\in
X\,{:}\,\mathopen{}(a_{i}^{*}a_{i})(x)>0\\}=s(U_{i})$, the equalities
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}$ and $a=\sum_{i=n+1}^{n+m}a_{i}^{*}a_{i}$
imply $V=\bigsqcup_{i=1}^{n}s(U_{i})$ and $V=\bigsqcup_{i=n+1}^{n+m}s(U_{i})$.
Hence the family $U_{i}\in\operatorname{Bis}(G)$ for $1\leq i\leq n+m$ has all
the desired properties, except that $U_{i}$ need not be compact. However,
since $G$ is ample, every $U_{i}$ is a union of some compact open bisections.
Since $V$ is compact and
$V=\bigsqcup_{i=1}^{n}s(U_{i})=\bigsqcup_{i=n+1}^{n+m}s(U_{i})$, we may, in
fact, replace each $U_{i}$ for $1\leq i\leq n+m$ by a finite union of compact
open bisections. This gives a compact open bisection.

Conversely, let $U_{i}\subseteq G$ for $1\leq i\leq n+m$ be a family of
bisection as in Definition 5.27. Let
$S^{\prime}\subseteq\operatorname{Bis}(G)$ be the family of open compact
bisections that trivialise the twist, that is, the restrictions of the
associated line bundle over $G$ to sets in $S^{\prime}$ are trivial. Note that
$S^{\prime}$ forms an inverse semigroup and a basis for the topology of $G$;
this holds for the family of all open bisections that trivialise the twist, by
the proof of [BussExel:Regular.Fell.Bundle]*Theorem 7.2, and for the family of
all compact open bisections because $G$ is ample. Since
$V=\bigsqcup_{i=1}^{n}s(U_{i})=\bigsqcup_{j=n+1}^{n+m}s(U_{j})$ is compact,
for each $i=1,\dotsc,n$ we may find a finite family of sets
$(U_{i,j})_{j=1}^{n_{i}}\subseteq S^{\prime}$ such that
$\bigcup_{j=1}^{n_{i}}U_{i,j}\subseteq U_{i}$ and
$V=\bigcup_{i=1}^{n}\bigcup_{j=1}^{n_{i}}s(U_{i,j})$. Since the bisections
$(U_{i,j})_{j=1}^{n_{i}}\subseteq U_{i}$ are closed and open, we may arrange
that the sets $(s(U_{i,j}))_{j=1}^{n_{i}}$ are pairwise disjoint. Then the
sets $(U_{i,j})_{i=1,j=1}^{n,n_{i}}$ are pairwise disjoint, and since
$\bigcup_{j=1}^{n_{i}}U_{i,j}\subseteq U_{i}$, for $i=1,\dotsc,n$, also
$(r(U_{i,j}))_{i=1,j=1}^{n,n_{i}}$ are pairwise disjoint. We put
$a_{i,j}\mathrel{\vcentcolon=}1_{U_{i,j}}$, for $i=1,\dots,n$,
$j=1,\dotsc,n_{i}$. By the choice of bisections in $S^{\prime}$, we may treat
$a_{i,j}$ as an element of the space $\textup{C}_{\textup{c}}(U_{ij})$ of
sections of the line bundle over $G$. By the construction of the Fell bundle
over $\widehat{A}\rtimes S$, by passing if necessary to smaller sets, we may
assume that each space $\textup{C}_{\textup{c}}(U_{i,j})$ is contained in
$\mathcal{E}_{t_{ij}}$ for some $t_{ij}\in S$. Hence
$a_{i,j}\in\mathcal{E}_{t_{ij}}$ for all $i,j$. Using the Fell bundle
structure, we get

$\sum_{i=1,j=1}^{n,n_{i}}a_{i,j}^{*}\cdot
a_{i,j}=\sum_{i=1,j=1}^{n,n_{i}}1_{s(U_{i,j})}=1_{V}=a.$

Similarly, we get $\sum_{i=n+1,j=1}^{n+m,n_{i}}a_{i,j}^{*}\cdot a_{i,j}=a$ and
$a_{i,j}^{*}a_{i^{\prime},j^{\prime}}=0$ for all
$(i,j)\neq(i^{\prime},j^{\prime})$. Hence $a$ is $\mathcal{E}$-paradoxical. ∎

######  Corollary 5.29.

Let $(G,\Sigma)$ be an essentially exact twisted groupoid where $G$ is ample
and residually topologically free with locally compact Hausdorff
$X\mathrel{\vcentcolon=}G^{0}$. If every compact open subset of $X$ is
$(2,1)$-paradoxical, then the essential $\textup{C}^{*}$-algebra
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ is purely infinite (and has the
ideal property).

###### Proof.

View $\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ as the essential crossed
product by an inverse semigroup action $\mathcal{E}$ on $\textup{C}_{0}(X)$ as
in Examples 3.21 and 3.22. The assertion follows from Proposition 5.28 and
Corollary 5.22. ∎

######  Remark 5.30.

When $G$ is Hausdorff, then
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)=\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)$
and $(G,\Sigma)$ is essentially exact if and only if it is inner exact. Thus
Corollary 5.29 generalises the pure infiniteness criteria in [Boenicke-
Li:Ideal, Rainone-Sims:Dichotomy], where the authors considered Hausdorff
ample groupoids without a twist. They proved, in addition, that if the type
semigroup associated to $G$ is almost unperforated, then the implication in
Corollary 5.29 may be reversed. We will generalise this and some other results
of Ma [Ma:Purely_infinite_groupoids] to étale twisted groupoids in the
forthcoming paper [Kwasniewski-Meyer:Type_semigroups].

## References


$S^{\prime}$-locally equivalent, i.e., the map $D_{n}^{S}F\to
D_{n}^{S^{\prime}}F$ is an $S^{\prime}$-local equivalence. For (2), since $F$
is reduced [We95, Corollary 8.3] implies that there is a commutative diagram

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

in which both rows are homotopy fibre sequences. The map $R_{n}^{S}F\to
R_{n}^{S^{\prime}}F$ is an $S^{\prime}$-local equivalence by part (1), and the
map $T_{0}^{S}F\to T_{0}^{S^{\prime}}F$ is also an $S^{\prime}$-local
equivalence since $F$ is reduced. An induction argument on the degree of
polynomials yields the result. ∎

We obtain corollaries for both stable and unstable Bousfield classes.

######  Lemma 9.1.6.

Let $E$ and $E^{\prime}$ be spectra and $F$ an orthogonal functor. If $\langle
E\rangle\leq\langle E^{\prime}\rangle$, then

  1. (1)

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

  2. (2)

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

######  Lemma 9.1.7.

Let $W$ and $W^{\prime}$ be based spaces and $F$ an orthogonal functor. If
$\langle W\rangle\leq\langle W^{\prime}\rangle$, then

  1. (1)

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

  2. (2)

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

####  9.2. The Telescope Conjecture

The height $n$ Telescope Conjecture relates the $T(n)$-localization and
$K(n)$-localization of spectra. There are numerous equivalent formalisations
of the conjecture see e.g., [BarthelChromaticConjectures, Proposition 3.6] and
we choose the following as it best suits any possible interaction with the
calculus.

######  Conjecture 9.2.1 (The height $n$ Telescope Conjecture).

Let $n\geq 0$. The Bousfield class of $T(n)$ agrees with the Bousfield class
of $K(n)$.

######  Lemma 9.2.2.

Let $n\geq 0$. The validity of the height $n$ Telescope Conjecture implies an
equality of model structures

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

###### Proof.

The Telescope Conjecture implies that the Bousfield class of $T(n)$ and the
Bousfield class of $K(n)$, agree, hence the result follows by Theorem 9.1.4. ∎

The following is an immediate corollary to Theorem 9.1.1.

######  Theorem 9.2.3.

Let $n\geq 0$. The height $n$ Telescope Conjecture is equivalent to the
statement that for every orthogonal functor $F$ the $K(n)$-local Weiss tower
of $F$ and the $T(n)$-local Weiss tower of $F$ agree.

This provides new insight into the the height $n$ Telescope Conjecture. For
example, to find a counterexample it now suffices to find an orthogonal
functor such that one corresponding term in the $K(n)$-local and $T(n)$-local
Weiss towers disagree. This can also be seen through the spectral sequences
associated to the local Weiss towers. The $K(n)$-local and $T(n)$-local Weiss
towers of an orthogonal functor $F$ produce two spectral sequences,

$\pi_{t-s}D_{s}^{K(n)}F(V)\cong\pi_{t-s}((S^{\mathbb{R}^{s}\otimes
V}\wedge\partial_{s}^{K(n)}F)_{hO(n)})\Rightarrow\pi_{\ast}\underset{d}{\operatorname{\mathrm{holim}}}~{}T_{d}^{K(n)}F(V),$

and,

$\pi_{t-s}D_{s}^{T(n)}F(V)\cong\pi_{t-s}((S^{\mathbb{R}^{s}\otimes
V}\wedge\partial_{s}^{T(n)}F)_{hO(n)})\Rightarrow\pi_{\ast}\underset{d}{\operatorname{\mathrm{holim}}}~{}T_{d}^{T(n)}F(V),$

These are closely related to the telescope conjecture as follows.

######  Lemma 9.2.4.

Let $F$ be an orthogonal functor. If the height $n$ Telescope Conjecture
holds, then for all $r\geq 1$, the $E_{r}$-page of the $T(n)$-local Weiss
spectral sequence is isomorphic to the $E_{r}$-page of the $K(n)$-local Weiss
spectral sequence. In particular, the homotopy limit of the $T(n)$-local Weiss
tower is levelwise weakly equivalent to the homotopy limit of the $K(n)$-local
Weiss tower.

###### Proof.

It suffices to prove the claim for $r=1$. The validity of the height $n$
Telescope Conjecture implies that there is a natural transformation
$L_{K(n)}\to L_{T(n)}$. This natural transformation induces a map
$D_{d}^{K(n)}F\to D_{d}^{T(n)}F$, which by Theorem 9.2.3 is an levelwise weak
equivalence. It hence suffices to show that the natural map $D_{d}^{K(n)}F\to
D_{d}^{T(n)}F$ induces a map on the $E_{1}$-pages of the spectral sequences,
that is, we have to show that the induced diagram

${{\pi_{t-s}D_{s}^{K(n)}F(V)}}$${{\pi_{t-s+1}D_{s+1}^{K(n)}F(V)}}$${{\pi_{t-s}D_{s}^{T(n)}F(V)}}$${{\pi_{t-s+1}D_{s+1}^{T(n)}F(V)}}$$\scriptstyle{d_{1}^{K(n)}}$$\scriptstyle{d_{1}^{T(n)}}$

commutes for all $s$ and $t$. This follows from the commutativity of the
induced diagram of long exact sequences induced by the diagram of homotopy
fibre sequences,

${{D_{s}^{K(n)}F(V)}}$${{T_{s}^{K(n)}F(V)}}$${{T_{s-1}^{K(n)}F(V)}}$${{D_{s}^{T(n)}F(V)}}$${{T_{s}^{T(n)}F(V)}}$${{T_{s-1}^{T(n)}F(V)}}$

and the construction of the $d_{1}$-differential in the homotopy spectral
sequence associated to a tower of fibrations. ∎

###  10\. Postnikov sections

The classical theory of Postnikov sections of based spaces is obtained by the
nullification with respect to the spheres, that is, given a based space $A$,
the $k$-th Postnikov section of $A$ is the nullification of $A$ at $S^{k+1}$,
i.e., $P_{k}A=P_{S^{k+1}}A$. Given a diagram of (simplicial, left proper,
combinatorial) model categories, Barwick [BarwickLeftRight, Section 5
Application 1] and Bergner [BergnerHomotopyLimits] develop a general machinery
for producing a model structure which captures the homotopy theory of the
homotopy limit of the diagram of model categories. Gutiérrez and Roitzheim
[GR16, Section 4] applied this to the study of Postnikov sections for model
categories, which recovers the classical theory when $\mathcal{C}$ is the Kan-
Quillen model structure on simplicial sets. We consider the relationship
between Postnikov sections and orthogonal calculus via our local calculus.

####  10.1. A combinatorial model for calculus

The current theory of homotopy limits of model categories requires that the
model categories in question be combinatorial, i.e., locally presentable and
cofibrantly generated. Since the category of based compactly generated weak
Hausdorff spaces is not locally presentable the Quillen model structure is not
combinatorial and hence none of our model categories for orthogonal functors
are either. We invite the reader to take for granted that all of our cellular
model categories may be replaced by combinatorial model categories by starting
with a combinatorial model for the Quillen model structure on based spaces,
and hence skip directly to Subsection 10.2.

We spell out the details of these combinatorial replacements here. We replace
compactly generated weak Hausdorff spaces with _$\Delta$ -generated spaces_; a
particular full subcategory of the category of topological spaces, which were
developed by Vogt [VogtConvenientHomotopy] and unpublished work of Smith,
which are surveyed by Dugger in [DuggerDeltaGeneratedSpaces]. The category of
$\Delta$-generated spaces may be equipped with a model structure analogous to
the Quillen model structure on compactly generated weak Hausdorff spaces.

######  Lemma 10.1.1.

There is a model category structure on the category of $\Delta$-generated
spaces with weak equivalences the weak homotopy equivalences and fibrations
the Serre fibrations. This model structure is combinatorial, proper and
topological.

###### Proof.

The existence of the model structure follows from [DuggerDeltaGeneratedSpaces,
Subsection 1.9]. The locally presentable (and hence combinatorial) property
follows from [FRDeltaGeneratedSpaces, Corollary 3.7] . ∎

The category of based $\Delta$-generated spaces is a convenient model category
for doing homotopy theory in the following sense, see
[DuggerDeltaGeneratedSpaces, Subsection 1.9].

######  Lemma 10.1.2.

The model category of based $\Delta$-generated spaces is Quillen equivalent to
the Quillen model structure on based compactly generated weak Hausdorff
spaces.

The combinatorial model for spaces transfers to categories of functors and we
obtain a projective model structure on the category of orthogonal functors
which is Quillen equivalent to our original projective model structure but is
now combinatorial.

A left or right Bousfield localization of a combinatorial model category is
again combinatorial, hence the $n$-polynomial, $n$-homogeneous and local
versions of these model categories are all combinatorial when we begin with
the combinatorial model for the projective model structure on orthogonal
fucntors.

######  Hypothesis 10.1.3.

For the remainder of this section, we will assume that all our model
structures are combinatorial, since they are all Quillen equivalent to
combinatorial model categories using the combinatorial model for based spaces.

####  10.2. The model structure of $k$-types in orthogonal functors

Denote by $I$ the set of generating cofibrations of the projective model
structure of orthogonal functors, and denote by $W_{k}$ the set
$I\Box\\{S^{k+1}\to D^{k+2}\\}$, that is, the set of maps of the form

$B\wedge S^{k+1}\coprod_{A\wedge S^{k+1}}A\wedge D^{k+2}\longrightarrow
B\wedge D^{k+2},$

where $A\to B$ is a map in $I$. The model category of $k$-types in
$\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ is the left
Bousfield localization of the projective model structure at
$I\Box\\{S^{k+1}\to D^{k+2}\\}$ used by Gutiérrez and Roitzheim [GR16] to
model Postnikov sections.

######  Proposition 10.2.1.

Let $k\geq 0$. Under Hypothesis 10.1.3, the model structure of $k$-types in
orthogonal functors is identical to the $S^{k+1}$-local model structure, that
is, there is an equality of model structures,

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

###### Proof.

It suffices to show that both model structures have the same fibrant objects
since the cofibrations in both model structures are identical. To see this,
note that by examining the pushout product we can rewrite the set $W_{k}$ as

$W_{k}=\\{\mathcal{J}_{0}(U,-)\wedge
S^{n+k+1}_{+}\longrightarrow\mathcal{J}_{0}(U,-)\wedge D^{n+k+2}_{+}\mid n\geq
0,U\in\mathcal{J}_{0}\\}.$

It follows by an adjunction argument that an orthogonal functor $Z$ is
$W_{k}$-local if and only if $\pi_{i}Z(U)$ is trivial for all $i\geq k+1$ and
all $U\in\mathcal{J}_{0}$. This last condition is equivalent to being
levelwise $S^{k+1}$-local. ∎

####  10.3. The model structure of $k$-types in spectra

######  Proposition 10.3.1.

Let $k\geq 0$. Under Hypothesis 10.1.3, there is an equality of model
structures between the model category of $k$-types in spectra, and the
stablisation of $S^{k+1}$-local spaces, that is,

$P_{k}{\sf{Sp}}:=L_{W_{k}}{\sf{Sp}}={\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}).$

###### Proof.

Both model structures can be described as particular left Bousfield
localizations of the stable model structure on spectra, hence have the same
cofibrations. The proof reduces to the fact that the model structures have the
same fibrant objects. To see this, note that the fibrant objects of
$P_{k}{\sf{Sp}}$ are the $k$-truncated $\Omega$-spectra, and the fibrant
objects of ${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ are the
levelwise $k$-truncated $\Omega$-spectra. Since both fibrant objects are
$\Omega$-spectra a connectivity style argument yields that an
$\Omega$-spectrum is $k$-truncated if and only if it is levelwise
$k$-truncated, and hence both model structures have the same fibrant objects.
∎

####  10.4. Postnikov reconstruction of orthogonal functors

The collection of $S^{k+1}$-local model structures on the category of
orthogonal functors assembles into a tower of model categories666A tower of
model categories is a special instance of a left Quillen presheaf, that is a
diagram of the form $F\colon\mathcal{J}^{\mathrm{op}}\to\sf{MCat}$ for some
small indexing category $\mathcal{J}$.

$\displaystyle{\sf{P}}_{\bullet}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto{\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}),$

where $\sf{MCat}$ denotes the category of model categories and left Quillen
functors. The homotopy limit of this tower of model categories recovers the
projective model structure on orthogonal functors. The existence of a model
structure which captures the homotopy theory of the limit of these model
categories follows from [GR16, Proposition 2.2]. In particular, the homotopy
limit model structure is a model structure on the category of sections777A
section $X_{\bullet}$ of the tower ${\sf{P}}_{\bullet}$ is a sequence
$\cdots\longrightarrow X_{k}\longrightarrow
X_{k+1}\longrightarrow\cdots\longrightarrow X_{0},$ of orthogonal functors,
and a morphism of sections $f\colon X_{\bullet}\to Y_{\bullet}$ is given by
maps of orthogonal functors $f_{k}\colon X_{k}\to Y_{k}$ for all $k\geq 0$
subject to a commutative ladder condition. of the diagram ${\sf{P}}_{\bullet}$
formed by right Bousfield localizing the injective model structure in which a
map of sections is a weak equivalence or cofibration if it is a levelwise weak
equivalence or cofibration respectively.

######  Lemma 10.4.1 ([GR16, Theorem 1.3 $\&$ Proposition 2.2]).

There is a combinatorial model structure on the category of sections of
${\sf{P}}_{\bullet}$ where a map $f_{\bullet}\colon X_{\bullet}\to
Y_{\bullet}$ is a fibration if and only if $f_{0}$ is a fibration in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{1}}\operatorname{\sf{Top}_{\ast}})$ and for
every $k\geq 1$ the induced map

${{X_{k}}}$${{Y_{k}\times_{Y_{k-1}}X_{k-1}}}$${{X_{k-1}}}$${{Y_{k}}}$${{Y_{k-1}}}$$\scriptstyle{f_{k-1}}$$\scriptstyle{f_{k}}$

indicated by a dotted arrow in the above diagram is a fibration in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$. A
section $X_{\bullet}$ is cofibrant if and only if $X_{n}$ is cofibrant in
${\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ and for every
$k\geq 0$, the map $X_{k+1}\to X_{k}$ is a weak equivalence in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$. A map
of cofibrant sections is a weak equivalence if and only if the map is a weak
equivalence in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for
each $k\geq 0$. We will refer to this model structure as the homotopy limit
model structure and denote it by
$\operatorname{\mathrm{holim}}{\sf{P}}_{\bullet}$.

######  Proposition 10.4.2.

Under Hypothesis 10.1.3 the adjoint pair

$\textstyle{\rm{const}:{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}{\sf
P}_{\bullet}:\lim}$

is a Quillen equivalence.

###### Proof.

The adjoint pair exists, and is a Quillen adjunction by [GR16, Lemma 2.4].

To see that the adjoint pair is a Quillen equivalence let $X_{\bullet}$ be a
cofibrant and fibrant section in the homotopy limit model structure. Showing
that $\rm{const}\lim X_{\bullet}\to X_{\bullet}$ is a weak equivalence is
equivalent to showing that the map $\lim X_{\bullet}\to X_{k}$ is a weak
equivalence in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for
all $k\geq 0$. This is in turn, equivalent to the map $(\lim
X_{\bullet})(U)\to X_{k}(U)$ being a weak equivalence in
$L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}$ for all $k\geq 0$. Since limits in
functor categories are computed levelwise, the fact that the unit is a weak
equivalence follows from [GR16, Theorem 2.5]. A similar argument, shows that
the counit is also a weak equivalence. ∎

####  10.5. Postnikov reconstruction for spectra with an $O(n)$-action

The aim is to show that similar reconstruction theorems may be obtained for
the $n$-homogeneous functors. We first start by investigating analogous
theorems for spectra and show that such reconstructions are compatible with
the zigzag of Quillen equivalences between spectra with an $O(n)$-action and
the $n$-homogeneous model structure.

######  Lemma 10.5.1.

The functor

$\displaystyle{\sf{P}}_{\bullet}^{{\sf{Sp}}}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto{\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}),$

defines a left Quillen presheaf.

###### Proof.

This follows from Proposition 10.3.1 and [GR16, Subsection 2.1] since the
stablisation of $S^{k+1}$-local spaces is precisely the model structure of
$k$-types in spectra. ∎

######  Remark 10.5.2.

Alternatively Lemma 10.5.1 may be proved by exhibiting that the adjoint pair

$\textstyle{\mathds{1}:{\sf{Sp}}(L_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$

is a Quillen adjunction. This fact follows from the facts that both model
structures have the same cofibrations and a $S^{k+1}$-local space is
$S^{k+2}$-local as $\langle\Sigma W\rangle\leq\langle W\rangle$ for all based
spaces $W$, see e.g., [BousfieldPeriodicity, §9.9].

This left Quillen presheaf is ‘convergent’ in the following sense.

######  Proposition 10.5.3.

Under Hypothesis 10.1.3 the adjoint pair

$\textstyle{\rm{const}:{\sf{Sp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}}:\lim}$

is a Quillen equivalence.

###### Proof.

The fact that the adjoint pair is a Quillen adjunction follows from [GR16,
Lemma 2.4].

The left adjoint reflects weak equivalences between cofibrant objects. Indeed,
if $X\to Y$ is a map between cofibrant spectra $X$ and $Y$, such that

$\mathrm{const}(X)\longrightarrow\mathrm{const}(Y),$

is a weak equivalence in
$\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}}$, then

$\mathrm{const}(X)\longrightarrow\mathrm{const}(Y),$

is a weak equivalence in
$\mathrm{Sect}(\mathbb{N},{\sf{P}}_{\bullet}^{{\sf{Sp}}})$ by the colocal
Whitehead’s theorem and the fact that the left adjoint is left Quillen and
thus preserves cofibrant objects. It follows that for each $k\in\mathbb{N}$,
the induced map

$\mathrm{const}(X)_{k}\longrightarrow\mathrm{const}(Y)_{k},$

is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$, that is, $X\to Y$ is a
weak equivalence in ${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for
all $k$. Unpacking the definition of a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ and using the fact that
the right adjoint is a right Quillen functor and hence preserves weak
equivalences between fibrant objects, we see that the induced map

$\lim~{}P_{k}X\longrightarrow\lim~{}P_{k}Y,$

is a weak equivalence in ${\sf{Sp}}$, and hence, so is the map $X\to Y$.

It is left to show that the derived counit is an isomorphism. Let
$Y_{\bullet}$ be bifibrant in
$\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}}$. The
condition that the counit applied to $Y_{\bullet}$ is a weak equivalence is
equivalent to asking for the map

$\lim_{\geq k}~{}P_{k}Y_{\bullet}\longrightarrow Y_{k},$

to be a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for all
$k\in\mathbb{N}$. The structure maps of $Y_{\bullet}$ induce a map of towers

${\cdots}$${{Y_{j}}}$${\cdots}$${{Y_{k+3}}}$${{Y_{k+2}}}$${{Y_{k+1}}}$${\cdots}$${{Y_{k+1}}}$${\cdots}$${{Y_{k+1}}}$${{Y_{k+1}}}$${{Y_{k+1}}}$$\scriptstyle{=}$

in which each vertical arrow is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$. This map of towers
induces a map

${0}$${{\lim^{1}_{\geq k}~{}\pi_{i+1}(Y_{\bullet})}}$${{\pi_{i}(\lim_{\geq
k}~{}Y_{\bullet})}}$${{\lim_{\geq
k}~{}\pi_{i}(Y_{\bullet})}}$${0}$${0}$${{\lim^{1}_{\geq
k}~{}\pi_{i+1}(Y_{k+1})}}$${{\pi_{i}(\lim_{\geq k}~{}Y_{k+1})}}$${{\lim_{\geq
k}~{}\pi_{i}(Y_{k+1})}}$${0}$

of short exact sequences. For $0\leq i<n$ the left and right hand side maps
are isomorphisms hence the map

$\lim_{\geq k}~{}Y_{\bullet}\longrightarrow Y_{k+1},$

is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for all $k$, and it
follows that the required map

$\lim_{\geq k}~{}Y_{\bullet}\longrightarrow Y_{k+1}\longrightarrow Y_{k},$

is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for all $k$. ∎

A similar justification to Lemma 10.5.1 provides a left Quillen presheaf

$\displaystyle{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto{\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})[O(n)],$

where ${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})[O(n)]$ is the
category of $O(n)$-objects in the category of $k$-types in spectra. This is
equivalent to the category of $k$-types in spectra with an $O(n)$-action.

######  Lemma 10.5.4.

Let $k,n\geq 0$. The model structure of the Borel stablisation of
$S^{k+1}$-local spaces with an $O(n)$-action is identical to the model
structure of $k$-types in the category of spectra with an $O(n)$-action, that
is, there is an equality of model structures

${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})[O(n)]=P_{k}({\sf{Sp}}[O(n)]).$

###### Proof.

Both model structures are identical to model structures transferred through
the same adjunction from identical model structures. ∎

As a corollary to Proposition 10.5.3, we obtain that the induced left Quillen
presheaf on spectra with an $O(n)$-action is also suitably convergent.

######  Corollary 10.5.5.

Under Hypothesis 10.1.3 the adjoint pair

$\textstyle{\rm{const}:{\sf{Sp}}[O(n)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}:\lim}$

is a Quillen equivalence.

####  10.6. Postnikov reconstruction for the intermediate categories

Our attention now turns to the intermediate categories. We construct an
analogous left Quillen presheaf and show that it is also convergent in a
fashion which interacts well with the convergent left Quillen presheaf for
spectra with an $O(n)$-action.

######  Lemma 10.6.1.

The functor

$\displaystyle{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto
L_{S^{k+1}}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}}),$

defines a left Quillen presheaf.

###### Proof.

As before, it suffices to show that there is an equality of model structures
between the $S^{k+1}$-local $n$-stable model structure and the model structure
of $k$-types in
${\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$. The
proof of which is completely analogous to the case for spectra, see Lemma
10.5.1. ∎

######  Remark 10.6.2.

Since the $S^{k+1}$-local $n$-stable model structure agrees with the model
structure of $k$-types, we will denote both model structure by
$P_{k}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$.

The homotopy limit of the left Quillen presheaf of Lemma 10.6.1 agrees with
the homotopy limit of the left Quillen presheaf of Lemma 10.5.1, in the sense
that the homotopy limit model categories are Quillen equivalent. In detail,
the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
59.48582pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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83.48582pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.48582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Sp}}[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.48584pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

of [BO13, §8] induces an adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
75.50989pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}^{\mathbb{N}}:\sf{Fun}(\mathbb{N},{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
99.50989pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
99.50989pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\sf{Fun}(\mathbb{N},{\sf{Sp}}[O(n)]):(\alpha_{n}^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
75.5099pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

where $(\alpha_{n}^{*})^{\mathbb{N}}=(\alpha_{n})^{*}\circ(-)$. This
adjunction in turn induces an adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.62474pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}^{\mathbb{N}}:\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.62474pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
56.62474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}:(\alpha_{n}^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.62476pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
######  Proposition 10.6.3.

Under Hypothesis 10.1.3 the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.62474pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}^{\mathbb{N}}:\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.62474pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
56.62474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}:(\alpha_{n}^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.62476pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence.

###### Proof.

Fibrations of the homotopy limit model structure of
${\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}$ are precisely the fibrations of the
injective model structure on the category of sections of
${\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}$ since the homotopy limit model
structure is a right Bousfield localization of the injective model structure.
A similar characterisation holds for the left Quillen presheaf
${\sf{P}}_{\bullet}^{\mathcal{J}_{n}}$, hence to show that the right adjoint
preserves fibrations it suffices to show that the left adjoint preserves
acyclic cofibrations of the injective model structure on the categories of
sections. To see this, note that the adjunction

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

is a Quillen adjunction and hence, so to is the induced adjunction on the
injective model structures on the categories of sections.

To show that the left adjoint preserves cofibrations it suffices to show that
cofibrations between cofibrant objects are preserved. As the homotopy limit
model structures are right Bousfield localizations [Hi03, Proposition
3.3.16(2)] implies that cofibrations between cofibrant objects are
cofibrations of the injective model structures on the categories of sections
which by the analogous reasoning as above are preserved by the left adjoint.
This yields that the adjunction in question is a Quillen adjunction.

To show that the adjunction is a Quillen equivalence notice that the right
adjoint reflects weak equivalences between cofibrant objects by the colocal
Whitehead’s Theorem [Hi03, Theorem 3.2.13(2)], and the fact that the induced
adjunction on the injective model structures on the categories of sections is
a Quillen equivalence since for
$B_{\bullet}\in{\sf{Sect}}(\mathbb{N},{\sf{P}}_{\bullet}^{\mathcal{J}_{n}})$
and
$X_{\bullet}\in{\sf{Sect}}(\mathbb{N},{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]})$,
a map $B_{\bullet}\to(\alpha_{n}^{*})^{\mathbb{N}}X_{\bullet}$ is a weak
equivalence if and only if for each $k\in\mathbb{N}$, the map
$B_{k}\to(\alpha_{n}^{*})^{\mathbb{N}}X_{k}$ is a weak equivalence of spectra,
which in turn happens if and only if the adjoint map $(\alpha_{n})_{!}B_{k}\to
X_{k}$ is an $n$-stable equivalence, which is precisely the condition that the
adjoint map $(\alpha_{n})_{!}^{\mathbb{N}}B_{\bullet}\to X_{\bullet}$ is a
weak equivalence.

It is left to show that the derived counit is an isomorphism. Let
$Y_{\bullet}$ be bifibrant in the homotopy limit model structure of the left
Quillen presheaf ${\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}$. Then the derived
counit

$(\alpha_{n})_{!}^{\mathbb{N}}~{}\widehat{c}~{}((\alpha_{n}^{*})^{\mathbb{N}}Y_{\bullet})\longrightarrow
Y_{\bullet},$

is a map between cofibrant objects, hence a weak equivalence in the homotopy
limit model structure if and only if a weak equivalence in the injective model
structure on the category of sections i.e., if and only if for each
$k\in\mathbb{N}$, the induced map

$(\alpha_{n})_{!}(\alpha_{n})^{*}Y_{k}\longrightarrow Y_{k},$

is a weak equivalence. This last is always a weak equivalence by [BO13,
Proposition 8.3]. ∎

As a corollary, we see that the left Quillen presheaf
${\sf{P}}_{\bullet}^{\mathcal{J}_{n}}$ is convergent.

######  Corollary 10.6.4.

Under hypothesis 10.1.3 the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
61.35667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{const}:{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
85.35667pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
85.35667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}:\lim}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
61.35667pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence.

###### Proof.

Consider the commutative diagram

${{{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{{\sf{Sp}}[O(n)]}}$${{\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}}}$${{\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}}}$$\scriptstyle{(\alpha_{n})_{!}}$$\scriptstyle{(\alpha_{n})^{\ast}}$$\scriptstyle{\rm{const}}$$\scriptstyle{\lim}$$\scriptstyle{\rm{const}}$$\scriptstyle{\lim}$$\scriptstyle{(\alpha_{n})_{!}^{\mathbb{N}}}$$\scriptstyle{(\alpha_{n}^{\ast})^{\mathbb{N}}}$

of Quillen adjunctions in which three out of the four adjoint pairs are
Quillen equivalences by [BO13, Proposition 8.3], Corollary 10.5.5 and
Proposition 10.6.3. It follows since Quillen equivalences satisfy the $2$-out-
of-$3$ property, that the remaining Quillen adjunction is a Quillen
equivalence. ∎

####  10.7. Postnikov reconstruction for homogeneous functors

Using the same approach as we have just employed from moving from spectra with
an $O(n)$-action to the intermediate categories we obtain similar results for
the homogeneous model structures. We choose to model $S^{k+1}$-local
$n$-homogeneous functors by the $S^{k+1}$-periodic $n$-homogeneous model
structures of Proposition 8.3.1.

######  Lemma 10.7.1.

The functor

$\displaystyle{\sf{P}}_{\bullet}^{\mathsf{Homog}^{n}}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longrightarrow{\mathsf{Homog}^{n}}(\mathcal{J}_{0},P_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}),$

defines a left Quillen presheaf.

###### Proof.

It suffices to show that the adjoint pair

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
52.64969pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.64969pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.64969pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
52.64969pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen adjunction. The adjoint pair

$\textstyle{\mathds{1}:\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$

is a Quillen adjunction since the composite of Quillen adjunctions is a
Quillen adjunction so the adjunction

$\textstyle{\mathds{1}:{\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$

is a Quillen adjunction, and by [Hi03, Proposition 3.3.18(1) $\&$ Theorem
3.1.6(1)], this composite Quillen adjunction extends to the $S^{k+2}$-local
$n$-polynomial model structure since $S^{k+1}$-local $n$-polynomial functors
are $S^{k+2}$-locally $n$-polynomial.

An application of [Hi03, Theorem 3.3.20(2)(a)] yields the desired result about
the $n$-homogeneous model structures. ∎

Similar proofs to Proposition 10.6.3 and Corollary 10.6.4 yeild the following
results relating the $n$-homogeneous model structure to the homotopy limit of
the tower of $S^{k+1}$-local $n$-homogeneous model structures.

######  Proposition 10.7.2.

Under Hypothesis 10.1.3 the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
51.07452pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\operatorname{\mathsf{res}}_{0}^{n}/O(n))^{\mathbb{N}}:\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathsf{Homog}^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
75.07452pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
75.07452pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}:(\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
51.07452pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence.

######  Corollary 10.7.3.

Under Hypothesis 10.1.3 the adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
52.02243pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{const}:\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.02243pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.02243pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathsf{Homog}^{n}}:\lim}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
52.02243pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

is a Quillen equivalence.

### References