# Formality of $\mathbb{E}_{n}$-algebras and cochains on spheres

Gijs Heuts  Mathematical Institute, Utrecht University, Budapestlaan 6, 3584
CD Utrecht, The Netherlands<EMAIL_ADDRESS>and  Markus Land
Mathematisches Institut, Ludwig-Maximilians-Universität München,
Theresienstraße 39, 80333 München, Germany<EMAIL_ADDRESS>
###### Abstract.

We study the loop and suspension functors on the category of augmented
$\mathbb{E}_{n}$-algebras. One application is to the formality of the cochain
algebra of the $n$-sphere. We show that it is formal as an
$\mathbb{E}_{n}$-algebra, also with coefficients in general commutative ring
spectra, but rarely $\mathbb{E}_{n+1}$-formal unless the coefficients are
rational. Along the way we show that the free functor from operads in spectra
to monads in spectra is fully faithful on a nice subcategory of operads which
in particular contains the stable $\mathbb{E}_{n}$-operads for finite $n$. We
use this to interpret our results on loop and suspension functors of augmented
algebras in operadic terms.

ML was supported by the Danish National Research Foundation through the
Copenhagen Centre for Geometry and Topology (DNRF151). GH was supported by an
ERC Starting Grant (no. 950048) and an NWO VIDI grant (no. 223.093).

###### Contents

  1. 1 Introduction
  2. 2 Suspensions and loops of $\mathbb{E}_{n}$-algebras
  3. 3 The stable $\mathbb{E}_{n}$-operads
  4. 4 Triviality of spheres

##  1\. Introduction

The notion of formality originates in rational homotopy theory, where a space
$X$ is called _formal_ if its $\mathbb{E}_{\infty}$-algebra
$C^{*}(X;\mathbb{Q})$ of rational cochains is equivalent, as an
$\mathbb{E}_{\infty}$-algebra, to the cohomology ring
$H^{*}(X;\mathbb{Q})$.111Classically, this is formulated via Sullivan’s
$\mathcal{A}_{\mathrm{PL}}$ functor as a zig-zag of quasi-isomorphisms of
CDGA’s. There are many interesting examples, including spheres, loop spaces,
and compact Kähler manifolds. One surprising aspect of rational formality of a
space $X$ is that it suffices to show that $C^{*}(X;\mathbb{Q})$ and
$H^{*}(X;\mathbb{Q})$ are equivalent as $\mathbb{E}_{1}$-algebras [Sal17,
CPRNW23].

If one changes coefficients from $\mathbb{Q}$ to a finite field
$\mathbb{F}_{p}$, the situation changes drastically. One of the simplest
examples, namely the $\mathbb{E}_{\infty}$-algebra
$C^{*}(S^{n};\mathbb{F}_{p})$ of cochains on a sphere, already fails to be
formal. Indeed, the cohomology of a sphere is a trivial square-zero algebra,
but the $\mathbb{E}_{\infty}$-structure of the cochain algebra is not trivial.
The latter is essentially a consequence of the fact that the operation
$\mathrm{Sq}^{0}$ acts nontrivially on $H^{n}(S^{n};\mathbb{F}_{p})$. Indeed,
the Steenrod squares can be interpreted as power operations which exist for
any $\mathbb{E}_{\infty}$-algebra over $\mathbb{F}_{p}$ and vanish on trivial
algebras (see Proposition 4.20 for a more general result).

Still, one can ask for a weaker notion of formality: does there exist a $k\geq
1$ such that $C^{*}(S^{n};\mathbb{F}_{p})$ and $H^{*}(S^{n};\mathbb{F}_{p})$
are equivalent as $\mathbb{E}_{k}$-algebras? The answer turns out to be yes if
$k\leq n$, independently of the coefficients. In fact, we will prove a
stronger statement. If $R$ is an $\mathbb{E}_{n+1}$-ring and $X$ a space, we
write $C^{*}(X;R)$ for the _$R$ -valued cochains on $X$_, defined as the limit
$\varprojlim_{X}R$ in the $\infty$-category
$\mathrm{Alg}_{\mathbb{E}_{n}}(\mathrm{Mod}_{R})$ of
$\mathbb{E}_{n}$-$R$-algebras. A choice of basepoint in $X$ equips
$C^{*}(X;R)$ with an augmentation to $R$, making it into an object of
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n}}(\mathrm{Mod}_{R})$. We shall
refer to a trivial square-zero extension in
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathrm{Mod}_{R})$ as an algebra
which is _$\mathbb{E}_{n}$ -trivial over $R$_, see Section 2.3 for the precise
definitions.

######  Theorem A.

Let $R$ be an $\mathbb{E}_{n+1}$-ring spectrum and $X$ a pointed space. Then
$C^{*}(\Sigma^{n}X;R)$ is $\mathbb{E}_{n}$-trivial over $R$.

Specializing to the case $X=S^{0}$ and $R=\mathbb{F}_{p}$ implies the
$\mathbb{E}_{n}$-formality of $C^{*}(S^{n};\mathbb{F}_{p})$. In fact, A itself
is a special case of a much more general statement about augmented
$\mathbb{E}_{n}$-algebras in stable $\infty$-categories:

######  Theorem B.

Let $\mathcal{C}$ be a stably $\mathbb{E}_{n}$-monoidal $\infty$-category and
$A$ an augmented $\mathbb{E}_{n}$-algebra in $\mathcal{C}$. Then the $n$-fold
loop object $\Omega^{n}A$ is $\mathbb{E}_{n}$-trivial.

Here the loops functor $\Omega$ is computed in the $\infty$-category
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ of augmented
$\mathbb{E}_{n}$-algebras in $\mathcal{C}$ and again, we refer to Section 2.3
for the precise definitions of trivial $\mathbb{E}_{n}$-algebras in
$\mathcal{C}$. A is the special case $\mathcal{C}=\mathrm{Mod}_{R}$ and
$A=C^{*}(X;R)$, using that $C^{*}(\Sigma^{n}X;R)\cong\Omega^{n}C^{*}(X;R)$.
Let us remark that Mandell has considered the same question and claims a
version of A in the case where $R$ is an ordinary commutative ring [Man09],
but we are not aware of a published reference of his, and that work of
Berger–Fresse [BF04] proves A again in the case where $R$ is an ordinary
commutative ring.

We also investigate to what extent A is sharp and first conjecture the
following.

######  Triviality Conjecture.

Let $n\geq 1$ and let $A$ be an $\mathbb{E}_{n+2}$-algebra in $\mathrm{Sp}$.
Then $C^{*}(S^{n};A)$ is $\mathbb{E}_{n+1}$-trivial over $A$ if and only if
$A$ is a $\mathbb{Q}$-algebra.

The interesting part of this conjecture is the _only if_ implication. We offer
a proof in many cases of interest:

######  Theorem C (See Theorems 4.11 and 4.12).

The Triviality Conjecture holds

  1. (1)

when $n\leq 2$,

  2. (2)

when $A$ is bounded below,

  3. (3)

when $A$ is the underlying $\mathbb{E}_{n+2}$-algebra of an
$\mathbb{E}_{\infty}$-algebra, and

  4. (4)

when there exists a prime $p$ such that $A\otimes\mathbb{F}_{p}$ or
$A\otimes\mathrm{KU}/p$ are non-zero.

We use two approaches to prove C. One is based on studying the operadic
structure maps of $C^{*}(S^{n};A)$ and the other is based on power operations.
The cases above are not unrelated, indeed 4 implies 2 and 3, see Lemma 4.13.

Let us now outline our approach to proving B. The key idea is to show that the
$\mathbb{E}_{n}$-algebra structure of $\Omega A$ only depends on the
$\mathbb{E}_{n-1}$-structure rather than the full $\mathbb{E}_{n}$-structure
of $A$. More precisely, we show in Corollary 2.7 that there exists a
commutative diagram as follows.

${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}).}$$\scriptstyle{\Omega}$$\scriptstyle{\mathrm{res}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}}$$\scriptstyle{\omega}$$\scriptstyle{\mathrm{res}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}}$$\scriptstyle{\omega}$

Here the vertical functors are the evident forgetful ones. The horizontal
functors $\omega$ are less obvious; they can be constructed as the right
adjoint to the bar construction. The lower triangle establishes the previously
mentioned key idea, and the upper triangles then allows us to inductively
deduce that $\Omega^{n}$ is equivalent to
$\omega^{n}\circ\mathrm{res}^{\mathbb{E}_{n}}_{\mathbb{E}_{0}}$. The proof of
B then boils down to relating $\omega^{n}$ with the trivial algebra functor,
which we do in Theorem 2.11.

We will offer an alternative interpretation of this square in terms of the
underlying operads. In Section 3 we show that the existence of the diagram
above is essentially equivalent to a diagram of operads in the
$\infty$-category of spectra as follows:

${\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}}$${\Sigma^{\infty}_{+}\mathbb{E}_{n+1}^{\mathrm{nu}}}$${S\Sigma^{\infty}_{+}\mathbb{E}_{n-1}^{\mathrm{nu}}}$${S\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}.}$$\scriptstyle{\sigma}$

Here the $S$ indicates _operadic suspension_ and $\sigma$ denotes the
_operadic suspension map_ ; we discuss both of these concepts in Section 3 as
well. The surprising result is that these two squares are essentially
equivalent: passing from the operadic square to a square of algebras is
immediate, but the converse is not. Again, we remark that in the $R$-linear
context, with $R$ an ordinary commutative ring, a square of operads as above
has been constructed by Fresse [Fre11, Theorem C] using point-set models. In
particular his results also provide a proof of B for $R$-linear categories.

To pass between the two diagrams above, we prove the following result about
the relation between operads and monads that could be of independent interest.

######  Theorem D.

The functors

$\mathrm{free}\colon\mathrm{SSeq}(\mathrm{Sp})\longrightarrow\mathrm{End}(\mathrm{Sp})\quad\text{
and
}\quad\mathrm{free}\colon\mathrm{Operad}(\mathrm{Sp})\longrightarrow\mathrm{Monad}(\mathrm{Sp})$

assigning to a non-unital symmetric sequence (or operad) $\mathcal{O}$ in the
$\infty$-category of spectra its corresponding free algebra endo-functor (or
monad) $\mathrm{free}_{\mathcal{O}}$ restricts to a fully faithful embedding
on the full subcategory containing the non-unital $\mathbb{E}_{n}$-operads,
for $n<\infty$, and their operadic (de)suspensions.

In fact, we single out a property of symmetric sequences, which we call having
_nilpotent Euler classes_ and show that the free functor is fully faithful on
symmetric sequences and operads with nilpotent Euler classes, and that the
$\mathbb{E}_{n}$-operads as well as their operadic (de)suspensions have
nilpotent Euler classes, see Theorem 3.8.

Finally, we outline the plan of this paper. In Section 2 we prove the basic
results on loops and suspensions of $\mathbb{E}_{n}$-algebras, in particular
establishing B and a dual version for $n$-fold suspensions. In Section 3 we
investigate the relation between operads and monads on the $\infty$-category
$\mathrm{Sp}$ and establish D. The final Section 4 is focused on the cochain
algebras of spheres and is mostly devoted to the inspection of our Triviality
Conjecture.

##  2\. Suspensions and loops of $\mathbb{E}_{n}$-algebras

###  2.1. Preliminaries

Let us denote by $\mathrm{Cat}_{\infty}^{\mathrm{wc}}$ the $\infty$-category
of $\infty$-categories which admit weakly contractible colimits and functors
which preserve weakly contractible colimits. Via Lurie’s tensor product
[Lur17, §4.8.1], $\mathrm{Cat}_{\infty}^{\mathrm{wc}}$ acquires a symmetric
monoidal structure. Throughout this section, we let $\mathcal{C}$ be an object
of $\mathrm{Alg}_{\mathbb{E}_{n}}(\mathrm{Cat}_{\infty}^{\mathrm{wc}})$ and
sometimes refer to it as an $\mathbb{E}_{n}$-monoidal category with weakly
contractible colimits. By definition, the tensor product of $\mathcal{C}$
preserves weakly contractible colimits in each variable. We denote the
monoidal unit in $\mathcal{C}$ by $\mathbf{1}$ and write
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})=\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{C})_{/\mathbf{1}}$
for the category of augmented $\mathbb{E}_{n}$-algebras. This is a pointed
category, with zero object given by the monoidal unit $\mathbf{1}$ of
$\mathcal{C}$. Moreover, Dunn additivity [Lur17, Theorem 5.1.2.2] induces the
equivalence
$\mathrm{Alg}_{\mathbb{E}_{n}}^{(\mathrm{aug})}(\mathcal{C})\simeq\mathrm{Alg}_{\mathbb{E}_{1}}^{(\mathrm{aug})}(\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C}))$.
We will need the following lemma.

######  Lemma 2.1.

Let $n\geq 1$ and $\mathcal{C}$ be in
$\mathrm{Alg}_{\mathbb{E}_{n}}(\mathrm{Cat}_{\infty}^{\mathrm{wc}})$ and
$0<k<n$. Then $\mathrm{Alg}_{\mathbb{E}_{k}}(\mathcal{C})$ is in
$\mathrm{Alg}_{\mathbb{E}_{n-k}}(\mathrm{Cat}_{\infty}^{\mathrm{wc}})$.
Moreover, the functor $\mathrm{res}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}$
admits a left adjoint
$\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}\colon\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{C})\rightarrow\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$.

###### Proof.

The first statement is [Lur17, Prop. 5.1.2.9]. For the second, [Lur17, Remark
5.2.2.10] then implies that the forgetful functor
$\mathrm{Alg}_{\mathbb{E}_{1}}(\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C}))\to\mathrm{Alg}_{\mathbb{E}_{0}}(\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C}))$
admits a left adjoint. This adjunction induces the required adjunction on
augmented objects, using that also
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{0}}(\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C}))\simeq\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n-1}}(\mathcal{C})$.
∎

We note that, as a consequence,
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ is cocomplete
since it admits weakly contractible colimits, as well as an initial
object.222This fact is also true for $n=0$. In particular, it admits a
suspension functor which we denote by $\Sigma_{\mathbb{E}_{n}}$ and warn the
reader that it is not compatible with the forgetful functors
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{k}}^{\mathrm{aug}}(\mathcal{C})$
for $k<n$.

######  Lemma 2.2.

For $A$ in $\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$, the
relative tensor product $\mathbf{1}\otimes_{A}\mathbf{1}$ exists and its
formation provides a functor

$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\longrightarrow\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{C}),\quad
A\mapsto\mathrm{Bar}(A):=\mathbf{1}\otimes_{A}\mathbf{1}$

called the Bar construction.

###### Proof.

We note again the equivalence
$\mathrm{Alg}_{\mathbb{E}_{n}}^{(\mathrm{aug})}(\mathcal{C})\simeq\mathrm{Alg}_{\mathbb{E}_{1}}^{(\mathrm{aug})}(\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C}))$
and that $\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C})$ is a monoidal category
with weakly contractible colimits by Lemma 2.1. In particular,
$\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C})$ is a monoidal category with
geometric realizations, so [Lur17, §4.4.2] provides a functor
$\mathrm{Bar}\colon\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C})$
which induces the required functor on augmented objects as clearly
$\mathrm{Bar}(\mathbf{1})\simeq\mathbf{1}$. ∎

Finally, we will also need the following lemma.

######  Lemma 2.3.

Let $\mathcal{C}$ be an $\mathbb{E}_{n}$-monoidal category with geometric
realizations. Then a commutative square

${A}$${B}$${A^{\prime}}$${B^{\prime}}$

in $\mathrm{Alg}_{\mathbb{E}_{n}}^{(\mathrm{aug})}(\mathcal{C})$ induces a
canonical map $A^{\prime}\otimes_{A}B\to B^{\prime}$ in
$\mathrm{Alg}_{\mathbb{E}_{n-1}}^{(\mathrm{aug})}(\mathcal{C})$.

###### Proof.

The assumptions on $\mathcal{C}$ imply that the relative tensor product can be
formed and that it provides a functor
$-\otimes_{A}B\colon\mathrm{RMod}_{A}(\mathcal{C})\to\mathrm{RMod}_{B}(\mathcal{C})$
left adjoint to the forgetful functor. This functor in addition inherits the
structure of an $\mathbb{E}_{n-1}$-monoidal functor from the
$\mathbb{E}_{n}$-structure on the map $A\to B$. It consequently induces a left
adjoint
$-\otimes_{A}B\colon\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathrm{RMod}_{A}(\mathcal{C}))\to\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathrm{RMod}_{B}(\mathcal{C}))$.
There is therefore an equivalence

$\mathrm{Map}_{\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathrm{RMod}_{B}(\mathcal{C}))}(A^{\prime}\otimes_{A}B,B^{\prime})\simeq\mathrm{Map}_{\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathrm{RMod}_{A}(\mathcal{C}))}(A^{\prime},B^{\prime})$

and the latter space contains the map $A^{\prime}\to B^{\prime}$ part of the
above diagram. We then use the forgetful functor
$\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathrm{RMod}_{B}(\mathcal{C}))\to\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathcal{C})$
induced from the unit of $B$ and arrive at the claimed canonical map. This
construction is natural in maps of squares, which shows that the just
constructed map is compatible with augmentations. ∎

###  2.2. Suspensions and loops of $\mathbb{E}_{n}$-algebras

The starting point of our paper is the following result.

######  Theorem 2.4.

Let $n\geq 0$ and $\mathcal{C}$ be in
$\mathrm{Alg}_{\mathbb{E}_{n+1}}(\mathrm{Cat}_{\infty}^{\mathrm{wc}})$. Then
there is a natural commutative diagram of colimit preserving functors as
follows:

${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$$\scriptstyle{\Sigma_{\mathbb{E}_{n}}}$$\scriptstyle{\mathrm{free}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}}$$\scriptstyle{\mathrm{Bar}}$$\scriptstyle{\mathrm{Bar}}$$\scriptstyle{\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}}$

Here, we assume $n\geq 1$ for the lower-left triangle.

######  Remark 2.5.

The proof we give in fact shows that for the lower-left triangle to commute as
claimed, it suffices that $\mathcal{C}$ is $\mathbb{E}_{n}$-monoidal.

###### Proof of Theorem 2.4.

We treat the upper-right triangle first. By Lemma 2.1, the $\infty$-category
$\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{C})$ lies in
$\mathrm{Alg}_{\mathbb{E}_{1}}(\mathrm{Cat}_{\infty}^{\mathrm{wc}})$. Using
the canonical equivalences
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\simeq\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{C})_{\mathbf{1}//\mathbf{1}}$
as well as
$\mathrm{Alg}_{\mathbb{E}_{1}}^{\mathrm{aug}}(\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{C}))\simeq\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n+1}}(\mathcal{C})$
for all $n\geq 0$, [Lur17, Cor. 5.2.2.13] says that the composite

$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\xrightarrow{\mathrm{free}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}}\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{aug}}(\mathcal{C})\xrightarrow{\mathrm{Bar}}\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{C})$

is given by the functor which takes an object $C$ to the pushout
$\mathbf{1}\amalg_{C}\mathbf{1}$. We note that the composite above canonically
factors through $\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$
since $\mathrm{Bar}(\mathbf{1})=\mathbf{1}$. The same formula then remains
true for this refined functor because the forgetful functor is conservative
and preserves pushouts. Since
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ is a pointed
category, with terminal object $\mathbf{1}$, the functor
$C\mapsto\mathbf{1}\amalg_{C}\mathbf{1}$ is the suspension functor
$\Sigma_{\mathbb{E}_{n}}$, showing the commutativity of the upper-right
triangle.

Now let us show that all functors in question preserve colimits. First, in any
pointed and cocomplete $\infty$-category, the suspension functor commutes with
colimits, hence $\Sigma_{\mathbb{E}_{n}}$ preserves colimits. Moreover, for
all $k\geq 1$, the functor $\mathrm{free}_{\mathbb{E}_{k-1}}^{\mathbb{E}_{k}}$
is a left adjoint and hence also preserves colimits. It then suffices to show
that for any $k\geq 1$, the functor
$\mathrm{Bar}\colon\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{k}}(\mathcal{C})\to\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{k-1}}(C)$
preserves small colimits. By construction, $\mathrm{Bar}$ preserves sifted
colimits, so in fact it suffices to show that it preserves the initial object
and binary coproducts. The initial object is given by the tensor unit
$\mathbf{1}$ and which is clearly preserved by $\mathrm{Bar}$. Next, we wish
to show that for any two objects $A,B$ of
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{k}}(\mathcal{C})$, the canonical map

$\mathrm{Bar}(A)\amalg\mathrm{Bar}(A^{\prime})\to\mathrm{Bar}(A\amalg
A^{\prime})$

is an equivalence. Since $\mathrm{Bar}$ preserves sifted colimits and every
augmented $\mathbb{E}_{k}$-algebra is a sifted colimit of algebras of the form
$\mathrm{free}_{\mathbb{E}_{k-1}}^{\mathbb{E}_{k}}(B)$ [Lur17, Prop. 4.7.3.14]
(to apply this proposition, note that the forgetful functor
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{k}}(\mathcal{C})\to\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{k-1}}(\mathcal{C})$
is conservative and preserves sifted colimits), it suffices to show that the
canonical map

$\mathrm{Bar}(\mathrm{free}_{\mathbb{E}_{k-1}}^{\mathbb{E}_{k}}(B))\amalg\mathrm{Bar}(\mathrm{free}_{\mathbb{E}_{k-1}}^{\mathbb{E}_{k}}(B^{\prime}))\longrightarrow\mathrm{Bar}(\mathrm{free}_{\mathbb{E}_{k-1}}^{\mathbb{E}_{k}}(B\amalg
B^{\prime}))$

is an equivalence. By the already established commutativity of the upper-right
triangle, this map is equivalent to the canonical map

$\Sigma_{\mathbb{E}_{k}}(B)\amalg\Sigma_{\mathbb{E}_{k}}(B^{\prime})\longrightarrow\Sigma_{\mathbb{E}_{k}}(B\amalg
B^{\prime})$

which is an equivalence since $\Sigma_{\mathbb{E}_{k}}$ preserves colimits as
we have already argued.

Finally, we show that the lower-left triangle commutes. To do so, we first
observe that the commutative (in fact pushout) diagram in
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n}}(\mathcal{C})$

${A}$${\mathbf{1}}$${\mathbf{1}}$${\Sigma_{\mathbb{E}_{n}}(A)}$

induces a canonical map
$\mathrm{Bar}(A)=\mathbf{1}\otimes_{A}\mathbf{1}\to\Sigma_{\mathbb{E}_{n}}(A)$
in $\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n-1}}(\mathcal{C})$, see Lemma
2.3 used for the above square for $A$, for $\mathbf{1}$, and the canonical map
from the former to the latter. By adjunction, we obtain a natural map
$\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}(\mathrm{Bar}(A))\to\Sigma_{\mathbb{E}_{n}}(A)$.
Source and target of this natural map are colimit preserving functors, as we
have just argued. The same reasoning as above implies that it therefore
suffices to show that this map is an equivalence in case
$A=\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}(B)$ in which case we may
again use the established commutativity of the upper-right triangle for $n-1$
in place of $n$. The map under investigation then becomes equivalent to the
canonical map

$\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}(\Sigma_{\mathbb{E}_{n-1}}(B))\longrightarrow\Sigma_{\mathbb{E}_{n}}(\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}(B))$

which is an equivalence because the free functor preserves colimits and
therefore commutes with suspensions. ∎

Applying Theorem 2.4 several times we obtain the following:

######  Corollary 2.6.

Let $\mathcal{C}$ be as in Theorem 2.4 and let $A$ be in
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$. Then there are
canonical equivalences

$\mathrm{Bar}^{k}(\mathrm{free}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+k}}(A))\simeq\Sigma^{k}_{\mathbb{E}_{n}}(A)\simeq\mathrm{free}_{\mathbb{E}_{n-k}}^{\mathbb{E}_{n}}(\mathrm{Bar}^{k}(A))$

where we assume $k\leq n$ for the right hand equivalence. In particular, the
$\mathbb{E}_{n}$-algebra $\Sigma^{n}_{\mathbb{E}_{n}}(A)$ is in the image of
the functor
$\mathrm{free}_{\mathbb{E}_{0}}^{\mathbb{E}_{n}}\colon\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$.

######  Corollary 2.7.

Let $\mathcal{C}$ be an $\mathbb{E}_{n+1}$-monoidal $\infty$-category and
assume that $\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})$ admits
a loop functor $\Omega$. Then there is a commutative diagram as follows:

${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$$\scriptstyle{\Omega}$$\scriptstyle{\mathrm{res}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}}$$\scriptstyle{\omega}$$\scriptstyle{\mathrm{res}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}}$$\scriptstyle{\omega}$

###### Proof.

First, we observe that also the categories
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ admit a loop
functor, since the forgetful functor to
$\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})$ preserves limits
and is conservative. Consider the Yoneda embedding
$\mathcal{C}\to\mathcal{P}(\mathcal{C})$. Day convolution provides an
$\mathbb{E}_{n+1}$-monoidal structure on $\mathcal{P}(\mathcal{C})$ which
preserves small colimits in each variable and such that the Yoneda embedding
canonically refines to a symmetric monoidal functor [Lur17, Cor. 4.8.1.12]. Up
to enlargening a universe, we may apply Theorem 2.4 to
$\mathcal{P}(\mathcal{C})$. In this case, all categories in question are
presentable, so that all colimit preserving functors are in fact left
adjoints. Passing to right adjoints, we obtain a commutative diagram

${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{P}(\mathcal{C}))}$${\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{aug}}(\mathcal{P}(\mathcal{C}))}$${\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{P}(\mathcal{C}))}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{P}(\mathcal{C}))}$$\scriptstyle{\Omega}$$\scriptstyle{\mathrm{res}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}}$$\scriptstyle{\omega}$$\scriptstyle{\mathrm{res}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}}$$\scriptstyle{\omega}$

where $\omega$ denotes the right adjoint of $\mathrm{Bar}$. We claim that this
diagram restricts to the one we aim to obtain. For this, we need to see that
all functors in question preserve algebras whose underlying object lies in
$\mathcal{C}\subseteq\mathcal{P}(\mathcal{C})$. This is clearly true for the
restriction functor and for $\Omega$ as well since the inclusion
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\subseteq\mathrm{Alg}_{\mathbb{E}_{n}}(\mathcal{P}(\mathcal{C}))$
preserves limits. Finally, we need to show that the composite

$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\subseteq\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{P}(\mathcal{C}))\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{aug}}(\mathcal{P}(\mathcal{C}))\longrightarrow\mathcal{P}(\mathcal{C}),$

where the last functor is the forgetful functor, lands in
$\mathcal{C}\subseteq\mathcal{P}(\mathcal{C})$. This follows from the
commutativity of the upper-right triangle since the forgetful functor above
factors through $\mathrm{res}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}$, whose
composite with $\omega$ is given by $\Omega$, so we can use that $\Omega$ has
the desired property. ∎

######  Corollary 2.8.

Let $\mathcal{C}$ be as in Corollary 2.7 and let $A$ be in
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$. There there are
canonical equivalences

$\mathrm{res}^{\mathbb{E}_{n+k}}_{\mathbb{E}_{n}}(\omega^{k}(A))\simeq\Omega^{k}(A)\simeq\omega^{k}(\mathrm{res}^{\mathbb{E}_{n}}_{\mathbb{E}_{n-k}}(A))$

where again, we assume $k\leq n$ for the right hand equivalence. In
particular, the $\mathbb{E}_{n}$-algebra $\Omega^{n}(A)$ is in the image of
the functor
$\omega^{n}\colon\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$.

Corollary 2.8 provides two useful observations. First, the second equivalence
shows that for an augmented $\mathbb{E}_{n}$-algebra $A$ the
$\mathbb{E}_{n}$-structure of $\Omega A$ only depends on the underlying
$\mathbb{E}_{n-1}$-structure of $A$. As mentioned in the introduction, the
special case where $A=C^{*}(X;R)$ is the $\mathbb{E}_{\infty}$-algebra of
$R$-valued cochains on a space $X$, for some commutative ring $R$, has been
claimed by Mandell [Man09]. Secondly, the first equivalence shows that the
algebra $\Omega A$ admits a natural $\mathbb{E}_{n+1}$-algebra structure
(which does, in general, depend on the full $\mathbb{E}_{n}$-structure of
$A$).

###  2.3. The stable case

We will be particularly interested in the case where $\mathcal{C}$ is stable,
in which case there is an equivalence between augmented and non-unital
algebras. Let us recall that a non-unital operad $\mathcal{O}$ is one with
$\mathcal{O}(0)=\varnothing$. There are then the non-unital versions
$\mathbb{E}_{n}^{\mathrm{nu}}$ of the $\mathbb{E}_{n}$-operads and they
satisfy

$\mathbb{E}_{n}^{\mathrm{nu}}(k)=\begin{cases}\varnothing&\text{ if }n=0\\\
\mathbb{E}_{n}(k)&\text{ else}\end{cases}$

and hence come with canonical morphisms
$\mathbb{E}_{n}^{\mathrm{nu}}\to\mathbb{E}_{n}$. For $\mathcal{C}$ an
$\mathbb{E}_{n}$-monoidal category, we write
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathcal{C})$ for
$\mathrm{Alg}_{\mathbb{E}_{n}^{\mathrm{nu}}}(\mathcal{C})$ for the category of
non-unital $\mathbb{E}_{n}$-algebras.

For an augmented algebra $A$, we write
$\mathfrak{m}(A)=\mathrm{fib}(A\to\mathbf{1})$ for its augmentation ideal. A
priori, it is simply an object of $\mathcal{C}$ but it canonically upgrades to
a non-unital $\mathbb{E}_{n}$-algebra. In fact, when $\mathcal{C}$ is stably
$\mathbb{E}_{n}$-monoidal,
$\mathfrak{m}\colon\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\to\mathcal{C}$
canonically enhances to an equivalence of $\infty$-categories [Lur17, Prop.
5.4.4.10]

$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\xrightarrow{\,\simeq\,}\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathcal{C})$

whose inverse we denote by $X\mapsto\mathbf{1}\oplus X$.

When $\mathcal{C}$ is stably $\mathbb{E}_{n}$-monoidal, there is a canonical
equivalence
$\mathcal{C}\simeq\mathrm{Sp}(\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}))$,
compatible with the restriction maps
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{k}}^{\mathrm{aug}}(\mathcal{C})$
whenever $k\leq n$ as well as compatible with lax $\mathbb{E}_{n}$-monoidal
and exact functors.333Informally, given such a functor
$f\colon\mathcal{C}\to\mathcal{D}$, it sends an augmented
$\mathbb{E}_{n}$-algebra $A\to\mathbf{1}_{\mathcal{C}}$ in $\mathcal{C}$ to
the augmented $\mathbb{E}_{n}$-algebra
$F(A)\times_{F(\mathbf{1}_{\mathcal{C}})}\mathbf{1}_{\mathcal{D}}$. This
follows from the canonical equivalence
$\mathrm{Mod}_{\mathbf{1}}^{\mathbb{E}_{n}}(\mathcal{C})\simeq\mathcal{C}$
[Lur17, Remark 7.3.5.3] and [Lur17, Theorem 7.3.4.13]. The composite of this
equivalence followed by the functor
$\Omega^{\infty}\colon\mathrm{Sp}(\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}))\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$
is the trivial $\mathbb{E}_{n}$-algebra functor

$\mathrm{tr}_{\mathbb{E}_{n}}\colon\mathcal{C}\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}),\quad
X\mapsto\mathbf{1}\oplus X$

which exhibits $\mathrm{tr}_{\mathbb{E}_{n}}(X)$ is the trivial square zero
extension on $X$. Informally speaking, it equips $X$ with the zero
multiplication. Note that the functor $\mathrm{tr}_{\mathbb{E}_{0}}$ is an
equivalence, as
$\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})\simeq\mathcal{C}_{\mathbf{1}/\mathbf{1}}\simeq\mathcal{C}$
is already stable. In what follows, we will also view
$\mathrm{tr}_{\mathbb{E}_{n}}$ as a functor on
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{0}}(\mathcal{C})$ via this
equivalence. If $\mathcal{C}$ is presentably stably $\mathbb{E}_{n}$-monoidal,
then $\mathrm{tr}_{\mathbb{E}_{n}}$ admits a left adjoint
$\Sigma^{\infty}_{\mathbb{E}_{n}}\colon\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\to\mathcal{C}$,
often referred to as the $\mathbb{E}_{n}$-homology of an augmented
$\mathbb{E}_{n}$-algebra.

######  Remark 2.9.

The trivial functors are compatible with forgetful functors. More precisely,
when $\mathcal{C}$ is $\mathbb{E}_{n}$-monoidal, it is also
$\mathbb{E}_{n-1}$-monoidal and the functor $\mathrm{tr}_{\mathbb{E}_{n-1}}$
is equivalent to the composite

$\mathcal{C}\xrightarrow{\mathrm{tr}_{\mathbb{E}_{n}}}\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n}}\xrightarrow{\mathrm{res}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}}\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{aug}}(\mathcal{C}),$

as follows from the compatibility of the equivalences
$\mathcal{C}\simeq\mathrm{Sp}(\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}))$
with the restriction maps as indicated above.

In particular, if $\mathcal{C}$ is symmetric monoidal, then
$\mathrm{tr}_{\mathbb{E}_{n}}$ is equivalent to the composite

$\mathcal{C}\xrightarrow{\mathrm{tr}_{\mathbb{E}_{\infty}}}\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{\infty}}(\mathcal{C})\xrightarrow{\mathrm{res}_{\mathbb{E}_{n}}^{\mathbb{E}_{\infty}}}\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{n}}(\mathcal{C}).$

For future reference, we record the following basic property of trivial
algebras.

######  Lemma 2.10.

Let $f\colon\mathcal{C}\to\mathcal{D}$ an exact and
lax-$\mathbb{E}_{n}$-monoidal functor between stably $\mathbb{E}_{n}$-monoidal
categories. Then the induced functor
$f\colon\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathcal{D})$
preserves trivial algebras.

###### Proof.

By assumption, $f\colon\mathcal{C}\to\mathcal{D}$ preserves finite limits. It
follows that the induced functor
$f\colon\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{D})$
also preserves finite limits. This implies that the following diagram
commutes.

${\mathcal{C}}$${\mathrm{Sp}(\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}))}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$${\mathcal{D}}$${\mathrm{Sp}(\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{D}))}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{D})}$$\scriptstyle{\simeq}$$\scriptstyle{f}$$\scriptstyle{\Omega^{\infty}}$$\scriptstyle{\mathrm{Sp}(f)}$$\scriptstyle{f}$$\scriptstyle{\simeq}$$\scriptstyle{\Omega^{\infty}}$

The claim then follows from the definition of the trivial functors as the
horizontal composites. ∎

The functors $\mathrm{tr}_{\mathbb{E}_{n}}$ and $\omega^{n}$ as studied above
are closely related:

######  Theorem 2.11.

Let $\mathcal{C}$ be a stably $\mathbb{E}_{n}$-monoidal $\infty$-category.
Then there is a canonical equivalence of functors
$\omega^{n}\simeq\Omega^{n}\circ\mathrm{tr}_{\mathbb{E}_{n}}\colon\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$.

###### Proof.

Again, we note that
$\mathrm{Alg}^{\mathrm{aug}}_{\mathbb{E}_{0}}(\mathcal{C})$ is canonically
equivalent to $\mathcal{C}$ via the augmentation ideal. Then we note that the
composite

$\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})\xrightarrow{\mathrm{tr}_{\mathbb{E}_{n}}}\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})\xrightarrow{\mathrm{res}^{\mathbb{E}_{n}}_{\mathbb{E}_{0}}}\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})$

is canonically equivalent to the identity. In particular, we have
$\omega^{n}\simeq\omega^{n}\circ\mathrm{res}^{\mathbb{E}_{n}}_{\mathbb{E}_{0}}\circ\mathrm{tr}_{\mathbb{E}_{n}}\simeq\Omega^{n}\mathrm{tr}_{\mathbb{E}_{n}}$
by Corollary 2.8. ∎

Combined again with Corollary 2.8 we obtain:

######  Corollary 2.12.

Let $\mathcal{C}$ be a stably $\mathbb{E}_{n}$-monoidal $\infty$-category. For
an augmented $\mathbb{E}_{n}$-algebra $A$ in $\mathcal{C}$ there is a natural
equivalence

$\Omega^{n}(A)\simeq\mathrm{tr}_{\mathbb{E}_{n}}(\Omega^{n}(\mathrm{res}^{\mathbb{E}_{n}}_{\mathbb{E}_{0}}(A))).$

By passing to left adjoints, we obtain the following corollary.

######  Corollary 2.13.

Let $\mathcal{C}$ be presentably stably $\mathbb{E}_{n}$-monoidal. For an
augmented $\mathbb{E}_{n}$-algebra $A$ in $\mathcal{C}$ there is a natural
equivalence

$\Sigma^{n}_{\mathbb{E}_{n}}(A)\simeq\mathrm{free}_{\mathbb{E}_{0}}^{\mathbb{E}_{n}}(\Sigma^{n}(\Sigma^{\infty}_{\mathbb{E}_{n}}A)).$

######  Remark 2.14.

By passing to left adjoints in Theorem 2.11 we obtain an equivalence
$\mathrm{Bar}^{n}(A)\simeq\Sigma^{n}\Sigma^{\infty}_{\mathbb{E}_{n}}A$ when
$\mathcal{C}$ is presentably $\mathbb{E}_{n}$-monoidal. This relation between
the $n$-fold Bar construction and $\mathbb{E}_{n}$-homology is well-known, see
e.g. [BM11, Theorem 1.3] or [GKRW21, Theorem 13.7], but often proved via an
explicit calculation, in contrast to the rather formal argument obtained here.

######  Remark 2.15.

Corollaries 2.13 and 2.12 exhibit a kind of ‘fast stabilization’ in the
$\infty$-category of $\mathbb{E}_{n}$-algebras: any $n$-fold loop object is
already an infinite loop object and dually, an $n$-fold suspension is already
an infinite suspension object. To put this into perspective, we note that for
a stable category $\mathcal{C}$ and a non-unital operad $\mathcal{O}$, the
stabilization of the $\infty$-category
$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$ is always equivalent to
$\mathcal{C}$ see again [Lur17, Theorem 7.3.4.13]. In this situation, it
follows formally that any compact object in
$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$, i.e., any finitely presented
$\mathcal{O}$-algebra is in the essential image of the functor
$\mathrm{free}_{\mathcal{O}}$ after finitely many suspensions. Corollary 2.6
shows that for $\mathcal{O}=\mathbb{E}_{n}^{\mathrm{nu}}$ one has a much
sharper statement; this happens after only $n$ suspensions, as well as for
_any_ non-unital $\mathbb{E}_{n}$-algebra rather than just finitely presented
ones.

Elaborating on Corollary 2.13, we obtain the following consequence, where we
again implicitly identify $\mathcal{C}$ with
$\mathrm{Alg}_{\mathbb{E}_{0}}^{\mathrm{aug}}(\mathcal{C})$. This was already
used in the PhD thesis of Shi [Shi23].

######  Corollary 2.16.

Let $\mathcal{C}$ be presentably stably $\mathbb{E}_{n}$-monoidal. Then the
costabilization
$\mathrm{Sp}(\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})^{\mathrm{op}})^{\mathrm{op}}$
of $\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ is equivalent
to $\mathcal{C}$. In particular, for any stable $\infty$-category
$\mathcal{D}$ the functor

$\mathrm{Fun}^{\mathrm{rex}}(\mathcal{D},\mathcal{C})\longrightarrow\mathrm{Fun}^{\mathrm{rex}}(\mathcal{D},\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})),\quad
F\mapsto\mathrm{free}^{\mathbb{E}_{n}}\circ F$

is an equivalence, where the superscript rex refers to functors which preserve
finite colimits.

###### Proof.

By [Lur17, Prop. 1.4.2.24] the costabilization of
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ may be computed as
the inverse limit of the system

${\cdots}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C}).}$$\scriptstyle{\Sigma_{\mathbb{E}_{n}}}$$\scriptstyle{\Sigma_{\mathbb{E}_{n}}}$

Cofinality and Corollary 2.13 implies that this agrees with the inverse limit
of the diagram

${\cdots}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathcal{C})}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathcal{C})}$${\mathcal{C}}$${\mathcal{C}}$$\scriptstyle{\Sigma^{n}_{\mathbb{E}_{n}}}$$\scriptstyle{\Sigma^{n}\Sigma^{\infty}_{\mathbb{E}_{n}}}$$\scriptstyle{\Sigma^{n}_{\mathbb{E}_{n}}}$$\scriptstyle{\Sigma^{n}\Sigma^{\infty}_{\mathbb{E}_{n}}}$$\scriptstyle{\mathrm{free}^{\mathbb{E}_{n}}}$$\scriptstyle{\mathrm{free}^{\mathbb{E}_{n}}}$

where the dashed composite is given by $\Sigma^{n}$ since the composite
$\Sigma^{\infty}_{\mathbb{E}_{n}}\mathrm{free}^{\mathbb{E}_{n}}$ is
canonically equivalent to the identity, as its right adjoint canonically
identifies with the identity. Applying cofinality again, the costabilization
of $\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathcal{C})$ agrees with the
inverse limit of the diagram

${\cdots}$${\mathcal{C}}$${\mathcal{C}.}$$\scriptstyle{\Sigma^{n}}$$\scriptstyle{\Sigma^{n}}$

All the functors in this diagram are equivalences since $\mathcal{C}$ is
stable and the corollary follows. ∎

##  3\. The stable $\mathbb{E}_{n}$-operads

The aim of this section is to explain how the results of Section 2 can be
reinterpreted as statements about the $\mathbb{E}_{n}$-operads themselves, at
least when viewed as operads in spectra. In particular, we will see that in
the special case of stable $\infty$-categories $\mathcal{C}$, Theorem 2.4 is
equivalent to the existence of a certain diagram of operad maps as follows:

${\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}}$${\Sigma^{\infty}_{+}\mathbb{E}_{n+1}^{\mathrm{nu}}}$${S\Sigma^{\infty}_{+}\mathbb{E}_{n-1}^{\mathrm{nu}}}$${S\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}.}$$\scriptstyle{\sigma}$

Here $\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$ is the operad in the
$\infty$-category of spectra obtained by taking the levelwise suspension
spectrum of the nonunital operad $\mathbb{E}_{n}^{\mathrm{nu}}$ in spaces.
Also, $S\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$ denotes the _operadic
suspension_ of $\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$, which we
discuss in Section 3.1, and $\sigma$ denotes the _suspension morphism_. For
the corresponding free algebra functors, this morphism can be described as the
canonical suspension map
$\sigma\colon\mathrm{free}_{\mathbb{E}_{n}}(X)\to\Sigma^{-1}\mathrm{free}_{\mathbb{E}_{n}}(\Sigma
X)$. The horizontal morphisms in the diagram are the standard inclusions. The
vertical arrows are a bit more exotic; they can be thought of as the ‘Koszul
dual’ maps to the standard inclusions, cf. Remark 3.13.

The main technical tool in this section is the result that there is no loss of
information in passing from the stable $\mathbb{E}_{n}$-operads to their
associated monads. More precisely, we show in Theorem 3.8 that there is a
class of operads $\mathcal{O}$ ‘with nilpotent Euler classes’ for which the
assignment

$\mathrm{Operad}(\mathrm{Sp})\longrightarrow\mathrm{Monad}(\mathrm{Sp}),\quad\mathcal{O}\mapsto\mathrm{free}_{\mathcal{O}}$

is fully faithful. In Section 3.2 we demonstrate that the stable
$\mathbb{E}_{n}$-operads and their operadic (de)suspensions satisfy the
conditions of that theorem.

It should be true that the square of operad maps above already exists
_unstably_ , i.e. in the setting of operads in pointed spaces without applying
$\Sigma^{\infty}$ anywhere. To work this out one would need a good handle on
the operadic suspension and a possible universal property in the unstable
case. We comment on existing work and what would need to be done to obtain
this sharpening in Section 3.5.

###  3.1. Preliminaries

We denote by
$\mathrm{SSeq}(\mathrm{Sp}):=\mathrm{Fun}(\mathrm{Fin}^{\simeq},\mathrm{Sp})$
the category of symmetric sequences in spectra. Here $\mathrm{Fin}^{\simeq}$
denotes the groupoid of finite sets and bijections, which is the free
symmetric monoidal category on a single generator. We will usually write
$A(n)$ for the value of a symmetric sequence $A$ on the set
$\\{1,\ldots,n\\}$. A symmetric sequence $A$ is called non-unital if $A(0)$ is
equivalent to $0$. The category $\mathrm{SSeq}(\mathrm{Sp})$ is equipped with
the Day convolution symmetric monoidal structure $\otimes$ based on the usual
symmetric monoidal structure on $\mathrm{Sp}$ and the disjoint union symmetric
monoidal structure on $\mathrm{Fin}^{\simeq}$. By construction, the fully
faithful functor $\iota\colon\mathrm{Sp}\to\mathrm{SSeq}(\mathrm{Sp})$,
sending $X$ to the symmetric sequence $\iota(X)(0)=X$ and $\iota(X)(n)=0$ for
$n>0$ is canonically symmetric monoidal. In particular,
$\mathrm{SSeq}(\mathrm{Sp})$ is tensored over $\mathrm{Sp}$ via this
inclusion.

The $\infty$-category $\mathrm{SSeq}(\mathrm{Sp})$ can be equipped with a
further monoidal structure, called the _composition product_ , obeying the
formula

$A\circ B=\bigoplus\limits_{n\geq 0}(A(n)\otimes B^{\otimes
n})_{h\Sigma_{n}},$

see [BCN23, Section 3.1] and [Hau22]. It follows that $A\circ\iota(X)$ is in
the essential image of $\iota$ for any symmetric sequence $A$, giving rise to
an action of $\mathrm{SSeq}(\mathrm{Sp})$ on $\mathrm{Sp}$, or equivalently, a
monoidal functor

$\mathrm{free}\colon\mathrm{SSeq}(\mathrm{Sp})\to\mathrm{End}(\mathrm{Sp}),\quad
A\mapsto[\mathrm{free}_{A}\colon X\mapsto\bigoplus_{n\geq 0}(A(n)\otimes
X^{\otimes n})_{h\Sigma_{n}}]$

where we view
$\mathrm{End}(\mathrm{Sp})=\mathrm{Fun}(\mathrm{Sp},\mathrm{Sp})$ as monoidal
via the composition of functors. It will be convenient to use the abbreviated
notation $D_{n}^{A}(X):=(A(n)\otimes X^{\otimes n})_{h\Sigma_{n}}$ for the
summands of $\mathrm{free}_{A}$.

For non-unital symmetric sequences $A$ and $B$, the above formula for the
composition product simplifies to the following. For a finite set $I$ we have

$(A\circ
B)(I)\simeq\bigoplus_{E\in\mathrm{Equiv}(I)}A(I/E)\otimes\bigotimes_{J\in
I/E}B(J),$

where the sum is indexed over the set of equivalence relations $E$ on $I$. It
will be important for us to make explicit the $\Sigma_{I}$-action on $(A\circ
B)(I)$. To that end, first note that $\Sigma_{I}$ acts on the set
$\mathrm{Equiv}(I)$ of equivalence relations on $I$ in the obvious fashion.
For $E\in\mathrm{Equiv}(I)$ let $G_{E}$ denote the stabilizer of the action so
that there are canonical induced maps $G_{E}\to\Sigma_{I/E}$ as well as
$G_{E}\to\Sigma_{J}$ for all $J\in I/E$. Then we have an equivalence of
$\Sigma_{I}$-objects

$(A\circ
B)(I)\simeq\bigoplus\limits_{E\in\mathrm{Equiv}(I)/\Sigma_{I}}\mathrm{ind}_{G_{E}}^{\Sigma_{I}}[A(I/E)\otimes\bigotimes_{J\in
I/E}B(J)]$

for the diagonal $G_{E}$ action on $A(I/E)\otimes\bigotimes_{J}B(J)$.

An _operad_ in $\mathrm{Sp}$ is defined to be an algebra object in the
monoidal $\infty$-category $\mathrm{SSeq}(\mathrm{Sp})$. We write
$\mathrm{Operad}(\mathrm{Sp})=\mathrm{Alg}(\mathrm{SSeq}(\mathrm{Sp}))$ for
the $\infty$-category of such. An operad is said to be non-unital when its
underlying symmetric sequence is non-unital. Since algebras in
$\mathrm{End}(\mathrm{Sp})$ are monads and $\mathrm{free}$ is monoidal, we
obtain a functor

$\mathrm{free}\colon\mathrm{Operad}(\mathrm{Sp})\longrightarrow\mathrm{Monad}(\mathrm{Sp}),\quad\mathcal{O}\mapsto\mathrm{free}_{\mathcal{O}}.$

We note that the monad $\mathrm{free}_{\mathcal{O}}$ associated to a non-
unital operad $\mathcal{O}$ is a reduced functor.

#### The operadic suspension

We write $\mathbb{S}^{1}[1]$ for the symmetric sequence that assigns the
spectrum $\mathbb{S}^{1}$ to a singleton and 0 to a set of any cardinality
other than 1. Observe that $\mathrm{free}_{\mathbb{S}^{1}[1]}$ is the
suspension functor $\Sigma\in\mathrm{End}(\mathrm{Sp})$. As follows from the
above formulas for the composition product, the object $\mathbb{S}^{1}[1]$ has
a monoidal inverse, namely $\mathbb{S}^{-1}[1]$ with
$\mathrm{free}_{\mathbb{S}^{-1}[1]}\simeq\Sigma^{-1}\in\mathrm{End}(\mathrm{Sp})$.
Conjugating by the invertible object $\mathbb{S}^{1}[1]$ therefore defines a
monoidal automorphism [BCN23, Lemma 3.10]

$S\colon\mathrm{SSeq}(\mathrm{Sp})\longrightarrow\mathrm{SSeq}(\mathrm{Sp}),\quad
A\mapsto\mathbb{S}^{-1}[1]\circ A\circ\mathbb{S}^{1}[1].$

######  Definition 3.1.

Since $S$ is monoidal it induces an equivalence
$S\colon\mathrm{Operad}(\mathrm{Sp})\to\mathrm{Operad}(\mathrm{Sp})$. For an
operad $\mathcal{O}$, we call $S\mathcal{O}$ its _operadic suspension_ and
$S^{-1}\mathcal{O}$ its _operadic desuspension_.

######  Remark 3.2.

Let us explicitly describe the underlying symmetric sequence of the operadic
suspension of a non-unital operad $\mathcal{O}$. Consider the standard
$n$-dimensional real representation $\mathbb{R}^{n}$ of the symmetric group
$\Sigma_{n}$ given by permuting the standard basis vectors of
$\mathbb{R}^{n}$. The diagonal $\Delta$ is a trivial subrepresentation; we
write $\rho_{n}=\mathbb{R}^{n}/\Delta$ for the quotient. As usual, we write
$S^{V}$ for the one-point compactification of a representation $V$ and
$\mathbb{S}^{V}$ for the suspension spectrum of $S^{V}$. There are then
equivalences of representation spheres

$\mathbb{S}^{\mathbb{R}^{n}}\simeq(\mathbb{S}^{1})^{\otimes
n}\quad\quad\text{and}\quad\quad\mathbb{S}^{\rho_{n}}\simeq\Sigma^{-1}(\mathbb{S}^{1})^{\otimes
n},$

where the symmetric group $\Sigma_{n}$ acts by permuting the factors on the
right-hand sides. As a consequence, we find equivalences
$S\mathcal{O}(n)\simeq\mathbb{S}^{\rho_{n}}\otimes\mathcal{O}(n)$. Applying
this for $\mathcal{O}=\Sigma^{\infty}_{+}\mathbb{E}_{\infty}^{\mathrm{nu}}$,
we find that $\mathbb{S}^{\rho}$, the symmetric sequence sending $n$ to
$\mathbb{S}^{\rho_{n}}$, is a non-unital operad and that $S\mathcal{O}$ is
obtained from the pointwise tensor product of the operads $\mathbb{S}^{\rho}$
and $\mathcal{O}$.

Clearly, the operad $\mathbb{S}^{\rho}$ ought to be in the image of the
functor $\mathrm{Operad}(\mathcal{S}_{*})\to\mathrm{Operad}(\mathrm{Sp})$.
Indeed, Arone–Kankaanrinta [AK14] and Ching–Salvatore [CS22] both give
explicit constructions of a _sphere operad_ in the category of pointed spaces
that should provide such a desuspension of $\mathbb{S}^{\rho}$. It would be
desirable to have a comparison between their work and the discussion we gave
above, but we do not pursue the matter here. We will discuss the unstable case
further in Section 3.5.

By construction, the monad associated to the operad $S\mathcal{O}$ is
described by

$\mathrm{free}_{S\mathcal{O}}\simeq\Sigma^{-1}\circ\mathrm{free}_{\mathcal{O}}\circ\Sigma.$

Moreover, since
$\mathrm{Alg}_{\mathcal{O}}(\mathrm{Sp})=\mathrm{Alg}_{\mathrm{free}_{\mathcal{O}}}(\mathrm{Sp})$,
there is a commutative square of $\infty$-categories

${\mathrm{Alg}_{\mathcal{O}}(\mathrm{Sp})}$${\mathrm{Alg}_{S\mathcal{O}}(\mathrm{Sp})}$${\mathrm{Sp}}$${\mathrm{Sp}}$$\scriptstyle{\mathrm{frgt}}$$\scriptstyle{\mathrm{frgt}}$$\scriptstyle{\Sigma}$

obtained by letting the automorphism $\Sigma\colon\mathrm{Sp}\to\mathrm{Sp}$
act on $\mathrm{Monad}(\mathrm{Sp})$ via conjugation. Both horizontal functors
in this diagram are equivalences, so the category of $S\mathcal{O}$-algebras
is equivalent to that of $\mathcal{O}$-algebras via the suspension functor on
the level of underlying objects.

To conclude this section, we discuss a _suspension morphism_
$\sigma\colon\mathcal{O}\to S\mathcal{O}$ of a non-unital operad
$\mathcal{O}$. Let us first describe the effect $\sigma$ on the free algebra
monads of $\mathcal{O}$ and $S\mathcal{O}$, respectively. To that end, let $T$
be a monad on $\mathrm{Sp}$ that is reduced, i.e. $T(0)\cong 0$. Then there
are commuting squares

${\mathrm{Alg}_{T}(\mathrm{Sp})}$${\mathrm{Alg}_{T}(\mathrm{Sp})}$${\mathrm{Alg}_{T}(\mathrm{Sp})}$${\mathrm{Alg}_{T}(\mathrm{Sp})}$${\mathrm{Sp}}$${\mathrm{Sp}}$${\mathrm{Sp}}$${\mathrm{Sp}}$$\scriptstyle{\Sigma_{T}}$$\scriptstyle{\mathrm{fgt}_{T}}$$\scriptstyle{\mathrm{fgt}_{T}}$$\scriptstyle{\Omega_{T}}$$\scriptstyle{\mathrm{free}_{T}}$$\scriptstyle{\Sigma}$$\scriptstyle{\mathrm{free}_{T}}$$\scriptstyle{\Omega}$

of left and right adjoints, respectively, with $\Sigma_{T}$ and $\Omega_{T}$
denoting the suspension and loops functor internal to the pointed
$\infty$-category $\mathrm{Alg}_{T}(\mathrm{Sp})$. The unit of that adjunction
supplies a map of monads on $\mathrm{Sp}$ of the form

$T\to\mathrm{fgt}_{T}\Omega_{T}\Sigma_{T}\mathrm{free}_{T}\simeq\Omega\mathrm{fgt}_{T}\mathrm{free}_{T}\Sigma=\Omega
T\Sigma$

where the equivalence follows from the commutativity of the squares. In the
specific case $T=\mathrm{free}_{\mathcal{O}}$ for a non-unital operad
$\mathcal{O}$, we find a map of monads

$\sigma\colon\mathrm{free}_{\mathcal{O}}\longrightarrow\Omega\mathrm{free}_{\mathcal{O}}\Sigma\simeq\mathrm{free}_{S\mathcal{O}}.$

We claim that this map arises by applying the free functor to a map of operads
$\mathcal{O}\to S\mathcal{O}$ for which we also write $\sigma$. We will not
need a general construction here, but we will provide it in the specific case
$\mathcal{O}=\mathbb{E}_{n}^{\mathrm{nu}}$ in Section 3.4 below, see also
Remark 3.3. In particular, the map $\sigma$ yields a commutative diagram as
follows:

${\mathrm{Alg}_{\mathcal{O}}(\mathrm{Sp})}$${\mathrm{Alg}_{S\mathcal{O}}(\mathrm{Sp})}$${\mathrm{Alg}_{\mathcal{O}}(\mathrm{Sp})}$${\mathrm{Sp}}$${\mathrm{Sp}}$${\mathrm{Sp}}$$\scriptstyle{\mathrm{fgt}_{\mathcal{O}}}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\mathrm{fgt}_{S\mathcal{O}}}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{fgt}_{\mathcal{O}}}$$\scriptstyle{\Omega}$

Here the equivalence
$\mathrm{Alg}_{\mathcal{O}}(\mathrm{Sp})\simeq\mathrm{Alg}_{S\mathcal{O}}(\mathrm{Sp})$
in the top right is the inverse of the equivalence described above. Thus, we
may interpret the loop functor on $\mathrm{Alg}_{\mathcal{O}}(\mathrm{Sp})$ as
restriction along the morphism $\sigma$, up to an equivalence of categories.

######  Remark 3.3.

We recall that the formation of Goodwillie derivatives forms a functor

$\partial_{*}\colon\mathrm{End}(\mathrm{Sp})\to\mathrm{SSeq}(\mathrm{Sp})$

from endofunctors of $\mathrm{Sp}$ to symmetric sequences in $\mathrm{Sp}$. If
this construction would be (lax) monoidal, then it would induce a functor from
monads in $\mathrm{Sp}$ to operads in $\mathrm{Sp}$. This monoidality is
essentially the statement of the chain rule for the derivatives of
endofunctors of $\mathrm{Sp}$, but it has not quite been established in this
form in the literature. A version of the chain rule that does accomplish this
is the subject of ongoing work of Blans and Blom [BB]. Moreover, in loc. cit.
a monoidal equivalence
$\partial_{*}\circ\mathrm{free}\simeq\mathrm{id}_{\mathrm{SSeq}(\mathrm{Sp})}$
is established. One could therefore _define_ the suspended operad
$S\mathcal{O}$ as $\partial_{*}(\Omega\mathrm{free}_{\mathcal{O}}\Sigma)$ and
the suspension morphism as
$\partial_{*}(\sigma\colon\mathrm{free}_{\mathcal{O}}\to\Omega\mathrm{free}_{\mathcal{O}}\Sigma)$.
We will show later that this construction is equivalent to our construction of
the map $\sigma\colon\mathcal{O}\to S\mathcal{O}$ in case
$\mathcal{O}=\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$; note that we do
not give a construction of $\sigma$ otherwise.

###  3.2. Symmetric sequences with nilpotent Euler classes

The functor

$\mathrm{SSeq}(\mathrm{Sp})\longrightarrow\mathrm{Fun}(\mathrm{Sp},\mathrm{Sp}),\quad
A\mapsto\mathrm{free}_{A}$

is well known not to be fully faithful. Take for instance
$A=\mathbb{E}_{\infty}$. Then there is a non-trivial map
$\mathrm{id}=D^{\mathbb{E}_{\infty}}_{1}\to D^{\mathbb{E}_{\infty}}_{2}$, as
the space of such natural transformations is equivalent to
$\Omega^{\infty}(\mathbb{S}^{-1})^{tC_{2}}\simeq\Omega^{\infty+1}\mathbb{S}_{2}$,444See
the discussion around (3) for this argument. which contains the non-trivial
element $\eta$. This induces a non-trivial self-map of
$\mathrm{free}_{\mathbb{E}_{\infty}}$ which does not come from a self-map of
the symmetric sequence $\mathbb{E}_{\infty}$. However, we will show in Theorem
3.8 that the functor $A\mapsto\mathrm{free}_{A}$ _does_ become fully faithful
after restricting its domain to a certain full subcategory of symmetric
sequences, the ones with _nilpotent Euler classes_ , to which this section is
devoted.

Let $T$ be a finite set and let us write $\rho_{T}=\mathbb{R}[T]/\Delta$ for
the reduced standard representation of the symmetric group $\Sigma_{T}$. The
inclusion of the origin into $\rho_{T}$ induces a $\Sigma_{T}$-equivariant map

$e_{T}\colon S^{0}\to S^{\rho_{T}}$

which we refer to as the _Euler class_.

######  Definition 3.4.

A symmetric sequence $A\in\mathrm{SSeq}(\mathrm{Sp})$ has _nilpotent Euler
classes_ if there exists a natural number $k$, called the order of nilpotence,
with the following property: for each $n\geq 2$ and each subset
$T\subseteq\\{1,\ldots,n\\}$ of cardinality at least two, the map

$A(n)\simeq A(n)\otimes S^{0}\xrightarrow{\mathrm{id}\otimes
e_{T}^{k}}A(n)\otimes S^{k\rho_{T}}$

is nullhomotopic in the $\infty$-category
$\mathrm{Fun}(B\Sigma_{T},\mathrm{Sp})$, where $A(n)$ is interpreted as an
object of $\mathrm{Fun}(B\Sigma_{T},\mathrm{Sp})$ via restriction along the
evident inclusion $\Sigma_{T}\subseteq\Sigma_{n}$. Note that $k$ should not
depend on $n$ or $T$. An operad $\mathcal{O}\in\mathrm{Operad}(\mathrm{Sp})$
is said to have nilpotent Euler classes if its underlying symmetric sequence
does.

######  Example 3.5.

  1. (1)

Suppose that $A$ is a symmetric sequence such that for every $n\geq 2$ the
spectrum $A(n)$ is a $\Sigma_{n}$-free object, e.g.
$A=\Sigma^{\infty}_{+}\mathbb{E}_{1}^{\mathrm{nu}}$. Then $A$ has nilpotent
Euler classes with order of nilpotence equal to $1$. Indeed, for any
$T\subseteq\\{1,\dots,n\\}$, the $\Sigma_{T}$ object $A(n)$ is a sum of free
$\Sigma_{T}$-objects, hence the claim follows from the fact that the Euler
class $e_{T}\colon S^{0}\to S^{\rho_{T}}$ is nonequivariantly null when
$|T|\geq 2$.

  2. (2)

The symmetric sequence underlying the operad
$\Sigma^{\infty}_{+}\mathbb{E}_{\infty}^{\mathrm{nu}}$ does not have nilpotent
Euler classes. For example, taking homotopy orbits of the
$\Sigma_{2}$-equivariant map
$e_{2}^{k}\colon\mathbb{S}^{0}\to\mathbb{S}^{k\rho_{2}}$ yields the standard
map
$\Sigma^{\infty}\mathbb{RP}^{\infty}_{+}\to\Sigma^{\infty}\mathbb{RP}^{\infty}_{k},$
which is not nullhomotopic for any (finite) value of $k$.

######  Lemma 3.6.

The class of non-unital symmetric sequences with nilpotent Euler classes is
closed under operadic (de)suspensions and contains the monoidal unit.
Moreover, if $A$ is a non-unital symmetric sequence with nilpotent Euler
classes, then so is its composition product $A\circ B$ with any non-unital
symmetric sequence $B$. In particular, the class of non-unital symmetric
sequences with nilpotent Euler classes forms a monoidal subcategory of
$\mathrm{SSeq}(\mathrm{Sp})$.

###### Proof.

The monoidal unit of $\mathrm{SSeq}(\mathrm{Sp})$ is the sequence that
associates $\mathbb{S}$ to the singleton and $0$ to any finite set of
cardinality other than one, and consequently has nilpotent Euler classes.

Now let $A$ be a symmetric sequence. Observe that the Euler classes

$(SA)(n)\xrightarrow{e^{k}_{T}}(SA)(n)\otimes S^{k\rho_{T}}$

for $SA$ are obtained from those of $A$ by smashing with the invertible object
$\mathbb{S}^{\rho_{n}}$ of $\mathrm{Fun}(B\Sigma_{n},\mathrm{Sp})$. Hence the
operadic suspension $SA$ of $A$ has nilpotent Euler classes if and only if $A$
itself does.

Finally, consider a composition $A\circ B$ where $A$ has nilpotent Euler
classes, say with order of nilpotence $k$. Let $T$ be a subset of
$I:=\\{1,\ldots,n\\}$. As in our discussion of the composition product above,
the restriction of the $\Sigma_{I}$-object

$(A\circ
B)(I)\cong\bigoplus_{E\in\mathrm{Equiv}(I)}A(I/E)\otimes\bigotimes_{J\in
I/E}B(J)$

to $\Sigma_{T}\subseteq\Sigma_{n}$ splits into summands corresponding to the
different orbits of $\mathrm{Equiv}(I)/\Sigma_{T}$. Consider an
$E\in\mathrm{Equiv}(I)$ with stabilizer $G_{E}^{T}\leq\Sigma_{T}$. Then it
will suffice to show that

$A(I/E)\otimes\bigotimes_{J\in
I/E}B(J)\xrightarrow{e^{k}_{T}}A(I/E)\otimes\bigotimes_{J\in I/E}B(J)\otimes
S^{k\rho_{T}}$

is $G_{E}^{T}$-equivariantly null, where we recall $k$ to be the order of
nilpotence of $A$. If $G_{E}^{T}$ is a non-transitive subgroup of
$\Sigma_{T}$, then this is immediate from the fact that the restriction of the
Euler class $e_{T}$ itself to $G_{E}^{T}$ is null: indeed, then the
representation $\rho_{T}$ admits a nonzero $G_{E}^{T}$-fixed point. If
$G_{E}^{T}$ is a transitive subgroup of $\Sigma_{T}$, then the restriction of
the equivalence relation $E$ to $T$ must be a partition into blocks of equal
size; let us say there are $a$ blocks of size $b$. Then
$G_{E}^{T}\cong\Sigma_{b}\wr\Sigma_{a}$ and $T/E\cong\\{1,\ldots,a\\}$. Write
$q\colon G_{E}^{T}\to\Sigma_{a}$ for the quotient by the normal subgroup
$\Sigma_{b}^{a}$.

The standard representation $\mathbb{R}^{T}$ of $\Sigma_{T}$, when restricted
to $\Sigma_{b}\wr\Sigma_{a}$, can be written as
$\mathbb{R}^{a}\otimes\mathbb{R}^{b}$ in an evident way. As a
$\Sigma_{b}$-representation, we can split $\mathbb{R}^{b}$ as
$\Delta\oplus\rho_{b}$, simply by averaging. Hence we find an isomorphism of
representations

$\mathbb{R}^{a}\otimes\mathbb{R}^{b}\cong\mathbb{R}^{a}\oplus(\mathbb{R}^{a}\otimes\rho_{b}).$

From this we obtain isomorphisms of $\Sigma_{b}\wr\Sigma_{a}$-objects

$\rho_{T}\cong\rho_{a}\oplus\bigoplus_{i=1}^{a}\rho_{b}\quad\quad\text{and
hence}\quad\quad S^{\rho_{T}}\cong S^{\rho_{a}}\otimes(S^{\rho_{b}})^{\otimes
a}.$

We now observe that the map $e_{k}^{T}$ under consideration admits a tensor
factor

$A(I/E)\to A(I/E)\otimes S^{k\rho_{a}}$

on which the subgroup $\Sigma_{b}^{a}$ acts trivially. More precisely, this
factor is in the image of $q^{*}$. Hence it suffices to show that this map is
$\Sigma_{a}$-equivariantly null. That this is so follows from the assumption
that $A$ has nilpotent Euler classes of order $k$, after identifying $T/E$
with its image in $I/E$ and $\Sigma_{a}$ with the corresponding subgroup of
the permutation group of $I/E$. ∎

Our main source of operads with nilpotent Euler classes is the following:

######  Proposition 3.7.

For each $m\geq 1$, the operad
$\Sigma^{\infty}_{+}\mathbb{E}_{m}^{\mathrm{nu}}$ has nilpotent Euler classes.

###### Proof.

We will show that $k=m$ satisfies the condition of Definition 3.4. Recall that
$\mathbb{E}_{m}^{\mathrm{nu}}(n)$ is (homotopy equivalent to) the
configuration space $\mathrm{Conf}_{n}(\mathbb{R}^{m})$ of $n$ points in
$\mathbb{R}^{m}$. The map

$\Sigma^{\infty}_{+}\mathbb{E}_{m}^{\mathrm{nu}}(n)\otimes
S^{0}\xrightarrow{e_{T}^{m}}\Sigma^{\infty}_{+}\mathbb{E}_{m}^{\mathrm{nu}}(n)\otimes
S^{m\rho_{T}}$

is the infinite suspension of the map of pointed $\Sigma_{T}$-spaces

$\mathrm{Conf}_{n}(\mathbb{R}^{m})_{+}\xrightarrow{e_{T}^{m}}\mathrm{Conf}_{n}(\mathbb{R}^{m})_{+}\wedge
S^{m\rho_{T}}.$

The right-hand side may be interpreted as the Thom space of the
$\Sigma_{T}$-equivariant (trivial) vector bundle
$\mathrm{Conf}_{n}(\mathbb{R}^{m})\times\rho_{T}^{\oplus m}$ over the
configuration space. The map $e_{T}^{m}$ then includes
$\mathrm{Conf}_{n}(\mathbb{R}^{m})$ as the zero-section of this bundle and
sends the disjoint basepoint $+$ to the basepoint ‘at infinity’. To show that
the map $e_{T}^{m}$ is equivariantly nullhomotopic, it suffices to construct a
nowhere vanishing equivariant section of this bundle or, in other words, a
nowhere vanishing equivariant function

$f=(f_{1},\ldots,f_{m})\colon\mathrm{Conf}_{n}(\mathbb{R}^{m})\longrightarrow\rho_{T}^{\oplus
m}.$

Indeed, multiplying such a section by a scalar $t$ and letting $t$ range from
0 to $\infty$ then defines a homotopy from the zero-section to the constant
map sending all of $\mathrm{Conf}_{n}(\mathbb{R}^{m})$ to $\infty$.

Let $\mathrm{Conf}_{T}(\mathbb{R}^{m})$ denote the space of configurations of
the set $T$ inside $\mathbb{R}^{m}$. Then it suffices to construct a nowhere
vanishing equivariant map

$f=(f_{1},\ldots,f_{m})\colon\mathrm{Conf}_{T}(\mathbb{R}^{m})\longrightarrow\rho_{T}^{\oplus
m},$

since the previous case follows after precomposing with the forgetful map
$\mathrm{Conf}_{n}(\mathbb{R}^{m})\to\mathrm{Conf}_{T}(\mathbb{R}^{m})$. For
$1\leq i\leq m$, let

$g_{i}\colon\mathrm{Conf}_{T}(\mathbb{R}^{m})\longrightarrow\mathbb{R}^{T}$

be the map taking the $i$th coordinate of every point in a given
configuration. Set $f_{i}$ to be the composition of $g_{i}$ with the quotient
map $\mathbb{R}^{T}\to\rho_{T}$. Since the $g_{i}$ cannot all take values in
the diagonal simultaneously, the map $f$ has no zeros. ∎

###  3.3. From operads to monads

The goal of this section is to prove the following:

######  Theorem 3.8.

The functors

$\mathrm{free}\colon\mathrm{SSeq}(\mathrm{Sp})\longrightarrow\mathrm{End}(\mathrm{Sp})\quad\quad\text{and}\quad\quad\mathrm{free}\colon\mathrm{Operad}(\mathrm{Sp})\longrightarrow\mathrm{Monad}(\mathrm{Sp})$

become fully faithful when restricted to the full subcategories of non-unital
symmetric sequences (resp. non-unital operads) with nilpotent Euler classes.

Our argument is inspired by [Heu21, Appendix B], where the corresponding
result is proved for symmetric sequences and operads in $T(n)$-local spectra
(where the hypothesis on nilpotent Euler classes is unnecessary). As in loc.
cit., we will use _dual Goodwillie calculus_ for endofunctors of
$\mathrm{Sp}$, which is nothing but Goodwillie calculus applied to the
opposite category $\mathrm{Sp}^{\mathrm{op}}$, cf. [Heu21, Appendix A].
Goodwillie calculus itself supplies for any functor
$F\colon\mathrm{Sp}\to\mathrm{Sp}$ a tower of approximations from the right

$F\to\cdots\to P_{n}F\to P_{n-1}F\to\cdots\to P_{1}F\to P_{0}F$

with the property that $F\to P_{n}F$ is the initial map from $F$ to an
$n$-excisive functor. Dual Goodwillie calculus then gives a cotower of
approximations from the left

$P^{0}F\to\cdots\to P^{n-1}F\to P^{n}F\to\cdots\to F$

with the property that $P^{n}F\to F$ is the terminal map to $F$ from an
$n$-excisive functor.555Since $\mathrm{Sp}$ is stable, there is no distinction
between $n$-excisive and $n$-coexcisive functors.

The key to our proof of Theorem 3.8 is the following observation:

######  Lemma 3.9.

If the symmetric sequence $B$ has nilpotent Euler classes, then the evident
inclusion

$\bigoplus_{k=1}^{n}D_{k}^{B}\longrightarrow\bigoplus_{k=1}^{\infty}D_{k}^{B}=\mathrm{free}_{B}$

induces an equivalence

$\bigoplus_{k=1}^{n}D_{k}^{B}\xrightarrow{\,\simeq\,}P^{n}\mathrm{free}_{B}.$

######  Remark 3.10.

The functor $P_{n}$ commutes with filtered colimits, making it straightforward
to calculate that $P_{n}\mathrm{free}_{B}$ is also given by the sum of the
first $n$ homogeneous layers. However, the dual approximation $P^{n}$
generally does _not_ commute with filtered colimits. Nonetheless, the lemma
shows that it commutes with the infinite direct sum $\bigoplus_{n\geq
1}D_{n}^{B}$ in case $B$ has nilpotent Euler classes.

We will prove Lemma 3.9 after we show how it implies Theorem 3.8:

###### Proof of Theorem 3.8.

By passing to algebra objects, Lemma 3.6 implies that it suffices to show that
the functor
$\mathrm{free}\colon\mathrm{SSeq}(\mathrm{Sp})\to\mathrm{End}(\mathrm{Sp})$ is
fully faithful when restricted to the full subcategory of symmetric sequences
with nilpotent Euler classes. To that end, we consider the commutative diagram

${\mathrm{Map}_{\mathrm{SSeq}(\mathrm{Sp})}(A,B)}$${\mathrm{Map}_{\mathrm{End}(\mathrm{Sp})}(\mathrm{free}_{A},\mathrm{free}_{B})}$${\prod\limits_{n\geq
1}\mathrm{Map}_{\mathrm{Sp}^{B\Sigma_{n}}}(A(n),B(n))}$${\prod\limits_{n\geq
1}\mathrm{Map}_{\mathrm{End}(\mathrm{Sp})}(D_{n}^{A},\mathrm{free}_{B})}$${\prod\limits_{n\geq
1}\mathrm{Map}_{\mathrm{End}(\mathrm{Sp})}(D_{n}^{A},D_{n}^{B})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

in which we aim to show that the upper horizontal map is an equivalence. The
left vertical map is an equivalence since $\mathrm{Fin}^{\simeq}$ is
equivalent to the coproduct of the categories $B\Sigma_{n}$ and the right
downward vertical map is an equivalence since $\mathrm{free}_{A}$ is the
coproduct of the functors $D^{A}_{n}$. The diagonal arrow is an equivalence by
Goodwillie’s classification of homogenous functors. It hence remains to show
that the lower right vertical arrow is also an equivalence. By definition, it
is given by the product over $n\geq 1$ of the following composites.

$\mathrm{Map}(D_{n}^{A},D_{n}^{B})\longrightarrow\mathrm{Map}(D_{n}^{A},\bigoplus_{k=1}^{n}D_{k}^{B})\longrightarrow\mathrm{Map}(D_{n}^{A},P^{n}\mathrm{free}_{B})\longrightarrow\mathrm{Map}(D_{n}^{A},\mathrm{free}_{B})$

The first map in this composite is an equivalence since for $k<n$, $D_{k}^{B}$
is $k$-excisive and $P_{k}(D^{A}_{n})=0$. The second map is an equivalence by
Lemma 3.9 and the third map is an equivalence since $D_{n}^{A}$ is
$n$-excisive. This completes the argument that $\mathrm{free}$ is fully
faithful on symmetric sequences with nilpotent Euler classes and hence the
proof of the theorem. ∎

###### Proof of Lemma 3.9.

It will suffice to show that
$P^{n}\bigl{(}\bigoplus_{k>n}D_{k}^{B}\bigr{)}\cong 0$. We can reduce further
to checking that the dual derivative
$D^{\ell}\bigl{(}\bigoplus_{k>n}D_{k}^{B}\bigr{)}$ vanishes for each $\ell\leq
n$. Generally, the dual derivative $D^{\ell}F$ of a functor
$F\in\mathrm{End}(\mathrm{Sp})$ may be constructed as follows. First one forms
the _cocross effect_ of $F$, which is the functor of $\ell$ variables defined
as the total cofibre

$\mathrm{cr}^{\ell}F(X_{1},\ldots,X_{\ell}):=\mathrm{tcof}(F\circ\mathcal{X}),$

where $\mathcal{X}$ is the cube

$\mathcal{X}\colon\mathcal{P}(\ell)\rightarrow\mathrm{Sp}\colon
T\mapsto\bigoplus_{t\in T}X_{t}.$

Here $\mathcal{P}(\ell)$ is the power set of $\\{1,\ldots,\ell\\}$ regarded as
a poset under inclusion. Now one colinearizes $\mathrm{cr}^{\ell}F$ in each
variable to obtain a multilinear functor $\partial^{\ell}F$ described by

$(\partial^{\ell}F)(X_{1},\ldots,X_{\ell}):=\varprojlim_{j}\Sigma^{j\ell}(\mathrm{cr}^{\ell}F)(\Sigma^{-j}X_{1},\ldots,\Sigma^{-j}X_{\ell}).$

Finally, the dual derivative is then

$D^{\ell}F(X)=\partial^{\ell}F(X,\ldots,X)^{h\Sigma_{\ell}}.$

We are analyzing the specific case $F=\bigoplus_{k>n}D_{k}^{B}$. To show that
$D^{\ell}F$ vanishes, we will argue that $\partial^{\ell}F$ vanishes. The
cocross effect $\mathrm{cr}^{\ell}F(X_{1},\ldots,X_{\ell})$ is easily seen to
be the direct sum over $k>n$ of the terms

$\Bigl{(}\bigoplus_{f\colon\mathbf{k}\to\mathbf{l}}B(k)\otimes\bigotimes_{i=1}^{\ell}X_{i}^{\otimes
f^{-1}(i)}\Bigr{)}_{h\Sigma_{k}}.$

Here the sum is indexed over surjections
$f\colon\\{1,\ldots,k\\}\to\\{1,\ldots,\ell\\}$. As a consequence, the inverse
system
$\Sigma^{j\ell}(\mathrm{cr}^{\ell}F)(\Sigma^{-j}X_{1},\ldots,\Sigma^{-j}X_{\ell})$
is the direct sum over $k>n$ of the homotopy $\Sigma_{k}$-orbits of the
inverse systems in $j$

$\bigoplus_{f\colon\mathbf{k}\to\mathbf{l}}B(k)\otimes\bigotimes_{i=1}^{\ell}\mathbb{S}^{-j\rho_{f^{-1}(i)}}\otimes\bigotimes_{i=1}^{\ell}X_{i}^{\otimes
f^{-1}(i)}.$

Here the spectrum $\bigotimes_{i=1}^{\ell}\mathbb{S}^{-j\rho_{f^{-1}(i)}}$
denotes a smash product of (virtual) representation spheres corresponding to
the stabilizer $\Sigma_{f^{-1}(1)}\times\cdots\times\Sigma_{f^{-1}(\ell)}$ of
a given surjection $f$. We conclude that the expression above is induced, as a
spectrum with $\Sigma_{k}$-action, from a Young subgroup of the form
$\Sigma_{k_{1}}\times\cdots\times\Sigma_{k_{\ell}}$, with
$k_{1}+\cdots+k_{\ell}=k$. Since $k>\ell$ there exists an $i$ with $k_{i}\geq
2$.

The conclusion of the lemma will follow if we argue that there exists a
constant $C$, independent of $k$ and $f$, such that each of the maps

$B(k)\otimes\bigotimes_{i=1}^{\ell}\mathbb{S}^{-C\rho_{k_{i}}}\to B(k)$

is null in $\mathrm{Fun}(B\Sigma_{k_{1}}\times\cdots\times
B\Sigma_{k_{\ell}},\mathrm{Sp})$. But this map admits a factor of the form

$B(k)\otimes\mathbb{S}^{-C\rho_{k_{i}}}\to B(k)$

with $k_{i}\geq 2$, which is null if $C$ is at least the order of nilpotence
of the Euler classes of $B$. ∎

###  3.4. The suspension morphism of $\mathbb{E}_{n}$

The main result of Section 2 was Theorem 2.4, providing two different
factorizations of the suspension functor on the $\infty$-category of
$\mathbb{E}_{n}$-algebras. We will now see how, in the stable context, this
statement can also be considered as one about the $\mathbb{E}_{n}$-operad
itself. In the following theorem we have abbreviated the notation
$\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$ to simply $\mathbb{E}_{n}$,
and similarly for the other operads.

######  Theorem 3.11.

There is an essentially unique commutative square

${\mathbb{E}_{n}}$${\mathbb{E}_{n+1}}$${S\mathbb{E}_{n-1}}$${S\mathbb{E}_{n}}$$\scriptstyle{\iota_{n}}$$\scriptstyle{\beta_{n}}$$\scriptstyle{\sigma}$$\scriptstyle{\beta_{n+1}}$$\scriptstyle{S\iota_{n-1}}$

of non-unital operads in $\mathrm{Sp}$ such that the resulting diagram of
$\infty$-categories of algebras

${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathrm{Sp})}$${\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{nu}}(\mathrm{Sp})}$${\mathrm{Alg}_{S\mathbb{E}_{n-1}}^{\mathrm{nu}}(\mathrm{Sp})}$${\mathrm{Alg}_{S\mathbb{E}_{n}}^{\mathrm{nu}}(\mathrm{Sp})}$

is equivalent to the diagram of Theorem 2.4 in the special case
$\mathcal{C}=\mathrm{Sp}$.

###### Proof.

Adding the free functors for the various algebra categories to the square of
Theorem 2.4 and identifying non-unital with augmented algebras, gives the
following commutative diagram of left adjoint functors.

${\mathrm{Sp}}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathrm{Sp})}$${\mathrm{Alg}_{\mathbb{E}_{n+1}}^{\mathrm{nu}}(\mathrm{Sp})}$${\mathrm{Alg}_{\mathbb{E}_{n-1}}^{\mathrm{nu}}(\mathrm{Sp})}$${\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{nu}}(\mathrm{Sp})}$$\scriptstyle{\mathrm{free}^{\mathbb{E}_{n}}}$$\scriptstyle{\mathrm{free}^{\mathbb{E}_{n+1}}}$$\scriptstyle{\mathrm{free}^{\mathbb{E}_{n-1}}\circ\Sigma}$$\scriptstyle{\mathrm{free}_{\mathbb{E}_{n}}^{\mathbb{E}_{n+1}}}$$\scriptstyle{\mathrm{Bar}}$$\scriptstyle{\Sigma_{\mathbb{E}_{n}}}$$\scriptstyle{\mathrm{Bar}}$$\scriptstyle{\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}}$

We should verify that the two triangles added to the square indeed commute.
This is clear for the upper one, whereas for the left one it follows from the
natural equivalences

$\mathrm{Bar}\circ\mathrm{free}^{\mathbb{E}_{n}}\simeq\mathrm{Bar}\circ\mathrm{free}_{\mathbb{E}_{n-1}}^{\mathbb{E}_{n}}\circ\mathrm{free}^{\mathbb{E}_{n-1}}\simeq\Sigma_{\mathbb{E}_{n-1}}\circ\mathrm{free}^{\mathbb{E}_{n-1}}\simeq\mathrm{free}^{\mathbb{E}_{n-1}}\circ\Sigma.$

Here the second equivalence is a consequence of Theorem 2.4 again and the
others are clear.

The left adjoints in the diagram above allow us to view all the participating
$\infty$-categories as monadic over $\mathrm{Sp}$ and the corresponding
diagram of monads on $\mathrm{Sp}$ is of the form

${\mathrm{free}_{\mathbb{E}_{n}}}$${\mathrm{free}_{\mathbb{E}_{n+1}}}$${\Omega\mathrm{free}_{\mathbb{E}_{n-1}}\Sigma}$${\Omega\mathrm{free}_{\mathbb{E}_{n}}\Sigma.}$$\scriptstyle{\iota_{n}}$$\scriptstyle{\sigma}$$\scriptstyle{\iota_{n-1}}$

By our earlier discussion on operadic suspension, we can identify the monads
on the bottom row with $\mathrm{free}_{S\mathbb{E}_{n-1}}$ and
$\mathrm{free}_{S\mathbb{E}_{n}}$ respectively. The conclusion of the theorem
now follows from Theorem 3.8 and the fact that all operads involved have
nilpotent Euler classes, cf. Proposition 3.7 and Lemma 3.6. ∎

######  Remark 3.12.

As indicated in Remark 3.3 one could also appeal to a monoidal structure on
the Goodwillie derivative
$\partial_{*}\colon\mathrm{End}(\mathrm{Sp})\to\mathrm{SSeq}(\mathrm{Sp})$ to
obtain the square of operads of Theorem 3.11 from the square of monads at the
end of the proof we just gave. The argument we have given above circumvents
the issue that this monoidal structure is not yet established and gives a
sharper conclusion, by the fully faithfulness established in Theorem 3.8.

######  Remark 3.13.

The ‘wrong-way morphism’ $\beta_{n}\colon\mathbb{E}_{n}\to S\mathbb{E}_{n-1}$
arising from the bar construction can be described in many ways. First of all,
maps like this were already considered by May [May06]. Ching–Salvatore [CS22]
also describe a map of operads $\mathbb{E}_{n}\to S\mathbb{E}_{n-1}$ arising
from the Koszul self-duality of $\mathbb{E}_{n}$. To be precise, they
construct a functor

$K\colon\mathrm{Operad}(\mathrm{Sp})\to\mathrm{Operad}(\mathrm{Sp})^{\mathrm{op}}$

forming the ‘Koszul dual’ $K\mathcal{O}$ of an operad $\mathcal{O}$. It is
given by the levelwise Spanier–Whitehead dual of the operadic bar
construction. Then they prove that there is an equivalence of operads in
$\mathrm{Sp}$ of the form

$K\mathbb{E}_{n}\cong S^{-n}\mathbb{E}_{n}.$

Under this equivalence (and the analogous one for $n-1$) the evident inclusion
$\iota_{n-1}\colon\mathbb{E}_{n-1}\to\mathbb{E}_{n}$ produces a map of operads
$\mathbb{E}_{n}\to S\mathbb{E}_{n-1}$; Ching–Salvatore include an explicit
description of this map in [CS22]. This morphism should be equivalent to the
$\beta_{n}$ we have constructed, although we will not pursue the matter here.
It is interesting to note that Ching–Salvatore’s map already exists _unstably_
, as a map of operads in pointed spaces. Since we have only dealt with the
operadic suspension and the suspension morphism for $\mathbb{E}_{n}$ in the
stable case, our methods cannot address this unstable version.

Moreover, as indicated in the introduction, [Fre11, Theorem C] also provides a
square of operads as in Theorem 3.11 after tensoring with an ordinary
commutative ring $R$, based on explicit point-set models. Again, our square
(after tensoring with $R$) should be equivalent to his, but we will not pursue
the matter here.

###  3.5. The suspension morphism for operads in pointed spaces

We have discussed the operadic suspension in the stable case, i.e. for operads
in $\mathrm{Sp}$, but in fact it should already exist in the context of
operads in the symmetric monoidal $\infty$-category of pointed spaces with
smash product, or generally in pointed symmetric monoidal $\infty$-categories
for which the tensor product is compatible with colimits in each variable. As
indicated earlier, for the case of pointed spaces there are several versions
of this in the literature. Arone–Kankaanrinta [AK14] construct a _sphere
operad_ $S^{\rho}$, of which the terms $S^{\rho}(n)$ are equivalent to the
representation spheres $S^{\rho_{n}}$ and the composition maps

$S^{\rho}(n)\wedge S^{\rho}(k_{1})\wedge\cdots\wedge S^{\rho}(k_{n})\to
S^{\rho}(k_{1}+\cdots+k_{n})$

are homotopy equivalences (or even homeomorphisms). For a general non-unital
operad in pointed spaces, one can then define its operadic suspension by
taking the levelwise smash product with this sphere operad. An alternative
construction of a sphere operad and the corresponding operadic suspension is
given by Ching–Salvatore [CS22].

In order to compare these constructions to each other and to the operadic
suspension we have discussed here it is desirable to characterize the operadic
suspension by universal properties, rather than focus on any specific
construction of it. In this brief section we point out what such properties
might be and invite the reader to take this up. We speculate that our
factorizations of the suspension morphism
$\sigma\colon\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}\to
S\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$ of Theorem 3.11 already
exist for the suspension morphism
$\sigma\colon(\mathbb{E}_{n}^{\mathrm{nu}})_{+}\to
S(\mathbb{E}_{n-1}^{\mathrm{nu}})_{+}$ of the (unstable) pointed
$\mathbb{E}_{n}$-operad, which is essentially obtained by means of the Euler
classes. If true, these factorizations should then provide alternative proofs
of Theorem 2.4 (on suspensions of $\mathbb{E}_{n}$-algebras) and Corollary 2.7
(on loops of $\mathbb{E}_{n}$-algebras).

We will now describe the desired properties of the operadic suspension.
Consider a presentable pointed symmetric monoidal $\infty$-category
$(\mathcal{C},\wedge)$ and assume that the tensor product commutes with
colimits in each variable separately. Let $\mathcal{O}$ be a non-unital operad
in $\mathcal{C}$. Then there should be a suspended operad $S\mathcal{O}$ with
terms

$S\mathcal{O}(n)=\mathcal{O}(n)\wedge S^{{\rho}_{n}}$

together with a canonical suspension morphism $\sigma\colon\mathcal{O}\to
S\mathcal{O}$ induced from the Euler classes $S^{0}\to S^{{\rho}_{n}}$. These
data should satisfy the following property:

  * (F)

There is a canonical equivalence in $\mathrm{End}(\mathcal{C})$ between
$\Sigma\mathrm{free}_{S\mathcal{O}}$ and $\mathrm{free}_{\mathcal{O}}\Sigma$.
Its adjoint map
$\mathrm{free}_{S\mathcal{O}}\to\Omega\mathrm{free}_{\mathcal{O}}\Sigma$
refines to a map of monads, making the composite

$\mathrm{free}_{\mathcal{O}}\xrightarrow{\mathrm{free}_{\sigma}}\mathrm{free}_{S\mathcal{O}}\longrightarrow\Omega\mathrm{free}_{\mathcal{O}}\Sigma$

equivalent to the suspension morphism of the reduced monad
$\mathrm{free}_{\mathcal{O}}$.

Let us briefly explain the relation between the factorization above and the
suspension-loop adjunction on the $\infty$-category
$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$ itself. Consider the following right
adjoint functors:

${\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\xrightarrow{\,\Omega\,}\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\xrightarrow{\mathrm{fgt}}\mathcal{C}.}$

The monad on $\mathcal{C}$ associated with the composed adjunction is
$\Omega\mathrm{free}_{\mathcal{O}}\Sigma$. The composite of right adjoints
must therefore factor over a right adjoint functor
$\gamma^{*}\colon\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\Omega\mathrm{free}_{\mathcal{O}}\Sigma}(\mathcal{C})$.
Combining this with property (F) above allows us to construct a diagram of
right adjoint functors

${\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})}$${\mathrm{Alg}_{\Omega\mathrm{free}_{\mathcal{O}}\Sigma}(\mathcal{C})}$${\mathrm{Alg}_{S\mathcal{O}}(\mathcal{C})}$${\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})}$${\mathcal{C}}$${\mathcal{C}.}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\mathrm{fgt}}$$\scriptstyle{\Omega}$$\scriptstyle{\mathrm{fgt}}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\mathrm{fgt}}$$\scriptstyle{\mathrm{fgt}}$$\scriptstyle{\Omega}$

In particular, this shows that the loops functor
$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\xrightarrow{\Omega}\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$
factors over the restriction
$\sigma^{*}\colon\mathrm{Alg}_{S\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$
along the suspension morphism. Dually, the suspension functor of
$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$ factors over the pushforward functor
$\sigma_{!}$. We note that in the case where $\mathcal{C}$ is stable, the
first two arrows on the top row of the diagram are equivalences of
$\infty$-categories and their composite coincides with the shift we have
considered earlier. However, in the general case these need not be
equivalences. For general $\mathcal{C}$, it would be desirable if
$S\mathcal{O}$, together with its suspension morphism
$\sigma\colon\mathcal{O}\to S\mathcal{O}$ were determined by property (F)
described above, or to give some other additional characterizing properties.
We invite the interested reader to take this up.

##  4\. Triviality of spheres

In this section we discuss some examples where our results can be applied and
we investigate in what sense our results are optimal.

###  4.1. Preliminaries

For this section, let us fix an $\mathbb{E}_{n+1}$-ring spectrum $A$, so that
$\mathrm{Mod}(A)$ is a presentably $\mathbb{E}_{n}$-monoidal
$\infty$-category. We write $\mathrm{Alg}_{\mathbb{E}_{n}}(A)$ for
$\mathrm{Alg}_{\mathbb{E}_{n}}(\mathrm{Mod}(A))$ and likewise for the
augmented and non-unital versions. In what follows, an object of
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(A)$ is called
$\mathbb{E}_{n}$-trivial over $A$ if it is equivalent to
$\mathrm{tr}_{\mathbb{E}_{n}}$ of its underlying augmented
$\mathbb{E}_{0}$-algebra. For a space $X$, we write $C^{*}(X;A)$ for the
“cochains of $X$ with coefficients in $A$”, i.e. for the $X$-indexed limit of
the constant diagram on $A$. A choice of a basepoint on $X$ makes $C^{*}(X;A)$
into an object of $\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(A)$. As a
consequence of Corollary 2.12, we have the following result.

######  Theorem 4.1.

Let $X$ be a pointed space and $n\geq 0$. Then $C^{*}(\Sigma^{n}X;A)$ is
$\mathbb{E}_{n}$-trivial over $A$. In particular, $C^{*}(S^{n};A)$ is
$\mathbb{E}_{n}$-trivial over $A$.

###### Proof.

The functor
$C^{*}(-;A)\colon(\mathcal{S}_{*})^{\mathrm{op}}\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(A)$
preserves pullbacks and hence sends $\Sigma^{n}X$ to $\Omega^{n}C^{*}(X;A)$.
The result then follows from Corollary 2.12. ∎

Let us assume for a moment that $A$ is a discrete commutative ring and let us
also denote by $A$ its associated Eilenberg–Mac Lane spectrum. Then
$H^{*}(\Sigma^{n}X;A)$ is also canonically an object of
$\mathrm{Alg}_{\mathbb{E}_{\infty}}^{\mathrm{aug}}(A)$, via the composite of
lax symmetric monoidal functors

(1)
$\mathrm{Mod}(A)\xrightarrow{\pi_{*}}\mathrm{Mod}(A)^{\heartsuit}_{\mathrm{gr}}\xrightarrow{\,H\,}\mathrm{Mod}(A).$

Here, $\mathrm{Mod}(A)^{\heartsuit}_{\mathrm{gr}}$ denotes the ordinary
symmetric monoidal category of graded discrete $A$-modules, whereas
$\mathrm{Mod}(A)$ is equivalent to $\mathcal{D}(A)$, the unbounded derived
category of $A$, and the second functor in the above display is the graded
Eilenberg–Mac Lane functor.

######  Definition 4.2.

Let $X$ be a space and $A$ a commutative ring. We say that $X$ is
_$\mathbb{E}_{n}$ -formal over $A$_ if there exists an equivalence
$C^{*}(X;A)\simeq H^{*}(X;A)$ in
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(A)$.

######  Remark 4.3.

The formal spaces from rational homotopy theory are, in the above notation,
the $\mathbb{E}_{\infty}$-formal spaces over $\mathbb{Q}$. This uses the
equivalence between commutative differential graded algebras over $\mathbb{Q}$
and $\mathrm{Alg}_{\mathbb{E}_{\infty}}(\mathcal{D}(\mathbb{Q}))$ [Lur17,
Prop. 7.1.4.11]. Surprisingly, a space which is $\mathbb{E}_{1}$-formal over
$\mathbb{Q}$ is already $\mathbb{E}_{\infty}$-formal over $\mathbb{Q}$ [Sal17,
CPRNW23] so that the notion of $\mathbb{E}_{k}$-formality over $\mathbb{Q}$ is
independent of $k$ once $k\geq 1$. This is very special to fields of
characteristic $0$. Indeed, as we shall see below, the same statement fails
for non-rational rings.

Our main result implies the following formality result.

######  Theorem 4.4.

Let $X$ be a pointed space and $n\geq 0$. If there exists an equivalence
$\tilde{C}^{*}(X;A)\simeq\tilde{H}^{*}(X;A)$ in $\mathrm{Mod}(A)$, then
$\Sigma^{n}X$ is $\mathbb{E}_{n}$-formal over $A$.

###### Proof.

We have observed above that
$C^{*}(\Sigma^{n}X;A)\simeq\mathrm{tr}_{\mathbb{E}_{n}}(\tilde{C}^{*}(X;A))$.
Hence it suffices to observe that also
$H^{*}(\Sigma^{n}X;A)\simeq\mathrm{tr}_{\mathbb{E}_{n}}(\tilde{H}^{*}(X;A))$.
This follows from the fact that the composite (1) commutes with the suspension
and loop functors. ∎

######  Example 4.5.

It is well-known in rational homotopy theory that suspensions are formal. One
way to prove this is to show that any suspension is rationally equivalent to a
wedge of spheres, and then to separately show that spheres are formal, and
that wedges of formal spaces are formal, see e.g. [Ber12, Theorem 10.28] under
additional finiteness assumptions. Theorem 4.4 gives the following non-
computational proof of the formality of suspensions. Assume that $A$ is a
commutative ring of global dimension $\leq 1$, e.g. a Dedekind domain or a
field. It is well-known that for any object $M\in\mathrm{Mod}(A)$, there
exists an equivalence $M\simeq\oplus_{n}\Sigma^{n}\pi_{n}(M)$, see e.g.
[HLN21, proof of Cor. 3.8] which applies verbatim to the case at hand. The
right hand side is the value of the composite (1) applied to $M$. In
particular $\Sigma X$ is $\mathbb{E}_{1}$-formal over $A$, and hence as
indicated above, $\mathbb{E}_{\infty}$-formal over $\mathbb{Q}$.

######  Example 4.6.

Suppose that $X$ is of finite $\mathbb{Z}$-homological type and $A$ is a
commutative ring. Then the canonical map
$C^{*}(X;\mathbb{Z})\otimes_{\mathbb{Z}}A\to C^{*}(X;A)$ is an equivalence.
The same statement for cohomology in place if cochains is, of course, not
correct in general. However, it is true if the cohomology of $X$ is in
addition free as a graded abelian group. In this case, the same is true for
$\Sigma^{n}X$ and reasoning as in Example 4.5 one deduces that $\Sigma^{n}X$
is $\mathbb{E}_{n}$-formal over any commutative ring $A$. Examples of such $X$
include spheres, projective spaces over $\mathbb{C}$ and $\mathbb{H}$, loop
spaces of spheres, free loop spaces of odd dimensional spheres, as well as
products of such spaces.

######  Remark 4.7.

There are many examples of formal spaces whose cochain algebras are not
trivial algebras, for instance compact Kähler manifolds different from
$S^{2}$. Our methods cannot be applied in this situation.

We single out the most basic cases of suspensions in the following corollary.

######  Corollary 4.8.

For $n\geq 0$, the $n$-sphere $S^{n}$ is $\mathbb{E}_{n}$-formal over any
commutative ring.

Coming back to interpreting formality of spheres in terms of triviality, one
may wonder whether our results are suboptimal in the sense that $n$-fold loop
objects are even trivial as $\mathbb{E}_{n+1}$-algebras. The above case of
cochains on suspensions gives a good test case for such questions.

######  Observation 4.9.

A consequence of Lemma 2.10 in the present context is the following. Suppose
$A$ and $B$ are $\mathbb{E}_{k+1}$-algebras in $\mathrm{Sp}$. If
$C^{*}(S^{n};A)$ is $\mathbb{E}_{k}$-trivial over $A$, then
$C^{*}(S^{n};A\otimes B)$ is $\mathbb{E}_{k}$-trivial over $A\otimes B$ and
over $B$. Indeed, this follows by applying Lemma 2.10 to the extension of
scalars functor $-\otimes
B\colon\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathrm{Mod}(A))\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathrm{Mod}(A\otimes
B))$ as well as to the restriction of scalars functor
$\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathrm{Mod}(A\otimes
B))\to\mathrm{Alg}_{\mathbb{E}_{n}}^{\mathrm{aug}}(\mathrm{Mod}(B))$.

The following result shows that, in general, the conclusion of Theorem 4.1 is
not sharp.

######  Lemma 4.10.

Let $A$ be a rational $\mathbb{E}_{m+1}$-ring spectrum where $m\geq n+1\geq
2$. Then $C^{*}(S^{n};A)$ is trivial as an $\mathbb{E}_{m}$-$A$-algebra.

###### Proof.

It follows from Example 4.5 that $C^{*}(S^{n};\mathbb{Q})$ is in particular
$\mathbb{E}_{m}$-trivial over $\mathbb{Q}$. Since the unit of $A$ factors via
an $\mathbb{E}_{m+1}$-map $\mathbb{Q}\to A$, the lemma follows from 4.9. ∎

We believe that the case of rational ring spectra is, however, the only case
in which our result is not sharp. Recall the following conjecture already
stated in the introduction:

######  Triviality Conjecture.

Let $n\geq 1$ and let $A$ be an $\mathbb{E}_{n+2}$-algebra in $\mathrm{Sp}$.
Then $C^{*}(S^{n};A)$ is $\mathbb{E}_{n+1}$-trivial over $A$ if and only if
$A$ is rational.

The rest of this section is devoted to an inspection of this conjecture. We
offer two approaches, one based on a study of the operadic structure maps and
one based on power operations. These two approaches yield the following two
results, respectively.666In fact, we use the operadic approach to conclude
only the version of statement 2 where $A$ is connective.

######  Theorem 4.11.

The Triviality Conjecture holds

  1. (1)

when $n\leq 2$,

  2. (2)

when $A$ is bounded below,

  3. (3)

when $A$ is the underlying $\mathbb{E}_{n+2}$-algebra of an
$\mathbb{E}_{\infty}$-algebra.

######  Theorem 4.12.

Suppose $A$ is an $\mathbb{E}_{n+2}$-algebra in $\mathrm{Sp}$ for which there
exists a prime $p$ such that $A\otimes\mathbb{F}_{p}$ or
$A\otimes\mathrm{KU}/p$ is non-zero. Then $C^{*}(S^{n};A)$ is not
$\mathbb{E}_{n+1}$-trivial over $A$. In particular, the Triviality Conjecture
holds for such $A$.

We will prove Theorem 4.11 in Section 4.2 and Theorem 4.12 in Section 4.3. The
two results are not unrelated. For instance, we record the following
implication.

######  Lemma 4.13.

Theorem 4.12 implies Theorems 4.11 2 and 4.11 3.

###### Proof.

First, suppose $A$ is bounded below and not rational. Then there exists a
prime $p$ such that $A\otimes\mathbb{F}_{p}$ is non-zero, so Theorem 4.12
implies that $C^{*}(S^{n};A)$ is not $\mathbb{E}_{n+1}$-trivial over $A$,
showing the first implication. To prove the second, suppose $A$ underlies an
$\mathbb{E}_{\infty}$-algebra and that
$A\otimes\mathbb{F}_{p}=0=A\otimes\mathrm{KU}/p$ for all primes $p$. We need
to show that $A$ is rational. To that end, as a consequence of [Hah22], we
find that $L_{K(n,p)}A=0$ for all $0<n\leq\infty$ and all primes.
Consequently, $K(n,p)\otimes A\otimes\mathrm{End}(\mathbb{S}/p)=0$ for all $p$
and all $0\leq n\leq\infty$.777Thanks to Achim Krause for reminding us of this
fact. The nilpotence theorem [HS98, Theorem 1] then implies that the ring
spectrum $A\otimes\mathrm{End}(\mathbb{S}/p)$ is null, implying that $p$ is
invertible in $A$ for all primes $p$; see [LMMT22, Lemma 2.3] for a similar
argument. ∎

###  4.2. Operadic structure maps

In this section, we continue to write $\mathbb{E}_{n}$ for the non-unital
$\mathbb{E}_{n}$-operad in spectra
$\Sigma^{\infty}_{+}\mathbb{E}_{n}^{\mathrm{nu}}$ and work towards the proof
of Theorem 4.11. Recall that for a spectrum $X$, a natural number $n$, and a
prime $p$, we write

$D_{p}^{\mathbb{E}_{n}}(X)=[\mathbb{E}_{n}(p)\otimes X^{\otimes
p}]_{h\Sigma_{p}}.$

We will need some relations between these extended powers for particular
values of $n$ and $X$. First observe that Theorem 3.11 yields commutative
squares of non-unital operads

${S^{-n+1}\mathbb{E}_{n-1}}$${S^{-n+1}\mathbb{E}_{n+1}}$${S^{-n+1}\mathbb{E}_{n-1}}$${S^{-n+1}\mathbb{E}_{\infty}}$${\mathbb{E}_{0}}$${\mathbb{E}_{2},}$${\mathbb{E}_{0}}$${\mathbb{E}_{\infty}.}$

The horizontal maps are the usual inclusions, the first three vertical maps
are $(n-1)$-fold compositions of the ‘wrong-way maps’ $\beta$, and the last
vertical morphism $S^{-n+1}\mathbb{E}_{\infty}\to\mathbb{E}_{\infty}$ is an
$(n-1)$-fold composition of suspension maps $\sigma$. If we take free algebras
on the spectrum $\mathbb{S}^{-1}$ and use that $\mathbb{E}_{0}(p)=0$, then the
squares above induce sequences

$\Sigma^{n}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})\longrightarrow\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})\longrightarrow\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})$

and

$\Sigma^{n}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})\longrightarrow\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})\longrightarrow\Sigma
D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-1})$

equipped with nullhomotopies of the composite maps. Sequences of this form
were already analyzed by Kuhn [Kuh82].

######  Lemma 4.14.

After $p$-localization, the two sequences above are cofiber sequences. As a
consequence the commutative square

${\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})}$${\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})}$${\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})}$${\Sigma
D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-1}),}$

again obtained from Theorem 3.11, is a pushout.

###### Proof.

If the two sequences are indeed cofiber sequences then the fibers of the
vertical maps in the square agree, implying that it is indeed a pushout. At
$p=2$ it is rather straightforward to see that the sequences are cofiber
sequences by explicitly identifying them with

$\mathbb{RP}_{-n}^{-2}\longrightarrow\mathbb{RP}_{-n}^{0}\longrightarrow\mathbb{RP}_{-1}^{0}$

and

$\mathbb{RP}_{-n}^{-2}\longrightarrow\mathbb{RP}_{-n}^{\infty}\longrightarrow\mathbb{RP}_{-1}^{\infty}.$

For an odd prime $p$ we argue using homology. Since we have already specified
nullhomotopies for the composite maps, it will suffice to show that taking
homology with $\mathbb{F}_{p}$-coefficients yields short exact sequences of
graded vector spaces. Write $\iota_{n}\in
H_{-n}(\mathbb{S}^{-n};\mathbb{F}_{p})$ for the fundamental class. According
to [GKRW21, §16] the homology of
$\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})$ has a basis
consisting of the classes

$\displaystyle Q^{s}(\iota_{n})\in
H_{2s(p-1)}(\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n});\mathbb{F}_{p})$
$\displaystyle\text{for }2s\geq-n,$ $\displaystyle\beta Q^{s}(\iota_{n})\in
H_{2s(p-1)-1}(\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n});\mathbb{F}_{p})$
$\displaystyle\text{for }2s>-n,$

with $Q^{s}$ and $\beta Q^{s}$ the Dyer–Lashof operations. In particular, for
$n=1$ we have classes $Q^{s}(\iota_{1})$ and $\beta Q^{s}(\iota_{1})$ for
$s\geq 0$ and the suspension morphism
$\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})\to\Sigma
D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-1})$ maps $Q^{s}(\iota_{n})$ to
$Q^{s}(\iota_{1})$ and similarly for $\beta Q^{s}$. Indeed, the suspension
arises from an $\mathbb{E}_{\infty}$-map
$\mathrm{free}_{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})\to\Omega^{n-1}\mathrm{free}_{\mathbb{E}_{\infty}}(\mathbb{S}^{-1})$
and is therefore compatible with Dyer–Lashof operations. Thus the induced map
in homology is a surjection with kernel spanned by the classes
$Q^{s}(\iota_{n})$ and $\beta Q^{s}(\iota_{n})$ with $s\leq-1$. This submodule
is precisely the image of the homology of
$\Sigma^{n}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})$ (cf. [GKRW21,
§16]).888Note that the top two operations $Q^{-1}$ and $\beta Q^{-1}$ are
labelled $\xi$ and $\zeta$ in loc. cit. Here we have used that the Browder
bracket vanishes for degree reasons. Indeed, the fundamental class $\iota_{n}$
has degree $-n$, whereas the bracket has degree $n$. These have the same
parity, so that graded antisymmetry of the bracket implies
${[\iota_{n},\iota_{n}]}=0$. Finally, we conclude that the second sequence is
indeed short exact on homology.

To deduce that the first sequence is a cofiber sequence, we consider the
square of the lemma again. We have already analyzed the homology of the
vertical map on the right. The homology of the left-hand map is obtained by
restricting to the submodules on classes $Q^{s}(\iota)$ and $\beta
Q^{s}(\iota)$ for which $s\leq 0$. The resulting sequence

$H_{*}(\Sigma^{n}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n}))\longrightarrow
H_{*}(\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n}))\longrightarrow
H_{*}(\Sigma D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1}))$

is still short exact, completing the proof. ∎

######  Remark 4.15.

The two cofiber sequences of Lemma 4.14 are special cases of the statement of
[Kuh82, Prop. 1.3], but the odd primary case in loc. cit. is in general not
correct as stated. We will only need the special cases that we established
above, and hence only give a proof of these.

######  Remark 4.16.

It is useful to observe that the $\mathbb{F}_{p}$-homology of $\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})$ is spanned by the classes
$Q^{0}(\iota_{1})$ and $\beta Q^{0}(\iota_{1})$, with the Bockstein acting as
indicated by the notation. It follows that $\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})$ is $p$-locally equivalent to the
Moore spectrum $\mathbb{S}^{-1}/p$ and that the wrong-way map

$\beta\colon\Sigma D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})\longrightarrow
D_{p}^{\mathbb{E}_{1}}(\mathbb{S}^{0})\cong\mathbb{S}^{0}$

may be identified with projection to the top cell, i.e., the integral
Bockstein map $\mathbb{S}^{-1}/p\to\mathbb{S}^{0}$.

######  Remark 4.17.

A variation of the previous remark is the following. The homology of $\Sigma
D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-1})$ is spanned by the classes
$Q^{s}(\iota)$ and $\beta Q^{s}(\iota)$ for $s\geq 0$, whereas the homology of
$D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{0})=\mathbb{S}^{0}_{h\Sigma_{p}}$ has
all the same classes except for the one class $\beta Q^{0}(\iota)$ in degree
-1. It follows that there is a $p$-local cofiber sequence

${\mathbb{S}^{-1}\longrightarrow\Sigma
D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-1})\longrightarrow\mathbb{S}^{0}_{h\Sigma_{p}}}$

whose connecting map $\mathbb{S}^{0}_{h\Sigma_{p}}\to\mathbb{S}^{0}$ can be
identified with the transfer for $\Sigma_{p}$.

If $X$ is a (non-unital) $\mathbb{E}_{n+1}$-algebra in $\mathrm{Sp}$, or more
generally in $\mathrm{Mod}(A)$ for some $\mathbb{E}_{n+2}$-ring spectrum $A$,
there are structure maps

$\mu_{p}^{n+1}\colon D_{p}^{\mathbb{E}_{n+1}}(X)\longrightarrow X.$

We shall be interested in these maps in the case where $X$ is the augmentation
ideal of $C^{*}(S^{n};A)$ for $A$ such an $\mathbb{E}_{n+2}$-algebra. In this
case, the structure maps are determined by the composition

(2) $\mu_{p}^{n+1}\colon
D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})\longrightarrow\mathbb{S}^{-n}\longrightarrow\Sigma^{-n}A,$

where the first is the structure map in case $A=\mathbb{S}$ and the second is
the $n$-fold desuspension of the unit of the ring spectrum $A$. We can shift
the first map to obtain a map

$\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})\longrightarrow\mathbb{S}^{0}$

which we still somewhat abusively denote $\mu_{p}^{n+1}$. The $n$-fold wrong-
way map $\beta^{n}$ of Theorem 3.11 provides a commutative square

${\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})}$${\mathbb{S}^{0}}$${D_{p}^{\mathbb{E}^{1}}(\mathbb{S}^{0})}$${\mathbb{S}^{0}.}$$\scriptstyle{\mu_{p}^{n+1}}$$\scriptstyle{\beta^{n}}$$\scriptstyle{\mu_{p}^{1}}$

The structure map $\mu_{p}^{1}$ is an isomorphism, since it arises from the
cochain algebra of $S^{0}$. The square therefore allows us to identify
$\mu_{p}^{n+1}$ with $\beta^{n}$, up to isomorphism. We write
$C(\mu_{p}^{n+1})$ for the cofiber of $\mu_{p}^{n+1}$. In particular, this
cofiber comes equipped with a map $\gamma\colon\mathbb{S}^{0}\to
C(\mu_{p}^{n+1})$. The evident factorization

$\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})\xrightarrow{\beta^{n-1}}\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\mathbb{S}^{0}$

of $\beta^{n}$ provides a cofiber sequence

$C(\beta^{n-1})\longrightarrow C(\mu_{p}^{n+1})\longrightarrow C(\beta).$

By Lemma 4.14, we have
$C(\beta^{n-1})\simeq\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})$
when $n\geq 2$ and $0$ for $n=1$. Moreover, as observed in Remark 4.16 we may
identify $C(\beta)$ with $\mathbb{S}^{0}$. The key to our proof will be the
following result.

######  Proposition 4.18.

Let $p$ be a prime and $n\geq 1$. Then after $p$-localization, the cofiber
sequence

$C(\beta^{n-1})\longrightarrow C(\mu_{p}^{n+1})\longrightarrow\mathbb{S}^{0}$

described above is canonically split. The induced map
$\mathbb{S}^{0}\xrightarrow{\gamma}C(\mu_{p}^{n+1})\simeq\mathbb{S}^{0}\oplus
C(\beta^{n-1})$ is then given by $(p,\theta_{n})$ for some
$\theta_{n}\in\pi_{0}(C(\beta^{n-1}))$.

###### Proof.

The case $p=2$ is rather straightforward to deal with directly, so let us do
it first. In this case the sequence of the lemma can be identified with the
right column in the following diagram, in which all rows and columns are
cofiber sequences:

${\mathbb{RP}_{-n}^{0}}$${\mathbb{RP}_{-1}^{0}}$${\Sigma\mathbb{RP}_{-n}^{-2}}$${\mathbb{RP}_{-n}^{0}}$${\mathbb{S}^{0}}$${\Sigma\mathbb{RP}_{-n}^{-1}}$${0}$${\mathbb{S}^{0}}$${\mathbb{S}^{0}.}$$\scriptstyle{\beta^{n-1}}$$\scriptstyle{\mu_{p}^{n+1}}$$\scriptstyle{2}$

The spectrum $\Sigma\mathbb{RP}_{-n}^{-1}$ is the Spanier–Whitehead dual of
$\mathbb{RP}^{n}_{+}$. The bottom cell of the latter splits off; by duality,
the projection $\Sigma\mathbb{RP}_{n}^{-1}\to\mathbb{S}^{0}$ to the top cell
admits a splitting as well. Furthermore, the diagram shows that the horizontal
map $\mathbb{S}^{0}\to\Sigma\mathbb{RP}_{n}^{-1}$ is indeed of degree 2 onto
that top cell.

We proceed to the case of an odd prime $p$. To prepare, consider the inverse
limit $\varprojlim_{n}\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})$.
By construction this is the evaluation of the functor

$F(X):=\varprojlim_{n}\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\Omega^{n}X)$

at $X=\mathbb{S}^{0}$. The functor $F$ is the universal exact (i.e.,
1-excisive) approximation _from the left_ of the functor
$D_{p}^{\mathbb{E}_{\infty}}$ and hence the subject of dual Goodwillie
calculus. We refer to $F$ as the _dual derivative_ of
$D_{p}^{\mathbb{E}_{\infty}}$. Dual to the usual case, where an
$n$-homogeneous functor has trivial linear approximation from the right, an
$n$-cohomogeneous functor such as $X\mapsto(X^{\otimes p})^{h\Sigma_{p}}$ has
no linear approximation from the left. Consequently, the fiber sequence

(3) $\Omega(X^{\otimes p})^{t\Sigma_{p}}\longrightarrow(X^{\otimes
p})_{h\Sigma_{p}}\longrightarrow(X^{\otimes p})^{h\Sigma_{p}}$

shows that the first map produces an equivalence on dual derivatives.
Moreover, the first term is an exact functor of $X$.999Again, we recall that
we are working in $p$-local spectra. The decisive point is that in $p$-local
spectra, $(-)^{t\Sigma_{p}}$ vanishes on (additively) $C_{p}$-induced objects.
Hence the sequence exhibits the first term as the dual derivative of
$D_{p}^{\mathbb{E}_{\infty}}$. We conclude that
$\varprojlim_{n}\Sigma^{n}D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-n})\cong(\mathbb{S}^{-1})^{t\Sigma_{p}}$.

Now take the inverse limit over $n$ of the squares of Lemma 4.14 to obtain a
pushout square

${\varprojlim_{n}\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})}$${(\mathbb{S}^{-1})^{t\Sigma_{p}}}$${\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})}$${\Sigma
D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{-1}).}$

It admits a map to the pushout square

${D_{p}^{\mathbb{E}_{1}}(\mathbb{S}^{0})=\mathbb{S}^{0}}$${D_{p}^{\mathbb{E}_{\infty}}(\mathbb{S}^{0})=\mathbb{S}^{0}_{h\Sigma_{p}}}$${\mathbb{S}^{0}}$${\mathbb{S}^{0}_{h\Sigma_{p}}}$$\scriptstyle{=}$$\scriptstyle{=}$

by using the limit of the $n$-fold wrong-way maps $\beta^{n}$ (resp. the
$n$-fold suspensions $\sigma^{n}$) in the top left (resp. the top right) and
the map $\beta$ (resp. $\sigma$) in the lower left (resp. the lower right)
corners. Taking the cofiber of this map of squares yields a further pushout
square

${\varprojlim_{n}C(\mu_{p}^{n+1})}$${(\mathbb{S}^{0})^{h\Sigma_{p}}}$${\mathbb{S}^{0}}$${\mathbb{S}^{0}.}$$\scriptstyle{\cong}$

where we have used Remark 4.16 to identify the lower left-hand corner and
Remark 4.17 for the lower right. We now argue that the right vertical map is
the canonical projection, which admits a canonical section. Indeed, to do so,
we note that by construction, its precomposition with the norm map
$\mathbb{S}_{h\Sigma_{p}}\to\mathbb{S}^{h\Sigma_{p}}$ is the transfer for
$\Sigma_{p}$. Associated to the norm fiber sequence for $\Sigma_{p}$ acting
trivially on $\mathbb{S}$, we obtain the fiber sequence

$\mathrm{map}(\mathbb{S}^{t\Sigma_{p}},\mathbb{S})\longrightarrow\mathrm{map}(\mathbb{S}^{h\Sigma_{p}},\mathbb{S})\longrightarrow\mathrm{map}(\mathbb{S}_{h\Sigma_{p}},\mathbb{S}).$

Recall that we implicitly work $p$-locally at all times. Then we find
$\mathbb{S}^{t\Sigma_{p}}\simeq\mathbb{S}_{p}$, see e.g. [NS18, Remark
IV.1.6]. Now, $\pi_{0}(\mathrm{map}(\mathbb{S}_{p},\mathbb{S}))=0$ as follows
from the fiber sequence $\tau_{\geq 1}\mathbb{S}\to\mathbb{S}\to\mathbb{Z}$,
the fact that $\tau_{\geq 1}\mathbb{S}$ is $p$-complete and has no $\pi_{0}$,
and that $\mathrm{Hom}(\mathbb{Z}_{p},\mathbb{Z}_{(p)})=0$. Finally, the
canonical map $\mathbb{S}^{h\Sigma_{p}}\to\mathbb{S}$ also gives the transfer
upon precomposition with the norm, showing the claim.

It remains to verify that the composite of the maps
$\mathbb{S}^{0}\to\varprojlim_{n}C(\mu_{p}^{n+1})\to\mathbb{S}^{0}$ we have
produced is of degree $p$. For this we can inspect the following commutative
diagram, analogous to the one for $p=2$ with which we started this proof:

${\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})}$${\Sigma
D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-1})}$${C(\beta^{n-1})}$${\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})}$${\mathbb{S}^{0}}$${C(\mu_{p}^{n+1})}$${0}$${\mathbb{S}^{0}}$${\mathbb{S}^{0}.}$$\scriptstyle{\beta^{n-1}}$$\scriptstyle{\beta}$$\scriptstyle{\mu_{p}^{n+1}}$$\scriptstyle{p}$

Here we have again applied Remark 4.16 to describe the middle column. ∎

We now turn to the proof of Theorem 4.11 1 and 3. It is based on the following
observation: The assumption that $C^{*}(S^{n};A)$ is
$\mathbb{E}_{n+1}$-trivial over $A$ implies that the composite

$\Sigma^{n}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})\stackrel{{\scriptstyle\mu^{n}_{p}}}{{\longrightarrow}}\mathbb{S}\longrightarrow
A$

is null. When $n\geq 1$, as a consequence of Proposition 4.18, the unit of
$A_{(p)}$ then factors as a composite

$\mathbb{S}\xrightarrow{(p,\theta_{n})}\mathbb{S}\oplus\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})\xrightarrow{(x+f)}A_{(p)}$

for some $x\in\pi_{0}(A)$ and
$f\colon\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})\to A_{(p)}$,
giving the relation

(4) $1=px+f\theta_{n}\in\pi_{0}(A)_{(p)}.$

###### Proof of Theorem 4.11.

As a warm-up, we will prove part 2 in case $A$ is connective. In this case,
the cell structure of $\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})$
reveals that there are no maps to $A_{(p)}$, showing that $f=0$. In
particular, we deduce from equation (4) that $p$ is invertible in
$\pi_{0}(A)_{(p)}$ and hence also in $\pi_{0}(A)$. As this holds for all
primes $p$ we deduce that $A$ is rational as claimed.

Next we deal with part 1. If $n=1$, then the domain of $f$ is contractible, so
the same argument as above applies. If $n=2$, then the domain of $f$ is given
by $\Sigma^{3}D_{p}^{\mathbb{E}_{1}}(\mathbb{S}^{-2})$ which is equivalent to
$\mathbb{S}^{-2p+3}$. In particular, we may think of $\theta_{2}$ as an
element in $\pi_{2p-3}(\mathbb{S})$. By Nishida’s nilpotence theorem [Nis73],
the map $\theta_{2}$ is smash-nilpotent. This implies that the element
$f\theta_{2}\in\pi_{0}(A)_{(p)}$ is nilpotent in the algebraic sense. In
particular, equation (4) implies that $p$ is invertible modulo nilpotent
elements, and hence is itself invertible. Again, we conclude that $A$ is
rational.

To show 3, we follow the same strategy and aim to show that
$f\theta\in\pi_{0}(A)_{(p)}$ is nilpotent. To do so, since $A$ is
$\mathbb{E}_{\infty}$, the solution of the May conjecture101010It is worth
mentioning that the solution of May’s conjecture we use here also makes use of
appropriate power operations. due to Mathew–Naumann–Noel [MNN15, Theorem B]
implies that it suffices to show that the image of $f\theta$ under the
Hurewicz maps $\pi_{0}(A)_{(p)}\to H_{0}(A;\mathbb{F}_{p})$ and
$\pi_{0}(A)_{(p)}\to H_{0}(A;\mathbb{Q})$ is trivial. This is the case because
$\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})\otimes k$ is a
coconnected spectrum for $k=\mathbb{Q}$ and $k=\mathbb{F}_{p}$ for all primes
$p$. ∎

######  Remark 4.19.

One might wonder whether the element $f\theta_{n}\in\pi_{0}(A)$ is always
nilpotent, in which case one would deduce from the above argument that $A$ has
to be rational. To that end, it would suffice to show that the map
$\theta_{n}$ is smash-nilpotent. However, unlike the cases $n=1,2$ considered
above, this is not true for $n\geq 3$ as we explain now, first focusing on the
prime $2$. To that end, note that if $\theta_{n}$ is smash-nilpotent, then
$K(1)_{*}(\theta_{n})$ is the zero map. We will argue that in fact this map is
non-zero. Indeed, consider the 2-local cofiber sequence

$\mathbb{S}\xrightarrow{(2,\theta_{n})}\mathbb{S}\oplus\Sigma\mathbb{RP}^{-2}_{-n}\longrightarrow\Sigma\mathbb{RP}^{0}_{-n}$

and take its long exact sequence in $\mathrm{KU}$-homology. The first map
induces an injective homomorphism and leads to a short exact sequence

$0\longrightarrow\mathrm{KU}_{0}(\mathbb{S})\xrightarrow{(2,\theta_{n})}\mathrm{KU}_{0}(\mathbb{S})\oplus\mathrm{KU}_{0}(\Sigma\mathbb{RP}^{-2}_{-n})\longrightarrow\mathrm{KU}_{0}(\Sigma\mathbb{RP}^{0}_{-n})\longrightarrow
0.$

From Adams’ calculation of the $K$-theory of projective spaces [Ada62, Theorem
7.3] it is easily deduced that this sequence is isomorphic to one of the form

$0\longrightarrow\mathbb{Z}\xrightarrow{(2,\bar{\theta}_{n})}\mathbb{Z}\oplus\mathbb{Z}/2^{k}\mathbb{Z}\longrightarrow\mathbb{Z}/2^{k+1}\mathbb{Z}\longrightarrow
0$

with $k=\frac{n-2}{2}$ if $n$ is even and $k=\frac{n-1}{2}$ if $n$ is odd and
where $\bar{\theta}_{n}$ is induced by $\theta_{n}$. In particular, if $n\geq
3$ then $k\geq 1$ and this sequence implies that $\bar{\theta}_{n}$ must hit a
generator of the group $\mathbb{Z}/2^{k}\mathbb{Z}$. Hence the reduction of
$\bar{\theta}_{n}$ mod 2 is nontrivial, from which it follows that
$\theta_{n}$ induces a nontrivial homomorphism in $\mathrm{KU}/2$-homology, or
equivalently in $K(1)$-homology.

A similar argument applies for all other primes $p$. In this case, we have the
$p$-local cofiber sequence

$\mathbb{S}\xrightarrow{(p,\theta_{n})}\mathbb{S}\oplus\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n})\longrightarrow\Sigma^{n+1}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n})$

whose long exact sequence in $\mathrm{KU}$-homology again induces a short
exact sequence as follows:

$0\longrightarrow\mathrm{KU}_{0}(\mathbb{S})\xrightarrow{(p,\theta_{n})}\mathrm{KU}_{0}(\mathbb{S})\oplus\mathrm{KU}_{0}(\Sigma^{n+1}D_{p}^{\mathbb{E}_{n-1}}(\mathbb{S}^{-n}))\longrightarrow\mathrm{KU}_{0}(\Sigma^{n+1}D_{p}^{\mathbb{E}_{n+1}}(\mathbb{S}^{-n}))\longrightarrow
0.$

By [BHK22, Theorem 6.1], this sequence is isomorphic to

$0\longrightarrow\mathbb{Z}\xrightarrow{(p,\theta_{n})}\mathbb{Z}\oplus\mathbb{Z}/p^{k}\mathbb{Z}\longrightarrow\mathbb{Z}/p^{k+1}\mathbb{Z}\longrightarrow
0$

for $k=\frac{n-2}{2}$ when $n$ is even and $k=\frac{n-1}{2}$ when $n$ is odd.
Again, we deduce that $K(1)_{*}(\theta_{n})$ is non-zero, showing that
$\theta_{n}$ is not smash-nilpotent. For a concrete example, take $n=3$. Then
we find that
$\theta_{3}\colon\mathbb{S}\to\Sigma^{4}D_{p}^{\mathbb{E}_{2}}(\mathbb{S}^{-3})\simeq\mathbb{S}^{-2p+2}/p$
and the above calculations imply that $\theta_{3}$ is given by a lift of a
generator of the $p$-torsion of $\pi_{2p-3}(\mathbb{S})$ to
$\pi_{2p-2}(\mathbb{S}/p)$.

###  4.3. Power operations

We separate the two cases of Theorem 4.12 in individual propositions.

######  Proposition 4.20.

Let $A$ be an $\mathbb{E}_{n+2}$-algebra in $\mathrm{Sp}$ with
$A\otimes\mathbb{F}_{p}\neq 0$. Then $C^{*}(S^{n};A)$ is not
$\mathbb{E}_{n+1}$-trivial over $A$.

###### Proof.

We first recall that any $\mathbb{E}_{n+1}$-$\mathbb{F}_{p}$-algebra $B$
admits a natural operation $Q^{0}\colon\pi_{-n}(B)\to\pi_{-n}(B)$ [Law20,
Remark 1.5.3]. This operation vanishes in case $B$ is
$\mathbb{E}_{n+1}$-trivial over $\mathbb{F}_{p}$. Let us then consider the
following commutative diagram

${\pi_{-n}C^{*}(S^{n};\mathbb{F}_{p})}$${\pi_{-n}C^{*}(S^{n};\mathbb{F}_{p})}$${\pi_{-n}C^{*}(S^{n};A\otimes\mathbb{F}_{p})}$${\pi_{-n}C^{*}(S^{n};A\otimes\mathbb{F}_{p})}$$\scriptstyle{Q^{0}}$$\scriptstyle{Q^{0}}$

where the horizontal arrows are given by the operation $Q^{0}$. It is a
fundamental fact about cochains of spaces that the upper horizontal arrow is
the identity [Law20, Example 1.5.9]. This shows that the lower horizontal map
is non-trivial, since the vertical maps are injective. As a consequence of 4.9
we find that $C^{*}(S^{n};A)$ is not trivial over $A$. ∎

######  Remark 4.21.

We note that the argument for Proposition 4.20 above shows that, more
generally, if $X$ is a pointed space with $H^{n}(X;\mathbb{F}_{p})\neq 0$ for
some $n>0$, then $C^{*}(X;\mathbb{F}_{p})$ is not a trivial square-zero
extension of $\mathbb{F}_{p}$ by the augmentation ideal of
$C^{*}(X;\mathbb{F}_{p})$ as an $\mathbb{E}_{n+1}$-$\mathbb{F}_{p}$-algebra.
Likewise, $X$ is not $\mathbb{E}_{n+1}$-formal over $\mathbb{F}_{p}$. Indeed,
we need to show that $C^{*}(X;\mathbb{F}_{p})$ is not equivalent to
$H^{*}(X;\mathbb{F}_{p})$ as $\mathbb{E}_{n+1}$-$\mathbb{F}_{p}$-algebra. To
see this we may use that $Q^{0}$ as an operation on
$\pi_{-n}(H^{*}(X;\mathbb{F}_{p}))$ is trivial (unless $n=0$), whereas it is
the identity on $\pi_{-n}(C^{*}(X;\mathbb{F}_{p}))$. Hence, in contrast to the
case of rational coefficients, with finite coefficients the only
$\mathbb{E}_{\infty}$-formal spaces are the ones whose cohomology is
concentrated in degree 0. In particular, the result of [CPRNW23] showing that
rational $\mathbb{E}_{1}$-formality implies rational
$\mathbb{E}_{\infty}$-formality does not extend to fields $k$ of positive
characteristic. Indeed, pick any space $X$ such that $H^{*}(X;k)$ is not
concentrated in degree $0$. Then $\Sigma X$ is $\mathbb{E}_{1}$-formal over
$k$ but not $\mathbb{E}_{\infty}$-formal over $k$. See [FC24] for further
aspects of such results in positive characteristic.

######  Proposition 4.22.

Let $A$ be an $\mathbb{E}_{n+2}$-algebra in $\mathrm{Sp}$ with
$A\otimes\mathrm{KU}/p\neq 0$. Then $C^{*}(S^{n};A)$ is not
$\mathbb{E}_{n+1}$-trivial over $A$.

###### Proof.

Again, the proof relies on appropriate power operations for $p$-complete
$\mathrm{KU}$-algebras111111In what follows $\mathrm{KU}$ is implicitly meant
$p$-complete. about which we shall use the following facts.

  1. (1)

For $i=0,1$, there are operations $\theta^{i}\colon\pi_{i}(A)\to\pi_{i}(A)$,
defined naturally for $p$-complete
$\mathbb{E}_{\infty}$-$\mathrm{KU}$-algebras121212In fact, more generally for
$K(1)$-local $\mathbb{E}_{\infty}$-rings. satisfying that for any spectrum
$X$, the diagram

${\mathrm{KU}^{-1}(X)}$${\mathrm{KU}^{-1}(X)}$${\mathrm{KU}^{0}(\Sigma
X)}$${\mathrm{KU}^{0}(\Sigma
X)}$$\scriptstyle{\theta^{1}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\theta^{0}}$

commutes. Moreover, for $X=S^{2n}$, we have
$\theta^{0}(\beta^{n})=p^{n-1}\beta^{n}$,131313This follows from the relation
$\psi^{p}(x)-x^{p}=p\cdot\theta(x)$ for all $x\in\mathrm{KU}^{0}(X)$. so also
the operation
$\theta^{1}\colon\mathrm{KU}^{-1}(S^{2n-1})\to\mathrm{KU}^{-1}(S^{2n-1})$ is
given by multiplication by $p^{n-1}$.

  2. (2)

For $i=0,1$, there are operations
$\theta_{r}^{i}\colon\pi_{i}(A)\to\pi_{i}(A/p^{r})$, defined naturally for
$p$-complete $\mathbb{E}_{2r+1-i}$-$\mathrm{KU}$-algebras, which when $A$ is a
$p$-complete $\mathbb{E}_{\infty}$-$\mathrm{KU}$-algebra, are given by the
composite

$\pi_{i}(A)\xrightarrow{\;\theta^{i}}\pi_{i}(A)\longrightarrow\pi_{i}(A/p^{r}).$

All these operations (when defined) vanish on trivial $p$-complete
$\mathrm{KU}$-algebras. Part (1) above is classical, see e.g. [Hop14], in
particular for $i=0$; for $i=1$ one can use the fiber sequence of Remark 4.17.
The proof of (2) follows the same strategy, [BHK22, §6.1] can be used for the
relevant calculations; see also [BHK22, §6.2] for discussions about power
operations for $p$-complete $\mathrm{KU}$-algebras.

Let us now denote by $A\hat{\otimes}\mathrm{KU}$ the $p$-completion of
$A\otimes\mathrm{KU}$. The assumption of the proposition is then equivalent to
the statement that $A\hat{\otimes}\mathrm{KU}$ is non-zero. Given this, let
$i\in\\{0,1\\}$ be such that $n=2r-i$ and consider the commutative diagram

${\mathrm{KU}^{-i}(S^{2r-i})}$${\mathrm{KU}^{-i}(S^{2r-i})/p^{r}}$${(A\hat{\otimes}\mathrm{KU})^{-i}(S^{2r-i})}$${(A\hat{\otimes}\mathrm{KU}/p^{r})^{-i}(S^{2r-i})}$$\scriptstyle{\theta^{i}_{r}}$$\scriptstyle{\theta^{i}_{r}}$

whose horizontal maps are induced by $\theta^{i}_{r}$. The vertical maps in
the above diagram are induced by the map $\mathrm{KU}\to
A\hat{\otimes}\mathrm{KU}$. It follows that the diagram is isomorphic to the
diagram

${\mathbb{Z}_{p}}$${\mathbb{Z}/p^{r}}$${\pi_{0}(A\hat{\otimes}\mathrm{KU})}$${\pi_{0}(A\hat{\otimes}\mathrm{KU}/p^{r})}$$\scriptstyle{\cdot
p^{r-1}}$

whose vertical maps are induced by the unit of
$\pi_{0}(A\hat{\otimes}\mathrm{KU})$. We claim that the lower horizontal map
is non-zero, showing that $C^{*}(S^{n};A)$ is not $\mathbb{E}_{n+1}$-trivial
over $A$. To see this, it suffices to show
$p^{r-1}\in\pi_{0}(A\hat{\otimes}\mathrm{KU})/p^{r}\subseteq\pi_{0}(A\hat{\otimes}\mathrm{KU}/p^{r})$
is non-zero. Aiming for a contradiction, assume that it is zero. Then
$p^{r-1}=p^{r}\cdot x$ for some $x$ in $\pi_{0}(A\hat{\otimes}\mathrm{KU})$
and hence $p^{r-1}(1-px)=0$. Since $n+2\geq 2r-2$, we find that
$A\hat{\otimes}\mathrm{KU}$ is a $K(1,p)$-local $\mathbb{E}_{2r-2}$-algebra,
so we deduce from [Hah17] that $(1-px)$ is nilpotent.141414In case $A$ is
$\mathbb{E}_{\infty}$, this also follows from the solution of May’s nilpotence
conjecture [MNN15] or a $\delta$-ring argument similar to [Bha17, Lemma 1.5]
using that $\pi_{0}$ of any $K(1)$-local $\mathbb{E}_{\infty}$-ring
canonically carries the structure of a $\delta$-ring [Hop14]. However, it is
also invertible since $(A\hat{\otimes}\mathrm{KU})/(1-px)$ is $p$-complete and
$[(A\hat{\otimes}\mathrm{KU})/(1-px)]/p=[(A\hat{\otimes}\mathrm{KU})/p]/(1-px)=0$.
This is in contradiction with the assumption that $A\hat{\otimes}\mathrm{KU}$
is non-zero. ∎

###  4.4. Free loop spaces

We finally discuss some applications of our formality results to free loop
spaces that might be interesting from the point of view of string topology of
spheres. Let us write $C^{*}(X)$ for $C^{*}(X;\mathbb{Z})$ to simplify
notation. We recall that for a connected, nilpotent space $X$ of finite
$\mathbb{Z}$-homological type with finite fundamental group, the free loop
space $LX$ as well as $X\times X$ are again of $\mathbb{Z}$-homological finite
type. Under this assumption, we have canonical equivalences in
$\mathrm{Alg}_{\mathbb{E}_{\infty}}(\mathbb{Z})$

$C^{*}(LX)\simeq C^{*}(X)\otimes_{C^{*}(X\times
X)}C^{*}(X)\simeq\mathrm{HH}(C^{*}(X))$

and that Hochschild homology (relative to $\mathbb{Z}$) is a functor
$\mathrm{HH}\colon\mathrm{Alg}_{\mathbb{E}_{n}}(\mathbb{Z})\to\mathrm{Alg}_{\mathbb{E}_{n-1}}(\mathbb{Z})^{\mathrm{B}\mathbb{T}}$.
The above composite equivalence is then in fact $\mathbb{T}$-equivariant. As a
consequence of Theorem 4.1, for $n\geq 1$, we obtain an equivalence of
$\mathbb{E}_{n-1}$-$\mathbb{Z}$-algebras with $\mathbb{T}$-action

$\mathrm{HH}(C^{*}(\Sigma^{n}X))\simeq\mathrm{HH}(H^{*}(\Sigma^{n}X)).$

Indeed, we have canonical equivalences
$C^{*}(\Sigma^{n}X)\simeq\Omega^{n}C^{*}(X)$ and
$H^{*}(\Sigma^{n}X)\simeq\Omega^{n}H^{*}(X)$, so Theorem 4.1 implies the claim
once we observe that there exists an equivalence of
$\mathbb{E}_{0}$-$\mathbb{Z}$-algebras $C^{*}(X)\simeq H^{*}(X)$ since the
global dimension of $\mathbb{Z}$ is one, see the argument in Example 4.5. In
particular, we obtain an equivalence of
$\mathbb{E}_{n-1}$-$\mathbb{Z}$-algebras with $\mathbb{T}$-action

$C^{*}(L\Sigma^{n}X)\simeq\mathrm{HH}(H^{*}(\Sigma^{n}X)),$

providing a reminiscence of formality for $L(\Sigma^{n}X)$.

As a nice explicit example, for $n\geq 2$ and a commutative ring $A$, we find
an equivalence of $\mathbb{E}_{n-1}$-$A$-algebras with $\mathbb{T}$-action

$C^{*}(LS^{n};A)\simeq\mathrm{HH}_{A}(A[x]/x^{2}),\quad|x|=-n$

where $\mathrm{HH}_{A}$ denotes Hochschild homology relative to $A$. For
instance for $A=\mathbb{Z}$, additively, $\mathrm{HH}(\mathbb{Z}[x]/x^{2})$ is
very easy to compute, because it is easy to resolve $\mathbb{Z}[x]/x^{2}$ as a
module over $\mathbb{Z}[y,z]/(y^{2},z^{2})$ (viewed as a graded commutative
ring with $|y|=|z|=-n$), where the module action comes from the ring map
sending both $y$ and $z$ to $x$, see e.g. [Lod98, E.4.1.8], but the above
equivalence also sheds some light on the coherent multiplicative structure in
$C^{*}(LS^{n};\mathbb{Z})$, more precisely its $\mathbb{E}_{n-1}$-algebra
structure. There are a number of papers surrounding the relation of the string
topology of $S^{n}$ with Hochschild (co)homology of $C^{*}(S^{n})$ or
$H^{*}(S^{n})$, in particular for $n=2$ and with coefficients
$\mathbb{F}_{2}$, see e.g. [Men09, PT23]. It would be interesting to find a
relation between these works and our formality result.

Finally, we mention that in [MNO24] the authors show, using the the above
relation to the free loop space, that $C^{*}(S^{2};\mathbb{F}_{2})$ is not
formal as a _framed_ $\mathbb{E}_{2}$-algebra. It is natural to then ask
whether $C^{*}(S^{n};k)$ is formal as a framed $\mathbb{E}_{n}$-algebra if and
only if $k$ is rational, but we will not address this here.

## References

  * [Ada62] J.F. Adams, _Vector fields on spheres_ , Annals of Mathematics (1962), 603–632.
  * [AK14] G. Arone and M. Kankaanrinta, _The sphere operad_ , Bull. Lond. Math. Soc. 46 (2014), no. 1, 126–132.
  * [BB] M. Blans and T. Blom, _On the chain rule in Goodwillie calculus_ , In preparation.
  * [BCN23] L. Brantner, R. Campos, and J. Nuiten, _PD operads and Explicit Partition Lie Algebras_ , arXiv:2104.03870v4, to appear in Mem. Amer. Math. Soc. (2023).
  * [Ber12] A. Berglund, _Rational homotopy theory_ , Available at the author’s homepage https://staff.math.su.se/alexb/rathom2.pdf, 2012.
  * [BF04] C. Berger and B. Fresse, _Combinatorial operad actions on cochains_ , Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1.
  * [Bha17] B. Bhatt, _Prismatic Cohomology. Lecture II: Delta rings_ , available at https://www.math.ias.edu/~bhatt/teaching/prismatic-columbia/lecture2-delta-rings.pdf, 2017\.
  * [BHK22] L. Brantner, J. Hahn, and B. Knudsen, _The Lubin-Tate Theory of Configuration Spaces: I_ , arXiv:1908.11321v3 (2022).
  * [BM11] M. Basterra and M. A. Mandell, _Homology of $E_{n}$ ring spectra and iterated $THH$_, Algebr. Geom. Topol. 11 (2011), no. 2, 939–981.
  * [CPRNW23] R. Campos, D. Petersen, D. Robert-Nicoud, and F. Wierstra, _Lie, associative and commutative quasi-isomorphism_ , Acta Math. (to appear) (2023).
  * [CS22] M. Ching and P. Salvatore, _Koszul duality for topological $E_{n}$-operads_, Proc. Lond. Math. Soc. (3) 125 (2022), no. 1, 1–60.
  * [FC24] O. Flynn-Connolly, _An obstruction theory for strictly commutative algebras in positive characteristic_ , arXiv:2404.16681 (2024).
  * [Fre11] B. Fresse, _Koszul duality of $E_{n}$-operads_, Selecta Math. (N.S.) 17 (2011), no. 2, 363–434.
  * [GKRW21] S. Galatius, S. Kupers, and O. Randal-Williams, _Cellular $\mathrm{E}_{\mathrm{k}}$-algebras_, arXiv:1805.07184v3, to appear in Astérisque (2021).
  * [Hah17] J. Hahn, _Nilpotence in $\mathbb{E}_{n}$-algebras_, arXiv:1707.00956 (2017).
  * [Hah22] by same author, _On the Bousfield classes of $H_{\infty}$-ring spectra_, arXiv:1612.04386v2 (2022).
  * [Hau22] R. Haugseng, _$\infty$ -operads via symmetric sequences_, Math. Z. 301 (2022), no. 1, 115–171.
  * [Heu21] G. Heuts, _Lie algebras and $v_{n}$-periodic spaces_, Ann. of Math. (2) 193 (2021), no. 1, 223–301.
  * [HLN21] F. Hebestreit, M. Land, and T. Nikolaus, _On the homotopy type of L-spectra of the integers_ , J. Topol. 14 (2021), no. 1, 183–214.
  * [Hop14] M. J. Hopkins, _$K(1)$ -local $E_{\infty}$-ring spectra_, Topological modular forms, Math. Surveys Monogr., vol. 201, Amer. Math. Soc., Providence, RI, 2014, pp. 287–302.
  * [HS98] M. J. Hopkins and J. H. Smith, _Nilpotence and stable homotopy theory. II_ , Ann. of Math. (2) 148 (1998), no. 1, 1–49.
  * [Kuh82] N. J. Kuhn, _The geometry of the James-Hopf maps_ , Pacific J. Math. 102 (1982), no. 2, 397–412.
  * [Law20] T. Lawson, _$E_{n}$ -spectra and Dyer-Lashof operations_, Handbook of homotopy theory, CRC Press/Chapman Hall Handb. Math. Ser., CRC Press, Boca Raton, FL, 2020, pp. 793–849.
  * [LMMT22] M. Land, A. Mathew, L. Meier, and G. Tamme, _Purity in chromatically localized algebraic K-theory_ , arXiv:2001.10425v4, to appear in J. Amer. Math. Soc. (2022).
  * [Lod98] J.-L. Loday, _Cyclic homology_ , second ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998.
  * [Lur17] J. Lurie, _Higher Algebra_ , Available at the author’s homepage http://www.math.ias.edu/~lurie, 2017.
  * [Man09] M. A. Mandell, _Towards formality_ , Slides available at https://mmandell.pages.iu.edu/talks/Austin3.pdf, 2009.
  * [May06] J Peter May, _The geometry of iterated loop spaces_ , vol. 271, Springer, 2006\.
  * [Men09] L. Menichi, _String topology for spheres_ , Comment. Math. Helv. 84 (2009), no. 1, 135–157.
  * [MNN15] A. Mathew, N. Naumann, and J. Noel, _On a nilpotence conjecture of J. P. May_ , J. Topol. 8 (2015), no. 4, 917–932.
  * [MNO24] R. McGowen, F. Naef, and B. O’Callaghan, _Non-formality of ${S^{2}}$ via the free loop space_, arXiv:2405.12047 (2024).
  * [Nis73] G. Nishida, _The nilpotency of elements of the stable homotopy groups of spheres_ , J. Math. Soc. Japan 25 (1973), 707–732.
  * [NS18] T. Nikolaus and P. Scholze, _On topological cyclic homology_ , Acta Math. 221 (2018), no. 2, 203–409.