Our goal is to prove the following theorem, which is equivalent to the first
bullet point of Theorem 6.1. The proof will closely follow Chapter 5 of
Gugenheim and May’s book [12], which proves the case where $U$ is the cobar
construction of an augmented connected algebra $A$.

######  Theorem 6.8.

Let $U$ be a differential graded algebra whose underlying algebra is Koszul.
The homology $H(U)$ is generated under matric Massey products by elements of
degree $n=-1.$

Our proof strategy is as follows. First, we will define a “derived” Koszul
condition for differential graded algebras, which we will show implies that
the homology is generated under matric Massey products by elements of degree
$n=-1.$ Then, we will show that any differential graded algebra whose
underlying algebra is Koszul (in the classical sense) is also Koszul in the
derived sense. To start, we define:

######  Definition 6.9.

Let $R$ be a field, and let $U$ be an augmented differential graded
$R$-algebra. We say $U$ is _Koszul_ if $R\otimes_{U}^{\mathbb{L}}R$ has
homology concentrated in degree $n=0.$

Here $R\otimes_{U}^{\mathbb{L}}R$ is the _derived_ tensor product, which
Gugenheim and May notate as $\operatorname{Tor}^{*}_{U}(R,R).$ We state the
two parts of our proof as separate theorems.

######  Theorem 6.10.

Let $U$ be a Koszul differential graded $R$-algebra. Then $HU$ is generated by
elements of degree $-1$ under matric Massey products.

######  Theorem 6.11.

Let $U$ be a differential graded $R$-algebra. If the underlying graded
$R$-algebra of $U$ is Koszul, then $U$ is as well.

The proof of Theorem 6.10 is essentially contained in Section 5 of [12],
although they do not state it in this generality. We sketch it here to
emphasize that the proof works for any Koszul differential graded $R$-algebra,
rather than just the cobar construction, and to foreshadow the proofs in the
next section. We start by generalizing the suspension map from definition 6.4
to include the presence of differentials.

######  Definition 6.12 (Definition 3.7 of [12]).

Let $R$ be a ring and $U$ be an augmented differential graded $R$-algebra with
augmentation ideal $IU.$ Denote by $E^{k}_{**}(R,U,R)$ the algebraic
Eilenberg-Moore spectral sequence computing the homology of
$R\otimes_{U}^{\mathbb{L}}R.$ The suspension map

$\sigma:H(IU)\to E^{\infty}_{1*}(R,U,R)$

is the top horizontal composition

$\textstyle{H(IU)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma}$$\textstyle{E^{\infty}_{1*}(R,U,R)}$$\textstyle{\operatorname{Tor}_{1*}^{HU}(R,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{2}_{1*}(R,U,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the $\sigma$ appearing on the left hand arrow is the classical map of
Definition 6.4.

Gugenheim and May then prove a generalization of Proposition 6.6:

######  Theorem 6.13 (Corollary 5.13 of [12]).

The kernel of the map $\sigma$ consists of the elements of $U$ which are
decomposable by matric Massey products.

###### Sketch of proof.

As a derived counterpart of Observation 6.5, Gugenheim and May show that the
image of the map

$H\left(IU\otimes_{U}^{\mathbb{L}}IU\right)\to H\left(IU\right)$

out of the derived tensor product is the set of decomposable in $H(IU)$ under
_matric Massey products._ The fiber sequence

$IU\to U\to R$

induces a fiber sequence

$IU\otimes_{U}^{\mathbb{L}}IU\to
IU\otimes^{\mathbb{L}}_{U}U\xrightarrow{\pi}IU\otimes_{U}^{\mathbb{L}}R,$

which then leads to an exact sequence

$H\left(IU\otimes_{U}^{\mathbb{L}}IU\right)\to
H\left(IU\right)\xrightarrow{\pi}H\left(R\otimes_{U}^{\mathbb{L}}IU\right).$

In particular, the kernel of the map $\pi$ must be the set of indecomposables.
Following the proof of the previous section, one sees that
$\ker\pi=\ker\sigma.$ ∎

The proof of Theorem 6.10 is then immediate. The Koszul condition implies that
$E^{\infty}_{1,n}(R,U,R)$ is concentrated in degree $-1$, so elements in any
other degree must land in the kernel of $\sigma$ and hence be decomposable.

In the remainder of this section, we prove Theorem 6.11. The argument is a
variant of the proof of Proposition 5.16 in [12] designed to avoid explicit
features of the cobar construction. To start, we place a filtration on an
arbitrary chain complex $X$.

######  Definition 6.14.

Let $X$ be a chain complex. The _inverse filtration_ on $X$ is the natural
filtration with

$F_{n}(X)_{i}=\begin{cases}X_{i}&\text{ for }i\leq-n\\\ 0&\text{ for
}i>-n.\end{cases}$

We call this the “inverse” filtration to contrast it with the canonical
filtration

$F_{n}(X)^{\text{can}}_{i}=\begin{cases}X_{i}&\text{ for }i>n\\\
\ker(d_{i})&\text{ for }i=n\\\ 0&\text{ for }<n.\end{cases}$

which runs in the opposite direction. While the canonical filtration leads to
a spectral sequence based on the homology of the DGA (traditionally called the
algebraic Eilenberg-Moore spectral sequence), the inverse filtration leads to
a spectral sequence based on the DGA’s underlying algebra.

###### Proof of Theorem 6.11.

Consider the inverse filtration on $U$, and note that the associated graded
complex of $F_{\bullet}(U)$ is equivalent to the underlying algebra $V(U)$ of
$U$ with zero differential. Applying 4.12 to with filtration on $U$ and the
trivial filtration $S\iota R$ on $R$, we obtain a spectral sequence

$E^{1}_{**}=\operatorname{Tor}_{*}^{V(U)}(R,R)\Rightarrow
H(R\otimes^{\mathbb{L}}_{U}R).$

Since $V(U)$ is Koszul, the $E^{1}$ page is concentrated in degrees $(n,-n)$,
so $H(R\otimes^{\mathbb{L}}_{U}R)$ must be concentrated in degree zero. ∎

###  6.4 Koszul conditions and generation in
$\operatorname{Fil}(\mathcal{C}).$

In this section we prove the second bullet point of Theorem 6.1. In
particular, we prove:

######  Theorem 6.15.

Let $E_{r}^{s,t}$ be a multiplicative spectral sequence over a field $R$, and
suppose the $E_{1}$ page can be chosen to be freely generated by elements in
degree $s=1$ in addition to a unital copy of $R$ in degree $s=t=0.$ Let
$E_{2,r}^{s,t}$ denote the set of elements in $E_{2}^{s,t}$ which survive to
the $E_{r}$ page. Then:

  * •

$E_{2,r}^{*,*}$ is generated under matric Massey products by elements in
degree $0<s<r.$

Our strategy is similar to that of the previous section. First, applying Lemma
7.4 if necessary, we may assume our spectral spectral sequence comes from a
ring object in $\operatorname{Fil}(\mathcal{C})$ for some stable
$\infty$-category $\mathcal{C}.$ Next, we define another “derived” variant of
the Koszul condition, which is essentially the definition of the previous
section constructed within the special fiber of the $\mathcal{D}^{1}$
deformation. We show that this condition implies the desired generation
condition on $E_{2,r}^{s,t}$, and then we show that a Koszul $E^{1}$ page
makes the spectral sequence Koszul in this derived sense.

Throughout this section, we use the notation $\otimes^{\mathbb{L}}$ to
emphasize that our tensor products are taken in $\mathcal{D}^{1}(\mathcal{C})$
rather than in $\operatorname{Fil}(\mathcal{C})$, and we use $n$-grading,
directing the reader who prefers $s$-grading to Warning 6.7. Our Koszul
condition is as follows.

######  Definition 6.16.

Let $R$ be a(n ordinary) field viewed as an object of $\mathcal{C}$, and let
$U$ be an augmented $R$-algebra in $\mathcal{D}^{1}(\mathcal{C}).$ We say $U$
is _Koszul_ if
$\pi_{nt}^{1}(R\otimes^{\mathbb{L}}_{U}R\otimes^{\mathbb{L}}C\tau^{2})$ is
concentrated in degree $0.$

The goal of this section is to prove two theorems:

######  Theorem 6.17.

Let $U$ be a Koszul $R$-algebra which is cofibrant in the $1$-projective model
structure on $\mathcal{F}il(\mathcal{C})$. Then for each $r$,
$\pi^{1}_{nt}(U\otimes C\tau^{r+1})$ is generated under (matric) smash Toda
brackets by elements with degree $1-r\geq n\geq-1.$

######  Theorem 6.18.

Suppose $U$ is a ring in $\operatorname{Fil}(\mathcal{C})$ which is cofibrant
in the 1-projective model structure, and $\pi^{0}_{**}(U\otimes C\tau)$ is
Koszul. Then $U$ is as well.

These will jointly imply Theorem 6.15, because by Lemma 5.24,
$E^{2,r}_{n,t}(U)$ is the image of the map

$\pi^{1}_{nt}(U\otimes C\tau^{r+1})\to\pi^{1}_{nt}(U\otimes C\tau^{2}),$

and this map preserves (matric) smash Toda brackets.

We begin with the following lemma.

######  Lemma 6.19.

We have an identification

$R\otimes^{\mathbb{L}}_{U}R\otimes^{\mathbb{L}}C\tau^{k+1}\to(R\otimes^{\mathbb{L}}C\tau^{k+1})\otimes^{\mathbb{L}}_{U\otimes^{\mathbb{L}}C\tau^{k+1}}(R\otimes^{\mathbb{L}}C\tau^{k+1}).$

###### Proof.

This is the same argument as the proof of Lemma 4.12. ∎

We now continue as before by defining a suspension map.

######  Definition 6.20.

Let $U$ be an augmented $R$-module in $\mathcal{D}^{1}(\mathcal{C})$ with
augmentation ideal $IU$, and let $E^{*}(U)$ denote the spectral sequence from
Theorem 4.13 with

$E^{2}_{rnt}(U)=\operatorname{Tor}_{r}^{\pi^{1}_{**}\left(U\otimes^{\mathbb{L}}C\tau^{k+1}\right)}\left(\pi^{1}_{**}(R\otimes^{\mathbb{L}}C\tau^{k+1}),\pi^{1}_{**}(R\otimes^{\mathbb{L}}C\tau^{k+1})\right)$

converging to

$\pi^{1}_{n-r,t+r}\left(R\otimes^{\mathbb{L}}_{U}R\otimes^{\mathbb{L}}C\tau^{k+1}\right).$

We define

$\sigma:\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1})\to
E^{\infty}_{1**}(U)$

as the top horizontal composition

$\textstyle{\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma}$$\textstyle{E^{\infty}_{1**}(U)}$$\textstyle{\operatorname{Tor}_{1}^{\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1})}(\pi^{1}_{**}(R\otimes^{\mathbb{L}}C\tau^{k+1}),\pi^{1}_{**}(R\otimes^{\mathbb{L}}C\tau^{k+1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{2}_{1**}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the $\sigma$ labelling the left hand arrow is the classical suspension
map from Definition 6.4.

As before, the first step to relating $\sigma$ with decomposables is to
identify indecomposables with a subspace of the homotopy groups of a tensor
product. In this setting, we have the following version of Observation 6.5.

######  Observation 6.21.

Let $U$ be a (not-necessarily commutative) $R$-algebra in
$\mathcal{D}^{1}(\mathcal{C})$ with an augmentation giving rise to a split
fiber sequence

$IU\to U\to R,$

and fix a right $U$-module $M$. Every element in the image of

$\pi^{1}_{**}(M\otimes^{\mathbb{L}}_{U}IU)\to\pi^{1}_{**}(M\otimes^{\mathbb{L}}_{U}U)$

is decomposable by nontrivial smash Toda brackets (by which we mean smash Toda
brackets such that the last factor comes from $\pi^{1}_{**}(IU),$ not just
$\pi^{1}_{**}(U).$)

###### Proof.

Fix a cofibrant replacement $X_{\bullet}$ of $IU$, taken in the 1-projective
model structure on $\operatorname{Fil}(\operatorname{Mod}_{U}).$ Combining the
results of section 3 with Theorem 5.16, we can view this resolution as a chain
complex $\operatorname{Ch}(X)_{\bullet}$. Note that we can build such a
replacement from any free $\pi^{1}_{**}(U)$ resolution of $\pi^{1}_{**}(IU)$,
so in particular we can assume the elements of $X_{\bullet}\otimes C\tau$ are
coproducts of sums of shifts of $U$, not merely of retracts of $U$.

Now, choose $\alpha\in\pi^{1}_{**}(M\otimes^{\mathbb{L}}_{U}IU)$. Since
$\operatorname{colim}X_{\bullet}=IU$ and spheres are compact by Theorem 3.33,
there must be some $n>0$ such that $\alpha$ lifts to some $\hat{\alpha}$ in
$\pi^{1}_{n^{\prime}t}\left(M\otimes^{\mathbb{L}}X_{-n}\right).$ But by
Theorem 5.16, we can identify $X_{-n}$ with
$\Sigma^{n}Z_{0}^{n}(\operatorname{Ch}(X)_{\bullet})$, which by Proposition
5.15 is equivalent to $C_{0}^{n}(\operatorname{Ch}(X)_{\bullet})).$ Since
$\operatorname{Ch}(X_{\bullet})$ is a complex of direct sums of shifts of $U$,
the maps $\operatorname{Ch}(X)_{k}\to\operatorname{Ch}(X)_{k-1}$ are matrix
maps with elements in $\pi^{1}_{**}(U)$. Call these matrices
$V_{2},\cdots,V_{n}$. Then, we can express $\alpha$ as a composition

$\textstyle{S^{n^{\prime},t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{\ell-1}Z_{0}^{\ell-1}(M\otimes_{U}^{\mathbb{L}}X|_{\operatorname{Ch}_{0}^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{C_{0}^{\ell-1}(M\otimes_{U}^{\mathbb{L}}X|_{\operatorname{Ch}_{0}^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\otimes_{U}IU}$

and therefore (using Definition 5.22) as an element of a nontrivial smash Toda
bracket. Composing this with the map $M\otimes_{U}IU\to M\otimes_{U}U$
exhibits the image of $\alpha$ as a nontrivial smash Toda bracket, as desired.
∎

With this observation, the counterpart of Proposition 6.6 is very similar to
the classical case.

######  Proposition 6.22.

The kernel of $\sigma$ consists of elements decomposable in terms of matric
Toda brackets.

###### Proof.

Let $\pi$ denote the composition

$\textstyle{\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1}\otimes^{\mathbb{L}}_{U\otimes^{\mathbb{L}}C\tau^{k+1}}U\otimes^{\mathbb{L}}C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1}\otimes^{\mathbb{L}}_{U\otimes^{\mathbb{L}}C\tau^{k+1}}R\otimes^{\mathbb{L}}C\tau^{k+1})}$

induced by the augmentation map.

The kernel of $\pi$ is the image of

$\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1}\otimes^{\mathbb{L}}_{U\otimes^{\mathbb{L}}C\tau^{k+1}}IU\otimes^{\mathbb{L}}C\tau^{k+1})$

in $\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1})$, which by Observation
6.21 is the collection of nontrivial matric Toda brackets. Now, note that
$E^{\infty}_{0nt}=R[\tau]/\tau^{k+1}$ is a direct summand of
$\pi^{1}_{**}(R\otimes^{\mathbb{L}}_{U}R\otimes C\tau^{k+1})$, so we may
extend $\sigma$ to a map

$\textstyle{E^{\infty}_{1nt}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{1}_{n-1,t+1}(R\otimes^{\mathbb{L}}C\tau^{k+1}\otimes^{\mathbb{L}}_{U\otimes^{\mathbb{L}}C\tau^{k+1}}R\otimes^{\mathbb{L}}C\tau^{k+1})}$$\textstyle{\pi^{1}_{nt}(IU\otimes^{\mathbb{L}}C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma^{\prime}}$

Now, since $\pi^{1}_{**}(R\otimes^{\mathbb{L}}C\tau^{k+1})=R[\tau]/\tau^{k}$
is a direct summand of $\pi^{1}_{**}(R\otimes^{\mathbb{L}}_{U}R)$, we obtain a
diagram whose rightmost column is exact.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{1}_{nt}(R\otimes^{\mathbb{L}}_{U}U\otimes
C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{1}_{n+1,t-1}(IU\otimes^{\mathbb{L}}C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\pi}$$\textstyle{\pi^{1}_{n+1,t-1}(R\otimes^{\mathbb{L}}_{U}R\otimes^{\mathbb{L}}C\tau^{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{1}_{n+1,t-1}(R\otimes^{\mathbb{L}}_{U}IU\otimes^{\mathbb{L}}C\tau^{k+1})}$

Comparing the definitions of $\pi$ and $\sigma$ shows that the diagram
commutes. Moreover, $0$ is the only element in both the image of $\sigma$ and
the image of $R[\tau]/\tau^{k}$, so it follows that $\ker\sigma=\ker\pi.$ ∎

To complete the proof, we need to constrain
$\pi_{1}(R\otimes^{\mathbb{L}}_{U}R\otimes^{\mathbb{L}}C\tau^{k+1}).$

######  Lemma 6.23.

Suppose $\pi^{1}_{**}(X\otimes^{\mathbb{L}}C\tau^{2})$ is concentrated in
degree $n=0$. Then $\pi^{1}_{**}(X\otimes^{\mathbb{L}}C\tau^{k+1})$ is
concentrated in degrees between $0$ and $1-k.$

###### Proof.

Filter $C\tau^{k+1}$ as

$\Sigma^{-k,0}C\tau\to\Sigma^{1-k,0}C\tau^{2}\to\cdots\to C\tau^{k}.$

Using Corollary 3.30 and Lemma 3.25, we can compute the cofiber in
$\mathcal{D}^{1}(\mathcal{C})$ of the map

$\Sigma^{-k+j-1,0}C\tau^{j}\to\Sigma^{-k+j,0}C\tau^{j+1}$

to be $\Sigma^{-k+j,0}C\tau^{2}\oplus\Sigma^{-k-1,1}C\tau$, which is weak
equivalent to $\Sigma^{-k+j,0}C\tau^{2}.$

Applying Theorem 4.12 to this filtration, we obtain a spectral sequence
converging to $\pi^{1}_{**}(X\otimes^{\mathbb{L}}C\tau^{k+1})$ with

$E^{2}_{rnt}=\pi^{1}_{nt}(X\otimes^{\mathbb{L}}\bigoplus_{r=1}^{k}\Sigma^{-k+r,0}C\tau^{2}),$

which is concentrated in degrees $1-k\leq n\leq 0.$ ∎

The proof of Theorem 6.17 is then immediate. Proposition 6.22 implies that any
element of $\pi^{1}_{**}(IU\otimes^{\mathbb{L}}C\tau^{k+1}$ outside the kernel
of sigma is a nontrivial smash Toda bracket. But Lemma 6.23 and the Koszul
condition imply that the kernel is trivial outside of degrees $1-k\leq n\leq
0,$ completing the proof.

Now, we prove Theorem 6.18.

######  Definition 6.24.

Let $U$ be an object of $\operatorname{Fil}(\mathcal{C}).$ The _inverse
filtration_ on $U$ is the natural object $F_{\bullet}(U)$ in
$\operatorname{Fil}(\operatorname{Fil}(\mathcal{C}))$ with

$\left(F_{n}(U)\right)_{i}=\begin{cases}U_{i}&\text{ for }i>n\\\ U_{n}&\text{
for }i\leq n.\end{cases}$

Note that if $U$ is cofibrant, the map $F_{n}(U)\to F_{n-1}(U)$ is a
cofibration, so this is also a filtration of $U$ in
$\mathcal{D}^{1}(\mathcal{C})$.

Moreover, in this case we can use Corollary 3.30 to compute the homotopy
cofiber of the map $F_{n+1}(U)\to F_{n}(U)$ in the 1-projective model
structure. The cofiber takes the form

$\cdots\to
X_{n+2}/X_{n+3}\xrightarrow{0}X_{n+1}/X_{n+2}\xrightarrow{0}X_{n}/X_{n+1}\xrightarrow{id}X_{n}/X_{n+1}\xrightarrow{id}X_{n}/X_{n+1}\to\cdots$

which is weak equivalent to

$\cdots\to 0\to 0\to
X_{n}/X_{n+1}\xrightarrow{id}X_{n}/X_{n+1}\xrightarrow{id}X_{n}/X_{n+1}\to\cdots$

which is precisely the cofiber of this map computed in
$\operatorname{Fil}(\mathcal{C}).$ This observation will help us in the
following proof:

###### Proof of Theorem 6.18.

We want to understand $\pi^{1}_{**}(R\otimes^{\mathbb{L}}_{U}R\otimes
C\tau^{2}).$ As before, this is equivalent to

$(R\otimes C\tau^{2})\otimes^{\mathbb{L}}_{U\otimes C\tau^{2}}(R\otimes
C\tau^{2}).$

Placing the inverse filtration on $U$, we find that the associated graded
complex $G_{n}(U)$ takes the form

$\displaystyle G_{n}(U)$ $\displaystyle=\left(F_{n}(U)\otimes
C\tau^{2}\right)/\left(F_{n+1}(U)\otimes C\tau^{2}\right)$
$\displaystyle=\left(F_{n}(U)/F_{n+1}(U)\right)\otimes C\tau^{2}$
$\displaystyle=\Sigma^{r,0}C\tau^{2}\otimes U_{r}/U_{r+1}.$

By Theorem 4.12, there is a spectral sequence with

$E^{1}_{n^{\prime}nt}=\pi^{0}_{n^{\prime}nt}\left(R\otimes
C\tau^{2}\otimes^{\mathbb{L}}_{G_{n}(U)}R\otimes C\tau^{2}\right).$

Here the right hand side is being computed in
$\operatorname{Fil}(\mathcal{D}^{1}(\mathcal{C})),$ with the index
$n^{\prime}$ coming from the $\operatorname{Fil}$-filtration and $n$ coming
from the $\mathcal{D}^{1}$-filtration. The sequence converges to
$\pi^{1}_{n,t}(R\otimes^{\mathbb{L}}_{U}R\otimes C\tau^{2}).$

To compute the $E^{1}$ page, we apply the spectral sequence of Theorem 4.13,
which takes the form

$E^{2}_{rn^{\prime}nt}=\operatorname{Tor}^{r}_{\pi_{**}^{0}(U)}(R,R)$

and converges to

$\pi^{0}_{n^{\prime}+r,n+r,t-r}\left(R\otimes
C\tau^{2}\otimes^{\mathbb{L}}_{G_{n}(U)}R\otimes C\tau^{2}\right).$

Looking closely at the filtration $G_{n}(U)$, we note that both $n$ and
$n^{\prime}$ are equal to the internal $n$-grading on the homotopy
groups/rings on the right hand side. By our Koszulity assumption, this $E^{2}$
page is concentrated in degrees $r=-n.$ In particular, we conclude that
$\pi^{1}_{**}(R\otimes^{\mathbb{L}}_{U}R\otimes C\tau^{2})$ and so $U$ is
Koszul. ∎

##  7 Additional lemmas

In this appendix we gather a handful of lemmas used at various points in the
text.

######  Lemma 7.1.

Let $A_{n}\to\cdots\to A_{0}$ and $B_{n}\to\cdots\to B_{0}$ be sequences of
maps (not chain complexes) of sets and suppose:

  1. 1.

$f$ maps the image of $A_{n}$ in $A_{1}$ isomorphically onto the image of
$B_{n}$ in $B_{1}$.

  2. 2.

$f$ maps the image of $A_{n-1}$ in $A_{0}$ isomorphically onto the image of
$B_{n-1}$ in $B_{0}$.

Then, $f$ maps the image of $A_{n}$ in $A_{0}$ isomorphically onto the image
of $B_{n}$ in $B_{0}.$

###### Proof.

Call the map from the image of $A_{n}$ in $A_{0}$ to the image of $B_{n}$ in
$B_{0}$ $f_{*}.$ The image of $A_{n}$ in $A_{0}$ is a subset of the image of
$A_{n-1}$ in $A_{0}$ (and the same for $B$), so $f_{*}$ is injective.
Similarly, the image of $A_{n}$ in $A_{0}$ is a quotient of the image of
$A_{n}$ in $A_{1}$, so $f_{*}$ is surjective. ∎

######  Lemma 7.2.

Suppose $f:X\to Y$ is $k$-exact and a $J^{k}$ injection. Then, the fiber $F$
of $f$ is $k$-acyclic.

###### Proof.

Consider the following diagram, noting that the vertical triples are fiber
sequences.

$\textstyle{F_{n-k}/F_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{i}$$\textstyle{\Sigma
F_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\Omega
X_{n-2k}/X_{n-k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{f}$$\textstyle{X_{n-k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\ell}$$\textstyle{X_{n-k}/X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{f}$$\textstyle{\Sigma
X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Omega
Y_{n-2k}/Y_{n-k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{Y_{n-k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell}$$\textstyle{Y_{n-k}/Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\Sigma
Y_{n}}$

Pick an arbitrary $a$ in $\pi_{t}(\Sigma F_{n}).$ To show that $\tau^{k}a=0$,
we need to show that $a$ lifts to $\pi_{t}(F_{n-k}/F_{n}).$ Since
$f_{*}(i_{*}(a))=0$ and $f$ is exact, there exists $b$ in
$\pi_{t}(X_{n-k}/X_{n})$ with $\delta_{*}b=i_{*}(a).$

Now, $f_{*}(b)$ might be nonzero. But
$\delta_{*}f_{*}(b)=f_{*}(\delta_{*}b)=0,$ so there exists $c$ in
$\pi_{t}(Y_{n-k})$ such that $\ell_{*}(c)=f_{*}(b).$ Now, since $f$ is exact,
there exist $y$ in $\pi_{t}(\Omega Y_{n-2k}/Y_{n-k})$ and $s$ in
$\pi_{t}(\Omega X_{n-2k}/X_{n-k})$ such that $f_{*}(s)=c+d_{*}y.$ Since $f$ is
a $J^{k}$ injection, there exists $x$ in $\pi_{t}(\Omega X_{n-2k}/X_{n-k})$
with $f_{*}(x)=y,$ so in particular $f_{*}(s-d_{*}x)=c.$

Then, $f_{*}(\ell_{*}(s-d_{*}x))=\ell_{*}(c)=f_{*}(b).$ But by definition,
$d\circ\ell=0$, so $d_{*}\ell_{*}(s-d_{*}x)=0.$ So if we set $t=b+\ell(dx-c),$
we have:

  * •

$f_{*}(t)=0$

  * •

$d_{*}(t)=i_{*}(a)$

This guarantees $t$ lifts to $u$ in $\pi_{t}(F_{n-k}/F_{n})$ with $d_{*}u=a$,
as desired! ∎

######  Lemma 7.3.

The cofiber map $\delta$ defined by the cofiber sequence

$\Sigma^{-k,0}C\tau^{k}\to C\tau^{2k}\to
C\tau^{k}\xrightarrow{\delta}\Sigma^{-k,1}C\tau^{k}$

is a derivation, in the sense that we have a commutative square

$\textstyle{C\tau^{k}\otimes
C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{1\otimes\delta+\delta\otimes
1}$$\textstyle{C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C\tau^{k}\otimes\Sigma^{-k,1}C\tau^{k}\oplus\Sigma^{-k,1}C\tau^{k}\otimes
C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{-k,1}C\tau^{k}}$

where the bottom horizontal map is the sum of two copies of $\mu$.

This lemma is essentially a standard step in proving that the spectral
sequence associated to a ring in $\operatorname{Fil}(\mathcal{C})$ is
multiplicative: we prove the corresponding statement about the representing
objects $C\tau^{k}$ rather than the spectral sequence itself. Our proof
closely follows appendix A.3 of [50]. We prove the Lemma over $\mathbb{F}_{2}$
to avoid signs, and refer the interested reader to the aforementioned appendix
to fill in the correct signs.

###### Proof.

We have a commutative square

$\textstyle{S^{-k,0}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\otimes
S^{0,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}\otimes
S^{0,0}}$

with an evident isomorphism to the commutative square

$\textstyle{S^{-2k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}}$

Let $PO$ denote the pushout

$\textstyle{S^{-k,0}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\otimes
S^{0,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{PO.}$

By standard results on total cofibers, we obtain a cofiber sequence

$PO\to S^{0,0}\otimes S^{0,0}\to C\tau^{k}\otimes C\tau^{k}.$

Now, the commutative square

$\textstyle{PO\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}\otimes
S^{0,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}}$

extends to a map of cofiber sequences

$\textstyle{PO\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}\otimes
S^{0,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\tau^{k}\otimes
C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{0,1}PO\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{0,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,1}}$

and a simple inspection shows that the given map is the multiplication map.

From here, the commutative square

$\textstyle{S^{-k,0}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{PO\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-2k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}}$

extends to a map of cofiber sequences

$\textstyle{S^{-k,0}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{PO\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\otimes
C\tau^{k}\oplus C\tau^{k}\otimes
S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-2k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{-k,0}C\tau^{k}}$

Pasting the rightmost square of the first cofiber sequence to the rightmost
square of the second cofiber sequence gives a commutative diagram

$\textstyle{C\tau^{k}\otimes
C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{0,1}PO\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma
S^{-k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-k,1}\otimes
C\tau^{k}\oplus C\tau^{k}\otimes
S^{-k,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{-k,1}C\tau^{k}}$$\textstyle{\Sigma^{-k,1}C\tau^{k}\otimes
C\tau^{k}\oplus
C\tau^{k}\otimes\Sigma^{-k,1}C\tau^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{-k,1}(\mu\oplus\mu)}$

where the bottom triangle comes from the unit axiom of a ring.

By inspection, the left vertical composition is $\delta\otimes 1\oplus
1\otimes\delta$, and the right vertical composition is $\delta$, giving us the
desired square. ∎

######  Lemma 7.4.

Any bigraded multiplicative spectral sequence over a field starting whose unit
is a permanent cycle is isomorphic, starting at the $E^{1}$ page, to the
spectral sequence associated to a filtered DGA.

###### Proof.

We can decompose the $E_{1}$ page into a vector space $S_{\infty}$ of
permanent cycles, a sequence of vector spaces $S_{i}$ of sources of nonzero
differentials on the $E_{i}$ page, and a sequence of vector spaces $T_{i}$ of
targets of nonzero differentials on the $E_{i}$ page. Note that we have a
canonical isomorphism $S_{i}\to T_{i}.$

Now, let $X$ be the filtered differential graded $R$-module

$X=S_{\infty}\oplus\mathbb{D}_{1}\otimes_{\mathbb{F}_{p}}S_{1}\oplus\mathbb{D}_{2}\otimes_{\mathbb{F}_{p}}S_{2}\oplus\cdots$

where $S_{\infty}$ has filtration zero and

$\mathbb{D}_{i}=\cdots\to 0\to R\to R\to 0\to\cdots$

where the first $R$ has filtration $0$ and the second lives in filtration $i$.

The spectral sequence associated to $X$ is by construction (additively)
isomorphic the one we started with. A multiplicative structure on our original
spectral sequence places a ring structure on the $E_{1}$ page, which is the
bigraded module obtained by forgetting the differential on $X.$ The Leibniz
rule says precisely that this multiplication preserves the differential on
$X$, so we obtain a DGA with the desired spectral sequence. ∎

##  8 Proof of Proposition 5.15

In this appendix, we provide a proof of Proposition 5.15. Throughout, we use
Lurie’s notation $\mathcal{D}_{x/}$ to denote the overcategory of maps in a
category $\mathcal{D}$ from an object $x$. We will prove the proposition at
the end of this appendix by writing down a diagram for which both
$C_{i}^{j}(X)$ and $\Sigma^{j-i}Z_{i}^{j}(X)$ are colimits. In order to do so,
we need four lemmas letting us manipulate poset-shaped diagrams without
changing their colimit. The first three are special cases of the
$\infty$-categorical version of Quillen’s theorem A, which states that a
functor $i:\mathcal{E}\to\mathcal{D}$ is cofinal if and only if for every
object $x\in\mathcal{D}$, the comma $\infty$-category $x/i$ (defined to be
$\mathcal{D}_{x/}\times_{\mathcal{D}}\mathcal{E}$) is a weakly contractible
simplicial set.

######  Lemma 8.1.

Let $\mathcal{D}$ be a partially ordered set, and let $\mathcal{E}$ be a
subcategory of shape $a\leftarrow b\to c.$ In order that the inclusion
$i:\mathcal{E}\to\mathcal{D}$ be final, it suffices to check that:

  * •

Every $d\in\mathcal{D}$ has a map to either $a$ or $c$.

  * •

$b$ is the (categorical) product of $a$ and $c$.

###### Proof.

For arbitrary $d\in\mathcal{D}$, the first condition implies that the comma
category $d/i$ is a nonempty subcategory of $\mathcal{E}$, and the second
implies that it cannot be $\\{a,c\\}.$ ∎

######  Lemma 8.2.

Let $\mathcal{D}$ be an ordinary category, and $d$ be an object of
$\mathcal{D}$ equipped with a morphism $f:d\to e$ such that every morphism out
of $d$ factors as a composition $g\circ f.$ Then, the inclusion
$i:\mathcal{D}-\\{d\\}\to\mathcal{D}$ is final.

###### Proof.

Pick an arbitrary $x\in\mathcal{D}$. If $x\neq d$, then the comma category
$x/i$ is just $(\mathcal{D}-\\{d\\})_{x/}$. If $x=d$, then
$x/i=(\mathcal{D}-\\{d\\})_{e/}$ by the factorization assumption. In either
case, $x/i$ is weakly contractible. ∎

######  Lemma 8.3.

Let $\mathcal{D}$ be a partially ordered set with objects $a,b,c$ such that:

  * •

There are maps $a\to c$ and $b\to c$.

  * •

There are no maps $a\to b$ or $b\to a$.

  * •

For any map $d\to a$, there exists a coproduct $b\coprod d$ in $\mathcal{D}.$

Let $\mathcal{E}$ be the partially ordered set extending $\mathcal{D}$ to
include a map $a\to b$

Then, the inclusion $i:\mathcal{D}\to\mathcal{E}$ is final. Moreover, any
functor $\mathcal{D}\to\mathcal{C}$ sending $a$ and $b$ to zero objects
extends to a functor $\mathcal{E}\to\mathcal{C}.$

###### Proof.

Consider an arbitrary element $d$ in $\mathcal{D}.$

If $d$ does not map to $a$, then the simplicial set $d/i$ is equivalent to
$D_{d/},$ which is of course contractible.

If $d$ does map to $a$, then $d/i$ is the (not disjoint) union $D_{d/}\cup
D_{b/}.$ The intersection is $D_{d\coprod b/},$ implying the union is weakly
contractible.

The rest of the claim follows from the universal properties of zero objects. ∎

The fourth lemma is a special case of [26] 4.2.3.10, and will give conditions
under which we can remove from a diagram objects which are already colimits of
certain subdiagrams.

######  Lemma 8.4.

Let $\mathcal{D}$ be a partially ordered set. For a given object $x$, let
${\mathcal{D}^{\prime}}_{/x}$ denote the sub-$\infty$-category of $D$
consisting of objects with a map to $x$, excluding $x$ itself.

Given an object $d$ and a functor $F$ from $\mathcal{D}$ into an
$\infty$-category $\mathcal{C}$ such that $F|_{\mathcal{D}_{/d}}$ exhibits
$F(d)$ as the colimit of $F|_{\mathcal{D}^{\prime}_{/d}}$, it follows that the
colimit of $F$ is the same as the colimit of $F|_{\mathcal{D}-\\{d\\}}.$

###### Proof.

This is a special case of HTT 4.2.3.10. Let $K$ be the nerve of
$\mathcal{D}-\\{d\\}$, and let $\mathcal{J}=\mathcal{D}.$ The functor
$\mathcal{D}\to\text{sSet}_{/K}$ is defined on objects as follows:

  * •

For $x\neq d$, send $x$ to $K_{/x}$.

  * •

For $x=d$, send $x$ to $\mathcal{D}^{\prime}_{/d}.$

Note that $\mathcal{J}_{\sigma}$ is automatically contractible, because for
$\sigma=x_{1}\to\cdots\to x_{n}$, $\mathcal{J}_{\sigma}$ is isomorphic to
$\mathcal{D}_{x_{n}/},$ so the hypotheses of HTT 4.2.3.10 are satisfied. The
lemma then follows because the induced diagram $q|_{N(\mathcal{j})}$ is
equivalent to $F$. ∎

###### Proof of Propositions 5.15.

Assume without loss of generality that $i=0$, and set $j$ equal to $n$ so we
can use $i$ and $j$ to index things throughout the proof. We will construct a
diagram $F:\mathcal{D}\to\mathcal{S}$ whose colimit is $C_{0}^{n}$, and use
the previous four lemmas to transform the diagram without changing the
colimit, until we end up with a diagram whose colimit is
$\Sigma^{n}Z_{0}^{n}.$

Construct a poset-shaped diagram $\mathcal{D}$ inductively as follows:

  * •

Start with the subdiagram $X_{n}\to X_{n-1}\to\cdots\to X_{0}$ of $X.$

  * •

Attach the pullback (and hence pushout) square

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.93471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{0}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.9694pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.70667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.9694pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
41.86942pt\raise-29.70667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.93471pt\raise-39.53998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.93471pt\raise-39.53998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{0}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

along the edge $X_{1}\to X_{0}$, (where we label individual zero objects
$0_{i}$ and $0^{i}$ to help keep track of them later.)

  * •

By the universal property, the map $X_{2}\to X_{1}$ lifts to $Z_{0}^{1},$ and
we can attach the pushout square

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.93471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{0}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.20554pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.60889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.20554pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
41.10555pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.93471pt\raise-40.4422pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.93471pt\raise-40.4422pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{0}^{1}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

  * •

Continue this pattern until we’ve attached the pushout square

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.21536pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{0}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.20554pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.30943pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.20554pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
41.3862pt\raise-28.50499pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
33.21536pt\raise-40.14275pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.21536pt\raise-40.14275pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{0}^{n-1}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

  * •

In the same way, we can attach the pushout square

$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n-1}^{n}}$

and factor $X_{n-1}\to X_{n-2}$ through a map $C_{n-1}^{n}\to X_{1}$. We
continue as before until we’ve attached the pushout square

$\textstyle{C^{n}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}^{n}}$

Let $\mathcal{D}\to\mathcal{S}$ be the diagram constructed above, _excluding_
$C_{0}^{n}.$ To give some idea what we mean, here’s $\mathcal{D}$ when $n=3$:

---
$\textstyle{0^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}}$$\textstyle{0_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{1}}$

Note that, by Lemma 8.1, the subcategory $0_{1}\leftarrow C_{1}^{n}\to X_{0}$
is final, so the colimit of this entire diagram is $C_{0}^{n}$, as we might
hope!

Now, for $1\leq j\leq n-1,$ note that the pushout square defining $C_{j}^{n}$
is final in $\mathcal{D}^{\prime}_{/C_{j}^{n}}$ by Lemma 8.1, so we can remove
the $C_{j}^{n}$ (in ascending order of $j$) by Lemma 8.4. This leaves us with
a diagram $\mathcal{B}$ containing no $C_{j}^{n}$’s, and with maps $0_{i}\to
0_{j}$ for $i>j$ and $0_{i}\to X_{k}$ for $i>k+1.$ In the case $n=3$, the
diagram looks like this:

---
$\textstyle{0^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}}$$\textstyle{0_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{1}}$

Now for $2\leq k\leq n$, we will lift the maps $0_{k}\to X_{k-2}$ to maps
$0_{k}\to 0^{k-1}.$ This can of course be done without changing the
commutativity of the diagram. To check that the new diagram has the same
colimit as the old, we need to check that, for arbitrary $d\in\mathcal{B}$
mapping to $0_{k}$, $0^{k-1}$ and $d$ have a coproduct in $\mathcal{D}$. We
check the three possible cases:

  * •

If $d=0_{k},$ then $0^{k-1}\coprod d$ is $X_{k-1}.$

  * •

If $d=0_{k+1}$ or $d=X_{k}$ then $0^{k-1}\coprod d$ is $Z_{0}^{k-1}.$

  * •

Otherwise, $0^{k-1}\coprod d=0^{k-1}$

We conclude by Lemma 8.3 with $(a,b,c)=(0_{k},0^{k-1},X_{k-2})$ that the
modified diagram, which we will call $W$, still has the same colimit. In the
case $n=3$, we have the following diagram:

---
$\textstyle{0^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}}$$\textstyle{0_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{1}}$

Now, we apply Lemmas 8.4 and 8.1 again to remove $X_{0}$, at which point
$X_{1}$ only has one arrow pointing out of it and we can remove it by Lemma
8.2. But now Lemmas 8.4 and 8.1 tell us we can remove $Z_{0}^{1}$, and Lemma
8.2 lets us remove $X_{2}.$ We can continue until we’ve removed all the
$X_{i}$ and all the $Z_{0}^{i}$ except for $Z_{0}^{n}.$

We’re left with a diagram containing only $Z_{0}^{n}$, $0^{i}$ and $0_{i}$,
with $Z_{0}^{n}$ initial and maps $0^{i}\to 0_{j}$, $0_{i}\to 0_{j}$,
$0^{i}\to 0^{j}$, and $0_{i}\to 0^{j}$ whenever $i>j.$

---
$\textstyle{0^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{0}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0^{1}}$$\textstyle{0_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{1}}$

Applying [26] 4.4.2.2 repeatedly shows that the colimit of this diagram is
$\Sigma^{n}Z_{0}^{n},$ as desired. ∎

## References

  * [1] Daniel C Isaksen, Guozhen Wang, and Zhouli Xu.  Stable homotopy groups of spheres: From dimension 0 to 90.  Publications mathématiques de l’IHÉS, 137(1):107–243, 2023\.
  * [2] Michael A Hill, Michael J Hopkins, and Douglas C Ravenel.  On the nonexistence of elements of kervaire invariant one.  Annals of Mathematics, pages 1–262, 2016.
  * [3] William Browder.  The kervaire invariant of framed manifolds and its generalization.  Annals of Mathematics, pages 157–186, 1969.
  * [4] Robert R Bruner.  Ext in the nineties.  Contemporary Mathematics, 146:71–71, 1993.
  * [5] Christian Nassau.  On the secondary steenrod algebra.  arXiv preprint arXiv:1011.2471, 2010.
  * [6] Dexter Edralin Chua.  The E3 Page of the Adams Spectral Sequence.  PhD thesis, Harvard University, 2022.
  * [7] Guozhen Wang and Zhouli Xu.  The triviality of the 61-stem in the stable homotopy groups of spheres.  Annals of Mathematics, 186(2):501–580, 2017.
  * [8] John SP Wang.  On the cohomology of the mod-2 steenrod algebra and the non-existence of elements of hopf invariant one.  Illinois Journal of Mathematics, 11(3):480–490, 1967.
  * [9] R Michael F Moss.  Secondary compositions and the adams spectral sequence.  Mathematische Zeitschrift, 115(4):283–310, 1970.
  * [10] J Peter May.  Matric massey products.  Journal of Algebra, 12(4):533–568, 1969.
  * [11] Stanley O Kochman.  A chain functor for bordism.  Transactions of the American Mathematical Society, 239:167–196, 1978.
  * [12] Victor KAM Gugenheim and J Peter May.  On the theory and applications of differential torsion products, volume 142.  American Mathematical Soc., 1974.
  * [13] Robert R Bruner.  An adams spectral sequence primer.  Department of Mathematics. Wayne State University. Detroit MI, pages 48202–3489, 2009.
  * [14] Joel M Cohen.  The decomposition of stable homotopy.  Annals of Mathematics, pages 305–320, 1968.
  * [15] Bogdan Gheorghe, Guozhen Wang, and Zhouli Xu.  The special fiber of the motivic deformation of the stable homotopy category is algebraic.  Acta Mathematica, 226(2):319–407, 2021.
  * [16] Piotr Pstragowski.  Synthetic spectra and the cellular motivic category.  Inventiones mathematicae, 232(2):553–681, 2023.
  * [17] Daniel Dugger and Daniel C Isaksen.  The motivic adams spectral sequence.  Geometry & Topology, 14(2):967–1014, 2010.
  * [18] Robert Burklund, Jeremy Hahn, and Andrew Senger.  On the boundaries of highly connected, almost closed manifolds.  arXiv preprint arXiv:1910.14116, 2019.
  * [19] Irakli Patchkoria and Piotr Pstragowski.  Adams spectral sequences and franke’s algebraicity conjecture.  arXiv preprint arXiv:2110.03669, 2021.
  * [20] Robert Burklund.  An extension in the adams spectral sequence in dimension 54.  Bulletin of the London Mathematical Society, 53(2):404–407, 2021\.
  * [21] Robert Burklund.  Synthetic cookware v2.0.  preparation, draft available on the author’s webpage.
  * [22] Aaron Mazel-Gee.  Goerss-Hopkins obstruction theory via model infinity categories.  University of California, Berkeley, 2016.
  * [23] John Michael Boardman and Rainer M Vogt.  Homotopy invariant algebraic structures on topological spaces, volume 347.  Springer, 1973.
  * [24] André Joyal.  Quasi-categories and kan complexes.  Journal of Pure and Applied Algebra, 175(1-3):207–222, 2002.
  * [25] André Joyal.  Notes on quasi-categories.  preprint, 2008.
  * [26] Jacob Lurie.  Higher topos theory.  Princeton University Press, 2009.
  * [27] Jacob Lurie.  Higher algebra. 2021.  Preprint, available at http://www. math. harvard. edu/~ lurie, 2021.
  * [28] Saul Glasman.  Day convolution for infinity-categories.  arXiv preprint arXiv:1308.4940, 2013.
  * [29] Bogdan Gheorghe, Daniel C Isaksen, Achim Krause, and Nicolas Ricka.  C-motivic modular forms.  Journal of the European Mathematical Society, 24(10):3597–3628, 2022.
  * [30] Jacob Lurie.  Rotation invariance in algebraic k-theory.  preprint, 2015.
  * [31] Denis-Charles Cisinski.  Higher categories and homotopical algebra, volume 180.  Cambridge University Press, 2019.
  * [32] Joana Cirici, Daniela Egas Santander, Muriel Livernet, and Sarah Whitehouse.  Model category structures and spectral sequences.  Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 150(6):2815–2848, 2020.
  * [33] James A Brotherston.  Monoidal model structures on filtered chain complexes relating to spectral sequences.  arXiv preprint arXiv:2402.09207, 2024.
  * [34] James A Brotherston.  A distributive lattice of model structures relating to spectral sequences.  arXiv preprint arXiv:2402.09893, 2024.
  * [35] Stefano Ariotta.  Coherent cochain complexes and beilinson t-structures, with an appendix by achim krause.  arXiv preprint arXiv:2109.01017, 2021.
  * [36] Mark Hovey.  Model categories.  Number 63. American Mathematical Soc., 2007.
  * [37] A Douady.  La suite spectrale d’adams: structure multiplicative, exp. 19.  Séminaire Henri Cartan, Ecole Normale Supérieure, vol, 11(2):1958–1959.
  * [38] Alice Petronella Hedenlund.  Multiplicative tate spectral sequences.  2021\.
  * [39] Gard Olav Helle.  Pairings and convergence of spectral sequences.  Master’s thesis, 2017.
  * [40] Gunnar Sjödin.  A set of generators for ext r (k, k).  Mathematica Scandinavica, 38(2):199–210, 1976.
  * [41] Anthony D Elmendorf, I Kriz, MA Mandell, and JP May.  Rings, modules, and algebras in stable homotopy theory.  In American Mathematical Society Surveys and Monographs, American Mathematical Society, 1995.
  * [42] Shaul Barkan.  Chromatic homotopy is monoidally algebraic at large primes.  arXiv preprint arXiv:2304.14457, 2023.
  * [43] Robert Burklund, Jeremy Hahn, and Andrew Senger.  Galois reconstruction of artin–tate r-motivic spectra.  arXiv preprint arXiv:2010.10325, 4, 2020.
  * [44] Bernhard Keller and Dieter Vossieck.  Aisles in derived categories.  Bull. Soc. Math. Belg. Sér. A, 40(2):239–253, 1988.
  * [45] Leovigildo Alonso Tarrío, Ana Jeremías López, and María José Salorio.  Construction of t-structures and equivalences of derived categories.  Transactions of the American Mathematical Society, 355(6):2523–2543, 2003.
  * [46] Piotr Pstragowski and Paul VanKoughnett.  Abstract goerss-hopkins theory.  Advances in Mathematics, 395:108098, 2022.
  * [47] Albert F Lawrence.  Matric Massey products and matric Toda brackets in the Adams spectral sequence.  PhD thesis, The University of Chicago, 1969.
  * [48] Eva Belmont and Hana Jia Kong.  A toda bracket convergence theorem for multiplicative spectral sequences.  arXiv preprint arXiv:2112.08689, 2021.
  * [49] CTC Wall.  Generators and relations for the steenrod algebra.  Annals of Mathematics, pages 429–444, 1960.
  * [50] William Balderrama.  Deformations of homotopy theories via algebraic theories.  arXiv preprint arXiv:2108.06801, 2021.


# Entropic Empirical Mode Decomposition

Sumit Kumar Ram1, Marta Molinas2
1Indian Institute of Science Education and Research, Kolkata, Mohanpur, Nadia,
India
<EMAIL_ADDRESS>12Dept. of Eng. Cybern., Norwegian Univ. of Sci. and
Tech., Trondheim, Norway.

<EMAIL_ADDRESS>
###### Abstract

> Empirical Mode Decomposition(EMD) is an adaptive data analysis technique for
> analyzing nonlinear and non stationary data[1]. EMD decomposes the original
> data into a number of Intrinsic Mode Functions(IMFs)[1] for giving better
> physical insight of the data. Permutation Entropy(PE) is a complexity
> measure[3] function which is widely used in the field of complexity theory
> for analyzing the local complexity of time series. In this paper we are
> combining the concepts of PE and EMD to resolve the mode mixing problem
> observed in determination of IMFs.

## Introduction

Empirical Mode Decomposition proposed by Huang Et al.[1] serves as a widely
used and effective method for analyzing nonlinear and non stationary data[5].
It decomposes the original data into a finite and often small number of well
behaved Hilbert transformable[8] intrinsic mode functions(IMFs). Due to its
adaptive nature and instantaneous frequency[8] determinable IMFs it gives a
sharp identification of the embedded structure and hence is highly efficient
in terms of analyzing physical reality[5] of the data.

”Mode mixing” is defined as any IMF consisting of oscillations of dramatically
disparate scales[2], mostly caused by intermittency of the driving mechanics.
Mode mixing leads to deceptive physical meaning of the mixed mode IMFs[5].
Hence mode mixing problem is the major drawback of the EMD and was first
illustrated by Huang et al.[5]

Permutation entropy proposed by Bandt et al[3] is an effective way of
measuring the complexity of the time series by comparing the neighboring
values around each point. Further Cao et al.[6] showed that it is an efficient
way for Detecting dynamical changes in time series. This shows the adaptive
nature[4] of the technique. Hence it can be concluded that the combination of
EMD and PE will maintain its adaptivity and can be used for adaptive data
analysis.

## Empirical Mode Decomposition

Empirical Mode Decomposition proposed by Huang et al. (1998)[1] is an
effective algorithm to decompose time series into the so-called Intrinsic Mode
Functions. These modes form a (quasi-)orthogonal set of basis functions that
is derived directly from the original data without a priory assumptions[1]
about their nature. In addition to orthogonality of the total set of basis
functions, each IMF should have the following property

  * •

Number of extreme values and zero crossings differ at most by one.

  * •

The ”local mean” value of the mode is zero .

In order to get the IMFs from the original data set we can follow the
following algorithm

  1. 1.

Start with a discrete time series x(t) with sampling period Ts.

  2. 2.

Compute all maximum (minimum) values of x(t), perform cubic spline
interpolation, $s_{max}(s_{min})$, for these values.

  3. 3.

Estimate the local mean of the time series as

$m(t)=\frac{1}{2}(s_{max}+s_{min})$

  4. 4.

Subtract m(t) from x(t) and repeat $step-1,2,3$ with the resulting time
series, until a stopping criterion is reached.

  5. 5.

Once the stopping criterion is reached name the function as first IMF
($c_{1}$). Subtract this IMF from the original signal and repeat the
$steps-1,2,3,4$ until getting the residue.

  6. 6.

Residue is characterized as the mode function which does not have enough
extremas for interpolation. This residual can be regarded as the global trend,
r(t), of the time series

Hence with the EMD algorithm we can decompose any time series into its
IMFs($c_{i}$) and residue($r(t)$) as

$x(t)=\sum_{i=1}^{n}c_{i}(t)+r(t)$

where n is the total number of IMFs obtained from the original time series.

## Permutation Entropy

The quantification of complexity of a time series plays a crucial role in
biomedical analysis[7] as well as in other branches of nonlinear sciences.
Various complexity measures have been proposed and is used to study the
complexity of time series data to get the useful information about the
temporal dynamical changes in time series. Some of the widely used complexity
measure are correlation dimension and the Kolmogorov-Sinai entropy and
Permutation entropy (PE). Permutation Entropy is a conceptually simple method
that allows the analysis and representation of a time series through a
symbolic sequence. The advantages of the PE method reside in its simplicity,
its robustness[3]. In addition, this method provides an extremely fast
computational algorithm and it can be applied to any type of time series.

For calculating the PE of a given time series we can proceed with the
following procedure.

Consider the measured scalar time series data $x(t_{i})$, for n = 1, 2,…,
where i denotes the i-th variable of the system. Firstly, we divide the long
time series into segments with fixed finite length $\tau$(for convenience we
have chosen the length $\tau$ to be 120 here) which can be denoted as
$\tau_{j}$ (j = 1, 2,….). These $\tau_{j}s$ are obtained by applying running
windows over our original time series. In our case we have chosen the step
size $1$ to have a better information about the complexity variation over
time. Further we use another sliding-window analysis to partition each
$\tau_{j}$ into short sequences of given length m. For convenience we have
taken the value of m to be 3 here. Fig. 1 describes the complete procedure of
sliding windows graphically. In this we for a segment of time series with
length $\tau_{i}$ , the total number of short sequences will be
$\tau_{i}-m+1$. Inside a short sequence, we distinguish it as a pattern by the
natural position order. It can be easyly observed that for our case $m=3$, we
can have six different possibilities such as

  * •

$x(t_{i})<x(t_{i+1})$

  * •

$x(t_{i})<x(t_{i+2})$,

  * •

$x(t_{i+1})<x(t_{i})$

  * •

$x(t_{i+1})<x(t_{i+2})$,

  * •

$x(t_{i+2})<x(t_{i})$

  * •

$x(t_{i+2})<x(t_{i+1})$

Using $p_{i}$ $(i=1,...,6)$ to denote the probability that one of the patterns
appears in the time series of a segment, we can define the permutation entropy
$H(\tau_{j})$ as

$H(\tau_{j})=-\sum_{i=1}^{3!}p_{i}ln(p_{i})$

For making $p_{i}$ roughly reflect the local structure of time series, the
number of short sequences in $\tau$ should not be too small,
i.e.,$N_{i}-m+1>100$[4].

Because the time series is always changing with time, $H(\tau_{j})$ changes
with time period interval $\tau_{j}$ and will be determined by the local
topological structure of the time series[4]. For determining the complexity
measure in terms of permutation entropy[3] around each point of our time
series we are calculating the permutation entropy by taking the running
windows of $\tau_{i}$ along the whole time series data.

(a) Sliding windows for m in $\tau_{i}$

(b) Sliding windows for $\tau_{i}$ in time series

Figure 1: Showing the sliding windows m and $\tau_{i}$ in $\tau_{i}$ and time
series respectively

For solving the mode mixing problem observed in the calculation of empirical
mode functions(IMF) using Empirical Mode Decomposition we can follow the
following algorithm.

  1. 1.

Calculate the Permutation Entropy of the time series for a particular
length($\tau$) with short sequence length(m) around each point.

  2. 2.

Extract the high entropic portions from the time series and run EMD in each
portion.

  3. 3.

Calculate the PE values for each IMFs.

  4. 4.

Compare the PE values of IMFs and the original time series and replace the
high entropic portions with appropriate IMFs.

  5. 5.

Now run EMD in both time series i.e, the repaired original data and the
intermittent extracted component to get the desired result.

### Detailed Methodology

As shown in the Fig.2 the intermittent portion of our signal can be easily
detected with the help of Permutation Entropy Algorithm. The increase in
complexity of the signal in the intermittent part can be easily observed by
the higher entropic value of the region. For detecting the onset and offset of
the noise in our original signal the following algorithm is applied

  * •

Calculation of permutation entropy around each point of our time series.

  * •

For detecting the region of increased permutation entropy

    * –

Taking the maxima envelop of Permutation Entropy.

    * –

Calculating the statistical mode of the maxima envelop of the permutation
entropy.

    * –

Considering the portions of the Entropy curve above the mode level and
extracting the data from the original time series at that particular portion.

This algorithm seems very specific for our problem but subsequently we will
have to come up with a more general algorithm to extract the noisy part(high
frequency part) from the time series.

Fig.2,3,4,5 represents the entire methodology for extracting the intermittent
part from the original signal.

Figure 2: Permutation Entropy for time series

Figure 3: Extracted time series based on high PE value

Figure 4: Empirical Mode Decomposition for the selected time series portions

Figure 5: Extracted Intermittent component and low frequency components

### Detecting the frequency variation

For our purpose we are particularly interested in detecting variation in
frequency and want to avoid the detection of any amplitude change in our time
series. But the above technique becomes problematic in certain cases where
both amplitude as well as frequency changes are detected by the permutation
entropy filter. So instead of running the PE directly on the time series it is
better to run it on the gradient of the time series. According to the
definition of gradient :

$grad(y)=\frac{\frac{dy}{dt}}{|{\frac{dy}{dt}}|}$

Where y and t are the data points and the time points in the time series. In
this way every sequence of the time series gets a value of $+1$ and $-1$ for
increasing and decreasing portions respectively. Afterwards applying PE
algorithm on the gradient set gives a better result than the previous one.

## Conclusion

Even though various modification to EMD has been proposed like EEMD[2] to
resolve the mode mixing problem observed in IMF determination. But most of
them seem computationally expensive and inappropriate for application in real
time data analysis. Proposed algorithm describes the selective implementation
of EMD/HHT which can adaptively determine the desired portions with the help
of local entropic value and extract useful information.

## Acknowledgement

The author would like to thank Department of Engineering Cybernetics,NTNU for
the financial support through Jens Balchen’s scholarship without which this
work might not have been possible.

## References

  * [1] Norden E Huang, Zheng Shen, Steven R Long, Manli C Wu, Hsing H Shih, Quanan Zheng, Nai-Chyuan Yen, Chi Chao Tung, Henry H Liu. Proc. R. Soc. Lond. A (1998) 454, 903-995
  * [2] Z. Wu and N. E. Huang. Adv. Adapt. Data Anal. 01, 1 (2009). DOI: 10.1142/S1793536909000047
  * [3] Christoph Bandt and Bernd Pompe: Phys. Rev. Lett. 88, 174102-Published 11 April 2002,doi:http://dx.doi.org/10.1103/PhysRevLett.88.174102
  * [4] Zonghua Liu 2004 Europhys. Lett. 68 19 doi:10.1209/epl/i2004-10173-x
  * [5] Norden E. Huang, Zheng Shen, and Steven R. Long, Annual Review of Fluid Mechanics Vol. 31: 417-457 (Volume publication date January 1999) DOI: 10.1146/annurev.fluid.31.1.417
  * [6] Yinhe Cao, Wen-wen Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively Phys. Rev. E 70, 046217-Published 27 October 2004. DOI:http://dx.doi.org/10.1103/PhysRevE.70.046217
  * [7] Massimiliano Zanin , Luciano Zunino, Osvaldo A. Rosso and David Papo: Entropy 2012, 14(8), 1553-1577; doi:10.3390/e14081553
  * [8] NORDEN E. HUANG et al, Adv. Adapt. Data Anal. 01, 177 (2009). DOI: 10.1142/S1793536909000096