We define and study coherent cochain complexes in arbitrary stable ,
following Joyal.
Our main result is that the of coherent cochain complexes in
a stable $\scrC$ is equivalent to the of complete filtered objects
in $\scrC$.

We then show how the Beilinson t-structure can be interpreted in light
of such equivalence, and analyze its behavior in the presence of symmetric
monoidal structures.
We also examine the relationship between the notion of (higher) Toda brackets
and coherent cochain complexes.
Finally, we prove how every coherent cochain complex gives rise to a
spectral sequence and illustrate some examples.

§ INTRODUCTION

Recall that a filtered chain complex (resp. a filtered spectrum) consists of a
$\bbZ$-indexed sequence

$$\cdots\to F^{n} \to F^{n-1} \to F^{n-2} \to\cdots$$

where each $F^i$ is a chain complex (resp. a spectrum) and the morphisms
between them are chain maps[in the case of chain
complexes it is common to allow only monomorphisms, but as one can always
replace a chain map by a monomorphic one up to quasi-isomorphism, this
extra condition will not be relevant to us]
(resp. morphisms of spectra).

Two of the motivating reasons for the study of filtered derived categories
and filtered spectra are the construction of spectral sequences, and
the existence of the Beilinson t-structure (firstly
introduced in [4]; see Definition <ref>).
The objective of this paper is to present a new perspective on such objects
(and more generally, filtered objects in stable ) that allows to
generalize at once both constructions in a homotopy coherent fashion,
and to gain some insight on other related constructions, like the obstruction
theory for the realizability of spectra with prescribed homotopy groups and
k-invariants (see Section <ref>).
This perspective will be realized by using coherent cochain complexes
(originally introduced in their bounded flavor in <cit.>,
in relation to the $\infty$-categorical Dold-Kan correspondence).

A coherent cochain complex in a stable $\scrC$ is a homotopy coherent
version of an ordinary cochain complex, and consists of a $\bbZ$-indexed
sequence of objects $C^i\in\scrC$ and differentials

$$\cdots\xto{\partial} C^{n} \xto{\partial} C^{n+1}
\xto{\partial} C^{n+2} \xto{\partial}\cdots$$

together with nullhomotopies $\partial^2 \simeq 0$, and further
coherence data making all the nullhomotopies mutually compatible.
Concretely, a coherent cochain complex will be defined as a pointed functor
from (the nerve of) a $1$-category having as objects the integers together with
an extra base point, and being generated by morphisms $\partial\colon n\to n+1$
such that $\partial^2 = 0$ (see Definition <ref> for a rigorous
For the sake of simplicity, in this introduction we will restrict to the case
of spectra, and the corresponding of coherent cochain complexes
$\coCh(\Sp)$, although the results in the rest of the paper are discussed in the
wider generality of stable equipped with some t-structure.

At their core, both the Beilinson t-structure and the spectral sequence
associated to a filtered spectrum are in a sense “blind” to the
information stored at the limit of the relevant filtered object.
As it is customary, we say that a filtered object $F$ is complete if
$\varprojlim F \simeq 0$, and we call the cofiber of the canonical map
$\varprojlim F \to \varinjlim F$ the completion of $\varinjlim F$.
If $F$ is a filtered spectrum,
the spectral sequence it generates only abuts to the completion of its colimit.
Something similar is true for the Beilinson t-structure, although saying
precisely what this means requires more work;
the failure of it being left complete is a good starting point:
all and only the constant objects are the ones that are $\infty$-connected
with respect to the Beilinson t-structure, and constant objects are in a precise
sense orthogonal to complete ones (see Proposition
<ref>). In Theorem <ref>, we prove that
the Beilinson t-structure can be glued out of its restriction to the full
subcategory of complete objects and the trivial t-structure on the full
subcategory of constant objects.

As it turns out, the of filtered spectra is equivalent to that
of coherent cochain complexes of spectra:

(see Theorems <ref>,
<ref> and <ref>)
There exists a symmetric monoidal adjunction

\begin{tikzcd}[column sep=huge]
\Fun(\bbZ\op,\Sp) \ar[r, shift left=1.1ex, "\aush"] &
\coCh(\Sp) \ar[l, shift left=1.1ex, "\imp"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

that restricts to an equivalence on complete filtered spectra.
Moreover, this adjunction sends the Day convolution symmetric monoidal
structure on filtered spectra, to a symmetric monoidal structure given
componentwise by

$$(C \tensor D)^n \simeq \tplus_{s+t=n} C^s \tensor D^t .$$

As already noted in <cit.>, given a filtered spectrum $F$ one can
construct a sequence of morphisms on suitable shifts of the graded pieces

\begin{equation}\label{eq-intro-gr}
\gr^nF[-n]\to\gr^{n+1}F[-n-1]
\end{equation}

such that all pairwise compositions are nullhomotopic, and the homotopies are
suitably compatible (this is in fact the first step for the construction of
the spectral sequence of $F$).
The left adjoint defined in the theorem sends $F$ to the coherent cochain

$$\cdots \to \gr^{-2}F[-2] \to \gr^{-1}F[-1] \to \gr^0 F \to \gr^1 F [1] \to
\gr^2 F [2] \to \cdots$$

where the differentials are precisely the maps of (<ref>).
Although above we only represented the components and the differentials of
the object $\aush F$, it also encodes much more data: $\coCh(\Sp)$ is
defined using a pointed diagram $1$-category and
pointed functors, hence its objects keep track also of the nullhomotopies
for the two-fold compositions of their differentials, and of all the
recursive coherences between them.
The compatibilities between all the nullhomotopies
can be made explicit by means of higher Toda brackets (see Section
<ref>); in order to do so, we use an explicit $\bbE_1$ presentation
of exterior algebras over $\bbZ$ due to Achim Krause, presented in
Appendix <ref>.
The left adjoint above is really a homotopy coherent version
of the construction of homotopy objects in Beilinson's t-structure: such
t-structure corresponds to the pointwise one along the left adjoint functor
discussed above; that is, for a filtered spectrum $F$,

$$\pi_n^B F \cong \pi_n^\text{lvl}\aush F,$$

where $\pi_n^\text{lvl} C$ denotes the functor applying
$\pi_n$ to all the components of $C$.

The right adjoint can be thought of as the functor
iteratively solving all the extension problems needed to reconstruct the
filtered object from its graded pieces and the differentials. Notice that,
in order for such a reconstruction to be possible in general, one really needs
to use all the information stored in the higher morphisms.
In fact, the differentials appearing in a coherent cochain complex encode
precisely the differentials of the $E_1$ page of a spectral sequence
abutting to the total homology of the complex (that is, the object
underlying its associated filtered object). The higher pages together with
their differentials can be recovered
from the complex itself, by means of an incarnation of Deligne's
décalage construction, as explained in Section <ref>.

§.§ Outline of the paper

In Section <ref>, we put the basis for the rest of the paper, by
the main definitions and some structural results about the at hand. In
particular, we show that for a stable $\scrC$,
both $\Fun(\bbZ\op,\scrC)$ and $\coCh\scrC$ sit in suitable recollements of
stable .

Section <ref> is the technical core of the paper, in which we
construct the adjunction between filtered spectra
and cochain complexes of spectra, proving the equivalence of the latter
with the of complete filtered spectra. Then, in Section
<ref>, we
generalize the result to all stable with sequential (inverse) limits.

In Section <ref>, we analyze the interplay between the
Beilinson t-structure and the levelwise t-structure on the relevant categories,
when the base $\scrC$ comes equipped with a t-structure. We continue the
analysis of with extra structure in Section <ref>, where
we study the interplay of the equivalence of Theorem
<ref> with symmetric monoidal structures, and the
compatibility of the relevant t-structures in the case where the base $\scrC$
is also equipped with a t-structure.

In Section <ref>, we use an explicit $\bE_1$ presentation
of exterior algebras over $\bbZ$, proven by Achim Krause in
Appendix <ref>, to show that $\coCh\scrC$ is the universal of $\bbZ\op$-indexed sequences of morphisms in $\scrC$ such that any pairwise
composition is trivial, as are all the possible Toda brackets.

In Section <ref>, we show how coherent cochain complexes encode
a form of obstruction theory, and we use the formalism developed there to
recover a spectral sequence from a coherent cochain complex in
Section <ref>.

Finally, in Section <ref>, we have a look at a few examples
from the recent literature that we believe benefit from the perspective of
coherent cochain complexes.

Some of the main tools for constructing the adjunction and proving
the equivalence of Theorem <ref> is
the stable nerve-realization paradigm. We
recollect and prove many results about the $\infty$-categorical incarnation
of this topic in Appendix <ref>.

§.§ Notational conventions

Throughout, we will use the following notations and terminology:

* We will denote by

the of presheaves on $\scrC$.

* We will denote by

the of spectra-valued presheaves on $\scrC$, and sometimes refer to
it as the of stable presheaves on $\scrC$.

* We will denote by

the Yoneda embedding, and by $\Yo_{\kern-0.2emC}$ the functor represented by an
object $C\in\scrC$.

* Given any functor $\Fun(\scrC,\scrD)$, we will denote by

the (fully faithful) functor obtained by precomposition with the terminal
functor $\scrC\to\Delta^0$, and by $\const_C$ its value at $C\in\scrC$.

* When considering $\bbZ$ or $\bbN$ as categories, we will always
implicitly assume that they are given the structure of a poset category,
with their usual ordering $\leq$.
We will use the notations $\bbZ^\delta$ and
$\bbN^\delta$ to denote
their underlying discrete categories.

* Unless otherwise stated, whenever we refer to $\bbZ$, $\bbZ\op$ or
$\bbZ^\delta$ as symmetric monoidal categories, we consider them endowed
with the symmetric monoidal structure induced by addition.

* We refer to a presentable endowed with a symmetric monoidal
structure whose tensor product preserves colimits in each variable as a
presentably symmetric monoidal .

* We refer to a stable endowed with a symmetric monoidal
structure whose tensor product is exact in each variable as a
stably symmetric monoidal .

* We refer to a presentable stable endowed with a symmetric monoidal
structure whose tensor product preserves colimits in each variable as a
stable presentably symmetric monoidal .

* We will most of the times omit the Eilenberg–MacLane functor from
the notations, when considering an Abelian group as a spectrum.

Throughout, when dealing with t-structures, we will use the homological
grading convention.
When dealing with spectral sequences, we will use the cohomological
Serre grading convention.
For the sake of clarity, we decided to stick with
the choice of working exclusively with decreasing filtered objects and
coherent cochain complexes.
The results of this paper translate immediately
to the case of increasing filtrations and coherent chain complexes,
with the caveat that the equivalence of Theorem <ref>
always inverts
the direction of the $\bbZ^\delta$-indexed arrows, hence complete
increasing filtered objects are equivalent to coherent chain

Throughout, we (mostly implicitly)
work in ZFC+U, where “U” is Tarski's axiom: “For each set $x$, there
exists a Grothendieck universe $U$ such that $x\in U$”.
Equivalently we assume the existence of infinitely many strongly inaccessible
cardinals, and we fix one as the cardinality of our universe of small sets.
As we are agnostic about the size of the universe fixed, all the arguments
that do not rely on dealing with a bigger universe hold regardless of
the actual size of the objects referred to as “small sets”.
In those few cases where we need to deal with more than one universe at a
time, we are going to be explicit about relative sizes.

§.§ Related works

Some of the structural results about filtered objects in stable contained in Section <ref> have been presented for the
case of filtered spectra in <cit.>.
The of filtered objects of a stable has also been studied in
<cit.>, and some results in Section
<ref> and Section <ref> overlap with
loc. cit.[the reader should note that in
[17] the authors refer to what we call
filtered objects as “sequences”, and reserve the term “filtered
objects” for what we call complete filtered objects].
In [27], Raksit considers an alternative formulation for coherent
chain complexes; although we don't dwelve into a detailed comparison,
the results of Section <ref>, should give a clear idea about
the relation
between the formulation in op. cit. and the one in this paper.
In [31], Walde proves the equivalence between $\Fun(\bbN,\scrC)$ and
$\Ch_{*\geq 0}(\scrC)$ through their equivalence with
A similar approach to the construction of the spectral sequence of a filtered
spectrum through the décalage functor
that does not use the language of coherent cochain complexes appeared
recently in Hedenlund's Ph.D. thesis [18].
The décalage functor introduced in Section <ref> is
further analyzed and discussed in forthcoming work by

§.§ Prerequisites

We assume the reader is familiar with the theory of as developed
in [22] and with the contents of [24]. In particular, we assume
the reader is thoroughly acquainted with the theory of t-structures as
developed in <cit.> (see also<cit.>).
Some of the terminology and notations in this paper differ from the ones used
in [22] and [24], but we use the same notations and terminology
of op. cit. for the concepts we do not explicitly recall or introduce

§.§ Aknowledgements

This paper is part of my Ph.D. project at the University of Münster.
I am very grateful to my advisor, Thomas Nikolaus, for suggesting
this topic and for his support and advice.
I would like to thank Benjamin Antieau, from whom I learned about the relation
between the Beilinson t-structure and Deligne's décalage, and Fosco
Loregian, for having introduced me to co/end calculus way before I could
appreciate its value.
I am also grateful to Achim Krause, Edoardo Lanari, Jonas McCandless and
Tashi Walde for the insightful discussions I shared with them while writing this

§ DEFINITIONS AND PRELIMINARIES

In this section we introduce and discuss a few key facts about coherent
cochain complexes and filtered objects in .
In particular, in Proposition <ref> and Proposition
<ref> we prove that both constructions figure in suitable
recollements (see Definition <ref>) of stable .

Let $\catCh$ be the pointed (ordinary) category given by

$$\operatorname{ob}\catCh = \bbZ \cup \{\pt\}$$

\begin{cases}
\{\id,\zero\} \quad &\text{if } m=n; \\
\{\de_n,\zero\} \quad &\text{if } m=n-1; \\
\{\zero\} \quad &\text{otherwise }.
\end{cases}$$

where $\pt\in\catCh$ is a zero object, and $\zero$ is the zero map.

Given any pointed $\scrC$

* the of coherent chain complexes in $\scrC$ is the full

spanned by pointed functors.

* The of coherent cochain complexes is the full

\Fun(\catCh\op,\scrC)$$

spanned by pointed functors.

* An object $C\in\coCh(\scrC)$ is bounded above if
there exists an $n\in\bbZ$ such that

$$C^k \simeq 0 \text{ for all } k>n.$$

* An object $C\in\coCh(\scrC)$ is bounded below if
there exists an $n\in\bbZ$ such that

$$C^k \simeq 0 \text{ for all } k<n.$$

* An object $C\in\coCh(\scrC)$ is bounded if it is
bounded above and bounded below.

Notice that $0\in\bbZ$ is an object of $\catCh$, but it is not its zero

Let $\scrA$ be an Abelian category, then $\coCh(\scrA)$ is the usual category
of cochain complexes of $\scrA$.

In what follows, we will exclusively focus our attention on coherent cochain

Given a coherent cochain complex $C\in\coCh{\scrC}$, we will denote $C(\de_n)$
by $\de^n_{C}$, or just by $\de^n$ if there is no risk of confusion.

Let $\scrC$ be any .

* The category $\Fild\scrC\coloneqq\Fun(\bbZ\op, \scrC)$ is called
the of (decreasing) filtered objects of $\scrC$.

* The category $\Fili\scrC\coloneqq\Fun(\bbZ,\scrC)$ is called the
of (increasing) filtered objects of $\scrC$
[the position of the arrows in the notation is meant to remind
the common convention of using upper indices for decreasing filtrations,
and lower indices for increasing filtrations].

* Given $F\in\Fild\scrC$, we call $F^{-\infty}\coloneqq\colim_i F^i$
the underlying object of $F$.

* Given $F\in\Fild\scrC$, we say that $F$ is complete if
$F^{+\infty}\coloneqq\lim_i F^i\simeq 0$.
We will denote by
the full subcategory
of $\Fild\scrC$ spanned by complete objects.

Throughout, we will concentrate on the case of descending filtrations.

Let $\scrC$ be an with cofibers and countable coproducts.
Precomposition with the inclusion map $\iota_n\colon\Delta^{\{n+1, n\}} \to
\bbZ\op$
induces functors

(ι_n)^*   →(Δ^1,)

F   ↦(F^n+1→F^n).

* We will define the $n$-th graded functor $\gr^n$ as the
composite functor

* We define the associated graded functor $\gr$ as the
composite functor

\prod_{n\in\bbZ}\scrC\xto{\bigoplus}\scrC.$$

We say that a map $\alpha\colon F\to G$ in $\Fild\scrC$ is a graded
equivalence if $\alpha\colon F\to G$ is such that $\gr(\alpha)$ is an
equivalence. Equivalently. if for all $n\in\bbZ$, the dotted map

\begin{tikzcd}
F^{n+1} \ar[d] \ar[r] & F^n \ar[d] \ar[r] & \gr^n F \ar[d, dashed]\\
G^{n+1} \ar[r] & G^n \ar[r] & \gr^n G
\end{tikzcd}

induced by universality of cofibers is an equivalence.

Let $\scrC$ be an . For $n,m\in\bbZ\cup\{+\infty,-\infty\}$, with
$n\leq m$, we use the notation

$$F^n/F^m\coloneqq\cofib\left(F^m\to F^n\right)$$

to denote the cofiber of the evident map.

One of the nice features of ordinary cochain complexes is the possibility to
write them as limits of bounded above (or below) ones. This feature is still
present in the coherent setting, as we now show.

Given any integer $n$, let $\catCh_{(-\infty,n]}$
denote the full subcategory of $\catCh$ spanned by
Let now $\scrC$ denote a pointed complete[or, more generally, such
that all the relevant Kan extensions exist and are pointwise]
. Then, each inclusion $\iota^n\colon\catCh_{(-\infty,n]}\to\catCh$
induces, by right Kan extension along it, an adjunction

\begin{tikzcd}[column sep=huge]
\coCh(\scrC)
\ar[r, shift left=1.1ex, "(\iota^n)^*"] &
\Fun^0\left(\catCh\op_{(-\infty,n]},\scrC\right).
\ar[l, shift left=1.1ex,"\iota^n_*"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

For any given $C\in\coCh(\scrC)$, we can compute explicitly the values of
As $\iota^n$ is fully faithful, the only values we need to determine are
the ones for $M>n$:

where the last limit is $0$ as the indexing category is pointed, and $C$
preserves the zero object.
If we now denote by $(-)^{\leq n} \coloneqq \iota^n_*(\iota^n)^*$, we have
that the unit of the adjunction $(\iota^n)^*\dashv\iota^n_*$ induces an
endofunctor of $\coCh(\scrC)$, denoted $(-)^{\leq n}$,
sending a complex $C$ to the complex $C^{\leq n}$, which in degree $m$
is given by

$$ \begin{cases}
C^m &\text{ if } m\leq n\\
0   &\text{ else.}
\end{cases} $$

Similarly, the inclusions $\iota^{n,n+1}\colon\catCh_{(-\infty,n]}\to
\catCh_{(-\infty,n+1]}$ induce adjunctions

\begin{tikzcd}[column sep=huge]
\Fun^0\left(\catCh\op_{(-\infty,n+1]},\scrC\right)
\ar[r, shift left=1.1ex, "(\iota^{n,n+1})^*"] &
\Fun^0\left(\catCh\op_{(-\infty,n]},\scrC\right).
\ar[l, shift left=1.1ex,"\iota^{n,n+1}_*"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

Notice that we have a natural equivalence

(\iota^{n+1}\circ\iota^{n,n+1})^* \simeq
(\iota^{n,n+1})^*(\iota^{n+1})^* \simeq (\iota^n)^*;

by passing to adjoints, we get a natural transformation

\begin{equation}\label{eq-compose}
(\iota^{n+1})^* \Rightarrow \iota^{n,n+1}_*(\iota^n)^*;
\end{equation}

if we now precompose (<ref>)
with $\iota^{n+1}_*$ (using that adjunctions compose), we get a natural

$$\upsilon^{n+1}\colon(-)^{\leq n+1}\Rightarrow(-)^{\leq n}.$$

By inspection, the above is given pointwise by

\begin{cases}
\id\colon C^m\to C^m &\text{ if } m\leq n \\
0\colon C^{n+1} \to 0 &\text{ if } m = n+1 \\
0\colon 0 \to 0 &\text{ if } m \geq n+2.
\end{cases}

Similarly to what we did in Construction <ref>, one can
truncate below a certan integer. It is also possible to consider
left Kan extensions along the $\iota^n$'s, and the induced counits
to obtain different truncations with $\cofib \partial^n$ in degree $n+1$,
for truncations above $n$, or $\fib \partial^n$ in degree $n-1$,
for truncations below $n$. In what follows, we won't need any of such

Let $\scrC$ be a pointed complete . Then, for any $C\in\coCh\scrC$

C\simeq\lim \left(\cdots \to C^{\leq n+1}\xto{\upsilon^{n+1}_C}
C^{\leq n} \to \cdots\right)

(where the $\upsilon^n$'s are the natural transformations defined in
Construction <ref>).

It follows from Proposition <ref> that limits in
$\coCh\scrC$ can be computed objectwise. As for any $m\in\bbZ$ the

\cdots \to \left(C^{\leq n+1}\right)^m\xto{\upsilon^n_C}
\left(C^{\leq n}\right)^m \to \cdots

is eventually constant on the left, the result follows.

Given any pointed $\scrC$,
the functor $\undch\colon\coCh(\scrC) \to \prod_{\bbZ} \scrC$ induced by
precomposition with $\bbZ^\delta\to\catCh\op$ is conservative.

As equivalences in are detected at the level of homotopy categories,
this is clear.

Let $\scrC$ be a complete and cocomplete semiadditive , and let
denote precomposition with $n\colon\Delta^0\to\catCh\op$.
The “evaluation at $n$” functor

\xto{n^*}\scrC,$$

admits both a left and a right adjoint.
Such adjoints are given objectwise by

\begin{cases}
X \quad &\text{if } m=n-1, n\\
0 &\text{else.}
\end{cases}

with the identity as the only nontrivial differential, and

\begin{cases}
X \quad &\text{if } m=n, n+1\\
0 &\text{else.}
\end{cases}

with the identity as the only nontrivial differential.

The functor $n^*$ admits both adjoints $n_!$ and $n_*$, given respectively
by left and right Kan extension.
As adjoint functors compose,
it follows from Proposition <ref> that the right
Kan extension $(\ev_n)_*$ is given by the composite $\ptize\circ n_*$,
and the left Kan extension $(\ev_n)_!$ is given by $\ptize\circ n_!$.
By Remark <ref>, $n_*$ is given by

n_*X^m= [\Map_{\catCh\op}(m,n),X] =
\begin{cases}
X \oplus X \quad &\text{if } m=n-1, n\\
X &\text{else.}
\end{cases}

with the differentials $n_*X^m\to n_*X^{m+1}$ determined by

$$\Map_{\catCh\op}(m+1,n) \xto{(\partial_{m}\op)^*} \Map_{\catCh\op}(m,n)$$

and thus given by

\begin{equation}\label{eq-complex-ext}
\begin{tikzcd}[ampersand replacement=\&]
\cdots \ar[r] \&
X \ar[r,"{\begin{pmatrix} \id \\ \id \end{pmatrix}}"] \&
X\oplus X \ar[r,"{\begin{pmatrix} \id & 0 \\ 0 & \id \end{pmatrix}}"] \&
X\oplus X \ar[r,"{\begin{pmatrix} \id & 0 \end{pmatrix}}"] \&
X \ar[r] \&
\cdots
\end{tikzcd}
\end{equation}

in degrees $n-2$ to $n+1$, and by identities elsewhere.

By Lemma <ref>, $n_!$ is given by

n_*X^m= \Map_{\catCh\op}(n,m) \tensor X =
\begin{cases}
X \oplus X \quad &\text{if } m=n,n+1\\
X &\text{else.}
\end{cases}

with the differentials $n_*X^m\to n_*X^{m+1}$ determined by

$$\Map_{\catCh\op}(n,m) \xto{(\partial_{m}\op)_*} \Map_{\catCh\op}(n,m+1)$$

and thus given by

\begin{equation}
\begin{tikzcd}[ampersand replacement=\&]
\cdots \ar[r] \&
X \ar[r,"{\begin{pmatrix} \id \\ 0 \end{pmatrix}}"] \&
X\oplus X \ar[r,"{\begin{pmatrix} \id & 0 \\ 0 & \id \end{pmatrix}}"] \&
X\oplus X \ar[r,"{\begin{pmatrix} \id & \id \end{pmatrix}}"] \&
X \ar[r] \&
\cdots
\end{tikzcd}
\end{equation}

in degrees $n-1$ to $n+2$, and by identities elsewhere.

By the explicit description of $\ptize$ given in Proposition
<ref>, we get the formulas for $(\ev_n)_*$ and

We now recall some definitions and facts about recollements,
which we will use extensively in the following sections; we consider only
recollements in the case of stable ,
but the theory holds in greater generality; see also
[12], [5] and <cit.>.

Let $\scrC$ be a stable , and let $i\colon\scrC_0\hookrightarrow\scrC$ and
$j\colon\scrC_1\hookrightarrow\scrC$ be full subcategories.
We say that
$\scrC$ is a recollement of the essential image of $i$ and
the essential image of $j$ if:

* Both $i$ and $j$ admit left adjoints:

\begin{tikzcd}
\scrC_0 \ar[r, hook, "i"'] &
\scrC \ar[l, shift right=0.6ex, bend right, "i_L"']
\ar[r, "j_L"]
\ar[from=r, hook', shift left=0.6ex, bend left, "j"]
\ar[l,phantom, shift right=1.2ex,
"\text{\rotatebox{-90}{$\dashv$}}"] &
\scrC_1
\ar[l,phantom, shift left=1ex, "\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

* The functor $j_L$, left adjoint to $j$, carries every object of
$\scrC_0$ to zero;

* If $\alpha$ is a morphism of $\scrC$ such that $i_L(\alpha)$ and
$j_L(\alpha)$ are equivalences, then $\alpha$ is an equivalence.

It follows from <cit.> that if $\scrC$ is a recollement of
$\scrC_0$ and $\scrC_1$, then we actually have the following adjunctions

\begin{tikzcd}[column sep=huge]
\scrC_0 \ar[r, hook, "i"' description] &
\scrC \ar[l, shift right=0.6ex, bend right, "i_L"']
\ar[from=r, hook, shift right=0.6ex, bend right, "(j_L)_!"']
\ar[r, "j_L" description]
\ar[from=r, hook', shift left=0.6ex, bend left, "j"]
\ar[l, shift left=0.6ex, bend left, "i_R"]
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
"\text{\scalebox{1}{\rotatebox{-90}{$\dashv$}}}"] &
\scrC_1
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
\end{tikzcd}

where $(j_L)_!$ is fully faithful, and $i_R$ is such that

$$ii_R \to \id_\scrC \to jj_L$$

is a co/fiber sequence.

Let $\scrC$ be a stable , and let $i\colon\scrC_0\to\scrC$ be a fully
faithful functor. The following are equivalent:

* The functor $i$ admits a left adjoint and a right adjoint;

* There exists a full subcategory $j\colon\scrC_1\hookrightarrow\scrC$,
closed under equivalences, such that $\scrC$ is the recollement of the
essential images of $i$ and $j$.

Moreover, if the conditions above hold, we can identify $\scrC_1$ with
the full subcategory $\scrC_0^\perp\subseteq\scrC$ spanned by those
objects $X\in\scrC$ such that for all $C\in \scrC_0$,

Recollements are strictly related to semiorthogonal decompositions.

Let $\scrC$ be a stable . A semiorthogonal decomposition of
$\scrC$ is the datum of two full subcategories $\scrC_0$ and $\scrC_1$ of
$\scrC$ such that

* $\scrC_1 \simeq \scrC_0^\perp$;

* Every object $C\in\scrC$ sits in a cofiber sequence

$$C_0\to C\to C_1$$

where $C_0\in\scrC_0$ and $C_1\in\scrC_1$.

It follows from Remark <ref> that every recollement
determines a semiorthogonal decomposition.

We learned the following argument from <cit.>, and
we present it here almost verbatim for the reader's convenience.

The commutative square

\begin{tikzcd}
\id_\scrC \ar[r, "\eta^i"] \ar[d, "\eta^j"'] & i i_L \ar[d, "i i_L \eta^j"] \\
j j_L \ar[r, "\eta^j j j_L"] & i i_L j j_L
\end{tikzcd}

is Cartesian in $\Fun(\scrC,\scrC)$.

It follows from Remark <ref> that the fiber of the
vertical maps is given by $\eta^i i i_R \colon i i_R \to i i_L i i_R$,
which, as $i i_L \simeq \id$, is an equivalence.

In particular, every recollement determines a “fracture square”

\begin{tikzcd}
C \ar[r] \ar[d] & i_L C \ar[d] \\
j_L C \ar[r] & i_L j_L C
\end{tikzcd}

for any object $C\in\scrC$.

We are now ready to prove the main results of this section.

Let $\scrC$ be a stable that is both complete and cocomplete.
The $\Fun(\catCh\op,\scrC)$ is the recollement of the essential images
of $\const\colon\scrC\to\Fun(\catCh\op,\scrC)$ and $\coCh(\scrC)$:

\begin{tikzcd}[column sep=huge]
\scrC \ar[r, hook, "\const"' description, pos=.7] &
\Fun(\catCh\op,\scrC)
\ar[l, shift right=0.6ex, bend right, "\colim" description]
\ar[from=r, hook, shift right=0.6ex, bend right, "i" description, pos=.4]
\ar[r, "\ptize" description]
\ar[from=r, hook', shift left=0.6ex, bend left, "i" description, pos=.4]
\ar[l, shift left=0.6ex, bend left, "\lim" description]
\ar[l,phantom, shift left=2ex,
"\text{\scalebox{1}{\rotatebox{-90}{$\dashv$}}}", pos=.3]
\ar[l,phantom, shift right=2ex,
"\text{\scalebox{1}{\rotatebox{-90}{$\dashv$}}}", pos=.3] &
\coCh(\scrC)
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
\end{tikzcd}

Moreover, the inclusion $i\colon\coCh(\scrC)\to\Fun(\catCh\op,\scrC)$ is both
left and right adjoint to a functor $\ptize$, given by

$$\ptize F(n)\simeq F(n)/F(\pt)$$

on objects.
Finally, the $\coCh(\scrC)$ is stable, complete and cocomplete;
if $\scrC$ is presentable, $\coCh(\scrC)$ is presentable as well.

The functor $\const$ admits a left and a right adjoint, given respectively
by left and right Kan extension along the terminal morphism (i.e. the
$\colim$ and $\lim$ functors). Hence, by Proposition
<ref>, $\Fun(\catCh\op,\scrC)$
is the recollement of (the essential image of) $\const$ and the full
subcategory $\const(\scrC)^\perp$ orthogonal to it.
We claim that

\begin{equation}\label{eq-orth-pstch}
\const(\scrC)^\perp\simeq\coCh(\scrC).
\end{equation}

To see this, recall that $\const(\scrC)^\perp$ is spanned by those objects
$F\in\Fun(\catCh\op,\scrC)$ such that for all $X\in\scrC$ the mapping space
$\Map_{\Fun(\catCh\op,\scrC)}(\const_X,F)$ is contractible. But, as

we see that (<ref>) holds.
We denote by $\ptize$ the left adjoint to the fully faithful inclusion

Now, $\coCh(\scrC)$
is stable by <cit.> (or just because of the fact that (co)limits
in functor categories are computed pointwise)
and, if $\scrC$ is presentable (as $\Fun(\catCh\op,\scrC)$ is presentable
by <cit.>) $\coCh(\scrC)$ is presentable by
<cit.>. By <cit.>, we see that

\begin{equation}\label{eq-pt-cof-cns}
\ptize F\simeq \cofib\left(\const_{F(\pt)}\to F\right)
\end{equation}

and in particular $\ptize F(n)\simeq F(n)/F(\pt)$.
The existence of a left adjoint for $\ptize$
follows from Remark <ref>.
To give a description, let's first note that, by (<ref>)
and the fully faithfulness of the inclusion $i$
we can write

and, as corepresentable functors commute with limits, the latter is
equivalent to

\to \Map_{\Fun(\catCh\op,\scrC)}\left(C[-1],X\right)\right)$$

which in turn, as $\const$ is right adjoint to $\colim$, can be computed as

\begin{equation}\label{eq-in-lemma-ptize}
\fib\left(\Map_{\scrC}\big(\colim C[-1],F(\pt)\big)
\to \Map_{\Fun(\catCh\op,\scrC)}\left(C[-1],X\right)\right).
\end{equation}

But, as the colimit of a coherent cochain complex is always $0$, we have

$$\Map_{\scrC}\big(\colim C[-1],F(\pt)\big)\simeq\pt$$

and thus (<ref>) is equivalent to

≃     Ω_(,)(C[-1],X)

≃     _(,)(C,X).

In particular, as $\ptize$ is both left and right adjoint to the inclusion,
we have that $\coCh(\scrC)$ is closed under all limits and colimits
in $\Fun(\catCh\op,\scrC)$, which is by hypothesis complete and cocomplete.

Let $\scrC$ be a stable that is both complete and cocomplete.
The $\Fild\scrC$ is the recollement of the essential images
of $\const\colon\scrC\to\Fild\scrC$ and $i\colon\cFild\scrC$:

\begin{tikzcd}[column sep=huge]
\scrC \ar[r, hook, "\const"' description] &
\Fild\scrC \ar[l, shift right=0.6ex, bend right, "\colim"' description, pos=.45]
\ar[from=r, hook, shift right=0.6ex, bend right, "L_!" description]
\ar[r, "L" description]
\ar[from=r, hook', shift left=0.6ex, bend left, "i" description]
\ar[l, shift left=0.6ex, bend left, "\lim" description, pos=.55]
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
"\text{\scalebox{1}{\rotatebox{-90}{$\dashv$}}}"] &
\cFild\scrC.
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
\end{tikzcd}

The left adjoint to the inclusion $i$
is given by Bousfield localization at the class of graded equivalences,
and computed as

$$LF^n\simeq F^n/F^{+\infty}.$$

Moreover, the $\cFild\scrC$ is stable, complete and cocomplete; if
$\scrC$ is presentable, $\cFild\scrC$ is presentable as well.

The proof is completely analogous to the proof of
Proposition <ref>. The only new element is the
identification of the local maps for the Bousfield localization determined
by $L$. In order to prove it,
let us notice that $L\alpha$ is an equivalence if and only
if $\cofib L\alpha \simeq L (\cofib \alpha) \simeq 0$, which in turn,
by <cit.>, is the case if and only if $\cofib \alpha$ is
essentially constant;
by inspection of the following diagram

\begin{tikzcd}
F^{n+1} \ar[r] \ar[d] & F^n \ar[r] \ar[d] & \gr^n F \ar[d] \\
G^{n+1} \ar[r] \ar[d] & G^n \ar[r] \ar[d] & \gr^n G \\
(\cofib \alpha)^{n+1} \ar[r] & (\cofib \alpha)^n &
\end{tikzcd}

where all the rows and columns are co/fiber sequences,
we see that this is the case if and only if $\alpha$ is a graded equivalence.

§ FILTERED SPECTRA AND COCHAIN COMPLEXES OF SPECTRA

Our goal in this section is to construct an equivalence between the $\rlywdhat{\Fild} \Sp$ and $\coCh(\Sp)$. In order to do so, we will first
construct a pair of adjoint functors between $\Fild\Sp$ and $\coCh(\Sp)$ using
the machinery of Appendix <ref>, and then prove that the right
adjoint is a fully faithful functor having $\cFild\Sp$ as its essential image.
Along the way, we compute explicitly the values of the pair of adjoints
constructed abstractly.

In order to construct the left adjoint, let us start with the following

Restriction along $\susinftyp\Yo\colon\bbZ\to\Pst\bbZ$ induces an equivalence

whose inverse is given by
associating to every functor $F\colon\bbZ\to\coCh(\Sp)$ its stable
realization functor (see Definition <ref>).

As, by Proposition <ref>, $\coCh(\Sp)$ is a stable
, this is just an application of Lemma <ref>, toghether
with the observation that $\Fild\Sp \simeq \Pst{\bbZ}$.

Let $\preaush\colon\bbZ\to\coCh(\Sp)$ be the functor defined on objects as
$\preaush(n)=\spch{n}{n}$, where

nn   →

m   ↦
[n]      if  m=n,

0      if  mn.

We define $\preaush$ on morphisms as follows.
First of all, let us note that, as

it is equivalent to determine a map in $\coCh(\Sp)$ or a map from $\Delta^1
\times\catCh\op$ landing in spectra, such that both restrictions to
$\{0\}\times\catCh\op$ and $\{1\}\times\catCh\op$ preserve zero objects
(recall that $\Fun^0(\catCh\op,\Sp)$ is a full subcategory of $\Fun(\catCh
\op,\Sp)$).
Let $\iota_{m}\colon\catCh\op_{[m,m+1]}\to\catCh\op$ be the inclusion functor
Then, we define $\preaush(\iota_{m})$ as the morphism
corresponding to the following left Kan extension

[d, "𝕀×ι_m"'] [r, "α"]

Δ^1×[ur, dashed, "β"']

where $\alpha$ is obtained from the defining square for the suspension

[m] [r] [d]     0 [d]

0 [r]     [m+1]

and extending this map $\Delta^1\times\Delta^1 \to \Sp$ to
$\Delta^1\times\catCh\op$ in the only possible way that sends $(0,\pt)$
and $(1,\pt)$ to the zero spectrum.
As $\id\times d_m$ is fully faithful, the values of $\beta(0,\pt)$ and
$\beta(1,\pt)$ are determined by $\alpha$, and are zero by construction.
By Corollary <ref>, this completes the construction of

Let[where $\aush$ stands for “Aushülen”, German for hull
shelling] $\aush\coloneqq\lvert - \rvert_{\preaush}^{\rm{st}}
\colon\Fild(\Sp)\to\coCh(\Sp)$ denote the stable
functor associated, by the equivalence of Lemma <ref>,
to the functor $\preaush$ given in
Construction <ref>.
Let us denote by[where $\imp$ stands for “Impilare”, Italian
for piling up] $\imp\coloneqq\nerve_{\preaush}^{\rm{st}}$ its right

\begin{tikzcd}[column sep=huge]
\Fild\Sp \ar[r, shift left=1.1ex, "\aush"] &
\coCh(\Sp) \ar[l, shift left=1.1ex,"\imp"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

(the reason for the hat in the notation will be clearified in Proposition
We will refer to $\aush$ as the shelling functor functor
and to the $\imp$ as the piling-up functor; we will use the terms
associated shelled complex and piled-up filtered object for
objects of the form $\aush F$ and $\imp C$, respectively.

It follows from Proposition <ref> that, for any coherent
cochain complex $C$, the filtered object $\imp C$ is given by

$$\imp C^{\istar} \simeq\map_{\coCh(\Sp)}(\spch{\istar}{\istar},C).$$

Our next goal is to show that $\aush$ factors through the localization
In order to do so, it will be useful to identify its graded pieces

There is a natural equivalence

of functors $\Fild(\Sp)\to\Sp$.

By virtue of Lemma <ref>, it suffices to prove that both
functors preserve colimits and agree on elements of the form
$\susinftyp \Yo_{-}$.

As both $\aush$ and $\ev_n$ admit right adjoints (see Proposition
<ref>), $\aush(-)^n\simeq\ev_n\circ \ \aush$ preserves
By Definition <ref>, $\gr^n$ is given by $\cofib\circ(\iota_n)^*$; by
<cit.>, $\cofib$ admits a right adjoint, and since $\Sp$ is
complete, $(\iota_n)^*$ admits a right adjoint, given by right Kan extension
along $\iota_n$.

It follows from Remark <ref> that for all $m\in\bbZ$

$$\aush(\susinftyp \Yo_m)\simeq\lvert \Yo_m \rvert_{\preaush} \simeq
\preaush(m)
\simeq\spch m m$$

(see Appendix <ref> for a detailed discussion)
and thus that

(_m)^n ≃
[n]      if  m=n,

0      if  mn,

whereas a direct check shows that

\gr^n\left(\susinftyp \Yo_m\right) \simeq
\begin{cases}
\bbS \quad &\text{if } m=n, \\
0 \quad &\text{if } m\ne n,
\end{cases}

concluding the proof.

The adjunction $\aush\dashv\imp$ given in Definition <ref>
factors as

\begin{tikzcd}[column sep=huge]
\Fild\Sp \ar[r, shift left=1.1ex, "L"] &
\cFild\Sp \ar[r, shift left=1.1ex, "\caush"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\ar[l, shift left=1.1ex] &
\coCh(\Sp). \ar[l, shift left=1.1ex,"\imp"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

through the localization of Proposition <ref>.

Let us first show that $\aush$ factors through the localization.
By virtue of Proposition <ref>,
it is enough to show that it sends maps inducing equivalences on associated
gradeds to equivalences. But, as per
Lemma <ref>,
the functor

is conservative.
It is thus enough to know that on each component, $\aush(-)^n$
sends local maps to equivalences, which is true by virtue of Lemma
As $\aush$ preserves colimits, and $\cFild\Sp$ is a full
subcategory of $\Fild\Sp$ closed under colimits (see Proposition
<ref>), we get that $\caush$ preserves colimits, and
hence admits a right adjoint.
Since adjoints compose, this just means that $\imp$ takes values
in the full subcategory of complete objects.

Our next goal is to prove that the induced adjunction $\caush\dashv\imp$
is an equivalence of . To this end, we need to prove a few key lemmata
before, and to get a better understanding of the functor $\imp$.

The functor $\caush$ is conservative.

Let $\alpha\colon F\to G$ be a map in $\cFild\Sp$ such that
$\caush(\alpha)\colon\caush F \to \caush G$
is an equivalence. As $\coCh(\Sp)$ is a full subcategory of a functor
category, equivalences are given pointwise, hence we have that, for all

$$\caush F^n\simeq \caush G^n.$$

But, by Lemma <ref>
this is just means that $\alpha$ induces
an equivalence on associated gradeds, hence, by
Proposition <ref>,
it is an equivalence in $\cFild\Sp$.

There is a natural equivalence
(with notation as in Proposition <ref>) of functors

It follows from the pointwise description of Proposition <ref>
that $(\ev_n)_*$ preserves all colimits.
As $\Sp\simeq\Pst{\Delta^0}\simeq\St(\Pre{\Delta^0})$,
Lemma <ref> implies that it is enough to check that they take
the same value on $\susinftyp\Yo_{\pt}\simeq\susinftyp\pt\simeq\bbS$. But
this follows immediately again from description given in Proposition

We have equivalences

$$\ptize \circ \susinftyp \Yo_n\simeq(\ev_n)_!\bbS\simeq(\ev_{n+1})_*\bbS$$

of objects in $\coCh(\Sp)$

The second equivalence is an instance of Lemma
<ref>. Regarding the first one, we will show that
they represent the same functor. In fact:

and, using Proposition <ref>:

concluding the proof.

Motivated by Lemma <ref>, we introduce the notation

to emphasize the connection with the pointed stabilization of the Yoneda

We have the following “density” result for elements of the form
$\evalfun_n\bbS$ in $\coCh(\Sp)$.

Given any $C\in\coCh(\Sp)$, there is a natural equivalence

\tensor\evalfun_n\bbS$$

where $\tensor$ denotes the canonical tensoring of stable over spectra
(see <cit.>).

Using Proposition <ref>

_n, D)

(_()(_n,C) _n, D)

Now, using the end formula for the space of natural transformations
(see <cit.>, <cit.>), we get that

hence both objects corepresent the same functor.

There is a co/fiber sequence in $\coCh(\Sp)$ given by

$$\spch{n-1}{n-1} \to \spch n n \to \evalfun_{n-1} \bbS[n]$$

where the first map is the structure map defined in Construction

By Proposition <ref>, the inclusion of $\coCh(\Sp)$ into
$\Pst\catCh$ preserves colimits, therefore we can understand the colimit
in the latter , where it can be computed pointwise (see
The result then follows by inspection of the following diagram

\begin{tikzcd}
\spch{n-1}{n-1} \ar[d] &[-25pt] = &[-25pt] \cdots \ar[r] & 0 \ar[r] \ar[d]
&\bbS[n-1] \ar[r] \ar[d] &  0 \ar[r] \ar[d] &  0 \ar[r] \ar[d] & \cdots \\
\spch n n \ar[d] &[-25pt] = &[-25pt] \cdots \ar[r] & 0 \ar[r] \ar[d] &
0 \ar[r] \ar[d] &  \bbS[n] \ar[r] \ar[d, equal] &  0 \ar[r] \ar[d]
& \cdots \\
\evalfun_{n-1} \bbS[n] &[-25pt] = &[-25pt] \cdots \ar[r] & 0 \ar[r] &
\bbS[n] \ar[r, equal] & \bbS[n] \ar[r] &  0 \ar[r] & \cdots .\\
\end{tikzcd}

The previous result lets us get an explicit description of the graded pieces
of $\imp$.

Let $C\in\coCh(\Sp)$; then

$$\gr^n(\imp C) \simeq C^n [-n].$$

By Lemma <ref>, we have a co/fiber sequence

$$\spch{n}{n} \to \spch{n+1}{n+1} \to \evalfun_{n} \bbS[n+1].$$

By applying $\map_{\coCh(\Sp)}(-,C)$ to it, we get

$$\map_{\coCh(\Sp)}(\evalfun_{n}\bbS[n+1],C)\to\imp C(n+1)\to\imp C(n),$$

(see Remark <ref>)
and, by Lemma <ref>,

(where the last equivalence follows from Proposition <ref>).

Let $C\in\coCh(\Sp)$. Then

* If $C^m \simeq 0$, then $\imp C^m \simeq \imp C^{m+1}$.

* If there exists an $N$ such that $C^m \simeq 0$ for all $m < N$,
then $\imp C^m \simeq \imp C^N$ for all $m \leq N$.

* If there exists an $N$ such that $C^m \simeq 0$ for all $m\geq N$,
then $\imp C^m \simeq \imp C^N$ for all $m \geq N$.

* If there exists an $N$ such that $C^m \simeq 0$ for all $m\geq N$,
then $\imp C^m \simeq 0$ for all $m \geq N$ and
$\imp C^{N-1} \simeq C^{N-1}[-N+1]$.

* For all $n\in\bbZ$, we have
that $\imp \spch n n \simeq \susinftyp\Yo_n$.

The first three points are straightforward consequences of Proposition
<ref>, whereas (<ref>) follows immediately
from (<ref>), together with the completeness of $\imp C$
(see Proposition <ref>).
Point (<ref>) follows from the definition of
$\spch n n$ together with (<ref>-<ref>).

The composite $\aush \imp$ is equivalent to $\id_{\coCh(\Sp)}$.

It follows from Remark <ref> that $\imp$ commutes with
arbitrary coproducts, and is thus cocontinuous. Hence, $\aush \imp$ is
cocontinuous as well.
By Lemma <ref>, we have that every coherent cochain
complex is canonically the colimit of elements in the essential image of
In particular, it is enough to check that the two functors agree on elements
of the form $\evalfun_{n}\bbS$.
Using Lemma <ref> and
Corollary <ref>.<ref>, we have

_n   ≃(n n →n+1n+1)

≃(_n →_n+1)

≃(n n →n+1n+1)

concluding the proof.

We can now prove the following theorem.

The adjunction

\begin{tikzcd}[column sep=huge]
\cFild\Sp \ar[r, shift left=1.1ex, "\caush"] &
\coCh(\Sp). \ar[l, shift left=1.1ex,"\imp"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

is an equivalence.

Putting together Proposition <ref>, and Proposition
<ref>, we get that $\caush\imp\simeq\id_{\coCh(\Sp)}$.
This, together with (the dual of) Proposition <ref>
implies that the counit
$\eps\colon\caush\imp\Longrightarrow \id_{\coCh{\Sp}}$ of the adjunction
$\caush\dashv\imp$ is an equivalence.
If we now consider the triangular identity

\begin{tikzcd}
\caush \ar[r, Rightarrow, "\caush \eta"]
\ar[dr, Rightarrow, "\id_{\caush}"'] &
\caush\imp\caush
\ar[d, Rightarrow, "\eps_{\caush}"] \\
& \caush
\end{tikzcd}

we see that, as both $\id_\caush$ and $\eps_\caush$
are equivalences, $\caush\eta$ is an equivalence as well;
but, as by Lemma <ref> $\caush$ is
conservative, then $\eta$ has to be an
equivalence, concluding the proof.

We conclude the section giving more information about the functor $\imp$;
we can leverage from Lemma <ref> our
understanding of it, by getting
a recursive description of its components, which in turn gives a complete
description of its values in the case of bounded above cochain complexes.

In the rest of the section, we will make free use of some results about
cubic diagrams in stable , as presented in [10], and refer to
op. cit. for all the related concepts, notations and terminology we
use and do not introduce here. The following fact is somewhat implicit in
[10], but as we will use it crucially, we report it
here for convenience.

Let $\scrC$ be a stable , let $a\in\bbN$ and let
$C:(\Delta^a)\op\to\scrC$ be a functor.
Moreover, if $(\Delta^1)^a$ denotes the $a$-fold product of $\Delta^1$ with
itself, and we denote by
$\overrightharp{v} =
(\overrightharp{v}_{a}, \overrightharp{v}_{a-1}, \ldots, \overrightharp{v}_1)$
the objects of $(\Delta^1)^a$ (where each $\overrightharp{v}_i$ can either
be $0$ or $1$),
let $F$ be the $a$-cube having for vertices:

F (\overrightharp{v}) =
\begin{cases}
C(a) \quad &\text{if } \overrightharp{v} = (0,\ldots,0)
\\[10pt]
\quad &\text{if } 0 < b \leq a
\text{, } \overrightharp{v}_i = 0 \text{ for } i > b
\text{ and } \overrightharp{v}_i = 1 \text{ for } i \leq b
\\[10pt]
0 &\text{else}
\end{cases}

where all the nonzero maps are determined by $C$
(that is, $F$ is an $a$-cube having the $C(i)$'s on its
“spine” and zero objects elsewhere).

\totcof F\simeq\cofib^a (C) \simeq
\underbrace{
\cofib \kern-.2em\Big(\kern-.4em\cdots
\kern-.2em\cofib \kern-.2em\Big(
\kern-.2em\cofib
}_{a \text{ times}}
\kern-.2em\Big(C(a) \to C(a-1) \Big)
\to C(1) \Big) \cdots \to C(0) \Big).

By <cit.>, one can extend $F$ to a coCartesian $(a+1)$-cube
$\widetilde F$
such that $\widetilde{F}|_{\{0\}\times\Delta^a} \simeq F$,
and with $\widetilde{F}|_{\{1\}\times\Delta^a}$
having $\totcof F$
as its terminal vertex, and $0$ elsewhere
(see op. cit. for a precise statement).
As $\widetilde F$ is coCartesian, by iteratively applying <cit.>,
$\cofib^a(\widetilde{F})$ is a coCartesian $1$-cube, i.e. an equivalence.
We conclude by observing that its source is given by $\cofib^a(F)$, and its
target is just $\totcof F$.

By virtue of Lemma <ref> and
Lemma <ref>, we see that

≃(n n , C)

≃((_n-1[n]→nn-1), C)

hence, by iteratively applying the above, we get that, for all $a\in\bbN$,
$\imp C^n$ is naturally equivalent to

-.2em^a  times
-.2em(   C^n-a →C^n-a[-n+a] )

→C^n-a+1[-n+a-1] ) ⋯→C^n-1[-n+1] ).

By stability, the above is equivalent to

-.2em^a  times
-.2em(   C^n-a →C^n-a[-n+a] )[-1]

→C^n-a+1[-n+a-1] )[-1] ⋯→C^n-1[-n+1] )[-1]

which, in turn, is equivalent to

-.2em^a  times
-.2em(   C^n-a[-a] →C^n-a[-n] )

→C^n-a+1[-n] ) ⋯→C^n-1[-n] ).

By <ref>,
the latter is equivalent to
the total cofiber of a suitable cube $F_a$, that is

$$\imp C^n \simeq \totcof F_a$$

and hence, by <cit.>,

$$\imp C^n \simeq \left(\totfib F_a\right)[a].$$

We can describe $F_a$ explicitly as follows;
if $(\Delta^1)^a$ denotes the $a$-fold product of $\Delta^1$ with itself,
and we denote by
$\overrightharp{v} =
(\overrightharp{v}_{a-1}, \overrightharp{v}_{a-2}, \ldots, \overrightharp{v}_0)$
the objects of $(\Delta^1)^a$ (where each $\overrightharp{v}_i$ can either
be $0$ or $1$)
$F_a$ is the $a$-cube having for vertices:

F_a (\overrightharp{v}) =
\begin{cases}
\imp C^{n-a}[-a] \quad &\text{if } \overrightharp{v} = (0,\ldots,0)
\\[10pt]
\quad &\text{if } 0 \leq b < a
\text{, } \overrightharp{v}_i = 0 \text{ for } i > b
\text{ and } \overrightharp{v}_i = 1 \text{ for } i \leq b
\\[10pt]
0 &\text{else.}
\end{cases}

As an example, in the case $a=3$, the formula above looks as follows

\imp C^n \simeq
\totcof \left(
\begin{tikzcd}
\imp C^{n-3}[-3] \ar[rr]\ar[dd]\ar[dr]&& C^{n-3}[-n] \ar[dr] \ar[dd] & \\
& 0 \ar[rr, crossing over] && C^{n-2}[-n] \ar[dd] \\
0 \ar[rr]\ar[dr]&& 0 \ar[dr]& \\
& 0 \ar[rr] \ar[from=uu, crossing over]&& C^{n-1}[-n] \\
\end{tikzcd}
\right).

The previous remark takes a particularly pleasant form when the complex is

Let $C\in\coCh(\Sp)$, and assume there exists an $N$ such that
$C^m \simeq 0$ for all $m>N$; then for all $a>0$

C^N-a ≃
-.2em^a  times
-.2em(    C^N-a[-N] →C^N-a+1[-N] )

→C^N-a+2[-N] ) ⋯→C^N[-N] ).

It follows from Remark <ref>, Corollary
and <cit.> that the $(a+1)$-cube
defined by

F (\overrightharp{v}) =
\begin{cases}
\imp C^{N-a}[-a] \quad &\text{if } \overrightharp{v} = (0,\ldots,0)
\\[10pt]
\quad &\text{if } 0 \leq b < a
\text{, } \overrightharp{v}_i = 0 \text{ for } i > b
\text{ and } \overrightharp{v}_i = 1 \text{ for } i \leq b
\\[10pt]
\imp C^N \simeq C^N[-N]
\quad &\text{if } \overrightharp{v} = (1,\ldots,1)
\\[10pt]
0 &\text{else}
\end{cases}

(where $\overrightharp{v} =
(\overrightharp{v}_{a}, \overrightharp{v}_{a-1}, \ldots,
\overrightharp{v}_0)$)
is coCartesian.
In particular, again by <cit.>,
$\imp C^{N-a}[-a]$ is the total fiber of $F|_{\Delta^a \times \{1\}}$;
the thesis follows from <cit.>.

In the cases $a=2$ and $a=3$, the previous Corollary specializes to

\imp C^{N-2} \simeq \totcof \left(
\begin{tikzcd}
C^{N-2} \ar[r] \ar[d] & C^{N-1} \ar[d] \\
0 \ar[r] & C^{N} \\
\end{tikzcd}
\right)[-N]

\imp C^{N-3} \simeq \totcof \left(
\begin{tikzcd}
C^{N-3} \ar[rr]\ar[dd]\ar[dr]&& C^{N-2} \ar[dr] \ar[dd] & \\
& 0 \ar[rr, crossing over] && C^{N-1} \ar[dd] \\
0 \ar[rr]\ar[dr]&& 0 \ar[dr]& \\
& 0 \ar[rr] \ar[from=uu, crossing over]&& C^{N}\\
\end{tikzcd}
\right)[-N]

The following remark, inspired by the “Gap objects” considered
in <cit.>, will be useful later, and sheds some light on the
relation between coherent cochain complexes and the objects called
$\bbZ$-complexes in op. cit.

Proposition <ref> and the calculus of total cofibers of cubic
diagrams allow us to understand also all the intermediate subquotients of
$\imp C$.
We can consider the diagram consisting of co/Cartesian squares

\begin{tikzcd}[column sep = 0.7em]
\imp C^n \ar[r] \ar[d]
& \imp C^{n-1} \ar[r] \ar[d]
& \imp C^{n-2} \ar[r] \ar[d]
& \imp C^{n-3} \ar[r] \ar[d]
& \imp C^{n-4} \ar[d] \ar[r] & \cdots\\
0 \ar[r]
& C^{n-1}[-n+1] \ar[r] \ar[d]
& \imp C^{n-2}/\imp C^{n} \ar[r] \ar[d]
& \imp C^{n-3}/\imp C^{n} \ar[r] \ar[d]
& \imp C^{n-4}/\imp C^{n} \ar[d] \ar[r] & \cdots\\
& 0 \ar[r]
& C^{n-2}[-n+2] \ar[r] \ar[d]
& \imp C^{n-3}/\imp C^{n-1} \ar[r] \ar[d]
& \imp C^{n-4}/\imp C^{n-1} \ar[d] \ar[r] & \cdots\\
&& 0 \ar[r]
& C^{n-3}[-n+3] \ar[r]\ar[d]
& \imp C^{n-4}/\imp C^{n-2} \ar[r]\ar[d] & \cdots\\
&&&\vdots & \vdots
\end{tikzcd}

from which we can deduce that, for any $n\in\bbZ$

$$\imp C^{n-2}/\imp C^{n}\simeq \cofib(\partial^{n-2})[-n+1];$$

moreover, we can proceed inductively and apply
<cit.> to the cofiber sequences

$$\imp C^{n-k+1}/\imp C^n \to \imp C^{n-k}/\imp C^n \to C^{n-k}[-n+k]$$

to identify
$\imp C^{n-k}/\imp C^n$ with the total cofiber of a cube having
the truncation

$$\left(C^{n-k}\to C^{n-k+1}\to \cdots \to C^{n-1}\right)[-n+1]$$

of $C[-n+1]$ on its “spine” and zeroes elsewhere; thus, by Proposition
<ref> we have

C^n-k/C^n ≃
-.2em^k-1  times
-.2em(    C^n-k →C^n-k+1 )

→C^n-k+2 ) ⋯→C^n-1 )[-n+1]

for any $k\geq 2$.

§ THE GENERAL EQUIVALENCE

In this section, we extend the result of Theorem <ref>
to all stable with sequential limits. Along the way, we prove that
the explicit formulas given for $\caush$ and $\imp$ in the previous section
hold also in the general case.

Let $\scrC$ be a small stable . Then, the following
equivalences hold:

\coCh\left(\Fun\left(\scrC,\Sp\right)\right)$$

\simeq\cFild\left(\Fun\left(\scrC,\Sp\right)\right).$$

We prove this for $\coCh\left(\Fun\left(\scrC,\Sp\right)\right)$, the
case of
$\cFild\left(\Fun\left(\scrC,\Sp\right)\right)$ being entirely
We have that, by definition of $\coCh\left(\Sp\right)$,

\begin{equation}\label{eq-funcats}
\Fun\left(\scrC,\coCh\left(\Sp\right)\right)
\simeq\Fun\left(\scrC,\Fun^0\left(\catCh\op,\Sp\right)\right).
\end{equation}

Now, as

\simeq\Fun\left(\scrC\times\catCh\op,\Sp\right)$$

we have that the of (<ref>)
are equivalent to the full subcategory
of $\Fun(\scrC\times\catCh\op,\Sp)$ spanned by functors that are pointed in
the second variable (that is, sending any pair
$(C,0)\in\scrC\times\catCh\op$ to $0$). Thus, as

\simeq\Fun\left(\catCh\op,\Fun\left(\scrC,\Sp\right)\right)$$

the of (<ref>) are in turn equivalent to
the full subcategory

\subset\Fun\left(\catCh\op,\Fun\left(\scrC,\Sp\right)\right)$$

spanned by
pointed functors, which, by definition, is $\coCh\left(\Fun\left(\scrC,
\Sp\right)\right)$.

Let $\scrC$ be a small stable . Then, the equivalence of
Theorem <ref> extends to an equivalence between
$\coCh(\Fun(\scrC,\Sp))$ and $\cFild(\Fun(\scrC,\Sp))$
which we will again denote by

\begin{tikzcd}[column sep=huge]
\cFild(\Fun(\scrC,\Sp)) \ar[r, shift left=1.1ex, "\caush"] &
\coCh(\Fun(\scrC,\Sp)). \ar[l, shift left=1.1ex,"\imp"]
\ar[l,phantom,"\sim" description]
\end{tikzcd}

One way to re-phrase Lemma <ref> and Corollary
<ref> is to say that objects in
$\coCh(\Fun(\scrC,\Sp))$ can be described as bifunctors
$(C,n)\mapsto G_C(n)$ such that $G_C(\pt)\simeq 0$ for any $C\in\scrC$,
and similarly
objects in $\cFild(\Fun(\scrC,\Sp))$ can be described as bifunctors
$(C,n)\mapsto F_C(n)$ such that $\lim_n F_C(n) \simeq 0$ for any $C\in\scrC$.
The equivalence of Corollary
is then given on objects by pointwise (in $\scrC$) postcomposition with
$\caush$ and $\imp$; that is, we have

$$\caush\Big((C,n)\mapsto F_C^n\Big) \simeq \Big((C,n) \mapsto
\caush\big(F_C^\bullet\big)^n\Big)$$

$$\imp\Big((C,n)\mapsto G_C^n\Big) \simeq \Big((C,n) \mapsto
\imp\big(G_C^\bullet\big)^n\Big).$$

Let $\scrC$ be a small stable , and let $\scrA\subseteq\Fun(\scrC,\Sp)$
be a full stable subcategory closed under sequential limits. Postcomposition
with the inclusion $\iota\colon\scrA\to\Fun(\scrC,\Sp)$ induces fully
faithful functors

By inspection, an object $F$ lies in the essential image of
$\iota_{\mathrm{Fil}}$ if and only if $F^n$ lies in $\scrA$ for all
integers $n$, and an object $C$ lies in the essential image
of $\iota_{\mathrm{Ch}}$ if and only if $C^n$ lies in $\scrA$ for all
integers $n$.

Let $\scrA\subseteq\Fun(\scrC,\Sp)$ be a full, stable subcategory closed
under sequential limits.

* If $F$ is a coherent cochain complex in $\cFild(\Fun(\scrC,\Sp))$ that
lies in the essential image of $\cFild(\scrA)$, then $\caush F$
lies in the essential image of $\coCh(\scrA)$.

* If $C$ is a coherent cochain complex in $\coCh(\Fun(\scrC,\Sp))$ that
lies in the essential image of $\coCh(\scrA)$, then $\imp C$
lies in the essential image of $\cFild(\scrA)$.

Let $F\colon (C,n)\mapsto F_C^n$ lie in (the essential image of)
By Lemma <ref> together with Remark
<ref>, we have that, for any integer $m$,
$(\caush F)^m$ is given by $\cofib\left(F_\bullet^{m+1}\to F_\bullet^m\right)$.
By Remark <ref>, each $F_\bullet^n$ lies in $\scrA$,
hence, as $\scrA$ is a stable subcategory each $(\caush A)^m$ lies in
$\scrA$, and thus $\caush F$ lies in the essential image of $\coCh(\scrA)$.

Let $G\colon (C,n)\mapsto G_C^n$ lie in (the essential image of)
By Lemma <ref>, $G$ can be expressed as the
limit of bounded above complexes:

G\simeq\lim_{j\in\bbZ\op}G^{\leq j};

notice that, by Remark <ref> and by the definition of
the $G^{\leq j}$'s given in Construction <ref>, all
the objects of the form $(G^{\leq j})^k_\bullet$ belong to $\scrA$.
By Remark <ref> together with Remark <ref>,
we have that, for any choice of integers $j$ and $m$,
$(\imp G^{\leq j})^m_\bullet$ is either zero or can be expressed as a
suitable total fiber for a finite diagram $K^{j,m}$ whose entries are either
zeroes or of the form $(G^{\leq j})^k_\bullet[j]$ for different values of $k$;
in particular, $G^{\leq j}$ is a finite limit of a diagram with values in
Putting everything together (and keeping in mind that by Proposition
<ref> evaluation commutes with limits),
we have that

(G)^n    ≃((lim_j G^≤j))^n

≃(lim_j (G^≤j))^n

≃lim_j (G^≤j)^n

≃lim_j (K^j,n)

is given by a sequential limit of finite colimits of elements of $\scrA$,
and thus lies in $\scrA$ for any given $n$. This in turn proves that
$\imp G$ lies in the essential image of $\cFild(\scrA)$.

Let $\scrA\subseteq\Fun(\scrC,\Sp)$ be a full, stable subcategory closed
under sequential limits. Then the equivalence between
$\cFild(\Fun(\scrC,\Sp))$ and $\coCh(\Fun(\scrC,\Sp))$ restricts to an

\cFild(\scrA) \simeq \coCh(\scrA).

It follows from Lemma <ref> that (in the
notation of Remark <ref>) $\caush\circ\iota_{\mathrm{Fil}}$
factors through $\iota_{\mathrm{Ch}}$ and $\imp\circ\iota_{\mathrm{Ch}}$
factors thorugh $\iota_{\mathrm{Fil}}$, i.e. there exist functors (which
we'll temporarily denote $A$ and $I$) such that
$\caush\circ\ifil\simeq\ich\circ A$ and $\imp\circ\ich\simeq\ifil\circ I$.
In particular, as

≃∘A ∘I

$A$ and $I$ are mutually inverse.

Let $\scrC$ be a stable having sequential limits. Then there exists
an equivalence of stable
\begin{tikzcd}[column sep=huge]
\cFild(\scrC) \ar[r, shift left=1.1ex, "\caush"] &
\coCh(\scrC). \ar[l, shift left=1.1ex,"\imp"]
\ar[l,phantom,"\sim" description]
\end{tikzcd}

Let $U_0$ be our universe of small sets, and let $U_1$ denote a
universe containing $U_0$ as an element. Let us denote by $\Sp_{U_1}$ the
stabilization of the of $U_1$-small spaces. The stable Yoneda embedding
(see Definition <ref>) provides a fully faithful
(see <cit.>) exact functor

The results now follows from Lemma <ref> applied
to $\stYo$.

In particular, by composing the equivalence of Theorem
<ref> with the adjunction $L\dashv i$ of Proposition
<ref>, we get an induced adjunction

\begin{tikzcd}[column sep=huge]
\Fild\scrC \ar[r, shift left=1.1ex, "\aush"] &
\coCh\scrC \ar[l, shift left=1.1ex,"\imp"]
\ar[l,phantom,"\text{\rotatebox{-90}{$\dashv$}}"]
\end{tikzcd}

for any stable $\scrC$ with sequential limits.

In the proof of Lemma <ref> we also showed
that the pointwise descriptions already given for $\caush$ and $\imp$ in
the case $\scrC=\Sp$ hold also in the general case of
Theorem <ref>;
that is, for $F\in\cFild\scrC$

$$\caush F^n\simeq\gr^n F[n]$$

and, for $C\in\coCh\scrC$,

$$\imp C^n \simeq \totfib F_a [a]$$

for a suitable cube $F_a$, given explicitly by

F_a (\overrightharp{v}) =
\begin{cases}
\imp C(n-a)[-a] \quad &\text{if } \overrightharp{v} = (0,\ldots,0)
\\[10pt]
\quad &\text{if } 0 \leq b < a
\text{, } \overrightharp{v}_i = 0 \text{ for } i > b
\text{ and } \overrightharp{v}_i = 1 \text{ for } i \leq b
\\[10pt]
0 &\text{else.}
\end{cases}

(see <ref> for details about the notation).
In particular, Remark <ref> generalizes as well,

C^n-k/C^n ≃
-.2em^k-1  times
-.2em(    C^n-k →C^n-k+1 )

→C^n-k+2 ) ⋯→C^n-1 )[-n+1]

for any $n\in\bbZ$ and $k\geq 2$.

§ COHERENT COCHAIN COMPLEXES AND BEILINSON $\MATHRM{T}$-STRUCTURES

In this section, we study the connection between the
pointwise t-structure on coherent cochain complexes
and the Beilinson t-structure on filtered objects and show how the
former can in some sense be interpreted as an easier to understand version of
the latter.

In fact, if $\Fild\scrC$ is equipped with the Beilinson t-structure,
then the full subcategory of complete filtered objects inherits a t-structure
from it[this fact can easily be proved directly, but will be an immediate
consequence of Theorem <ref>].
It turns out that such inherited t-structure
is equivalent to the one obtained by carrying the pointwise t-structure
on $\coCh\scrC$ along the equivalence of Theorem <ref>;
moreover, the Beilinson t-structure on $\Fild\scrC$
is in some sense characterized by this property and by carrying “trivial
information” on essentially constant objects
(see Theorem <ref> for a precise statement).
In particular, $\Fild\scrC$ and $\cFild\scrC$ have the same heart (Remark

Let $\scrC$ be a stable equipped with a t-structure. We define
the pointwise t-structure on $\coCh\scrC$ to be the one defined

()_≥0    = {C∈ |  ∀n  C^n ∈_≥0}

()_≤0    = {C∈ |  ∀n  C^n ∈_≤0}.

We will denote the truncation functors for this t-structure by
$\tau^\text{lvl}_{\geq n}$ and $\tau^\text{lvl}_{\leq n}$, and the
homotopy objects by $\pi^\text{lvl}_n$ for all $n\in\bbZ$.

It follows immediately from the definitions that $\coCh\scrC$ has precisely
the same separatedness and completeness properties that $\scrC$ has.

Let $\scrC$ be a stable with sequential limits.
The transferred t-structure on $\cFild\scrC$ is the t-structure
$(\imp(\coCh\scrC)_{\geq 0},\imp(\coCh\scrC)_{\leq 0})$ transferred
along the equivalence of Theorem <ref>.

As a direct consequence of the definitions, we have that

\simeq\coCh\left(\scrC^\heartsuit\right).$$

The following definition is a straightforward generalization of the
t-structure first introduced by Beilinson in [4].

Let $\scrC$ be a stable having all sequential limits[this
assumption is likely superfluous; see Remark <ref>],
equipped with a right separated[unlike the previous one, this
assumption is crucial; see Remark <ref>]
t-structure $(\scrC_{\geq 0}, \scrC_{\leq 0})$.

The Beilinson t-structure $(\Fild_{\geq 0}\scrC, \Fild_{\leq 0}\scrC)$
on $\Fild\scrC$ is defined as follows:

$\bullet$ $\Fild_{\geq 0}\scrC$ is the full subcategory spanned by
the objects $F\in\Fild\scrC$ such that

$$\gr^i(F)\in \scrC_{\geq -i} \text{ for all } i.$$

$\bullet$ $\Fild_{\leq 0}\scrC$ is the full subcategory spanned by
the objects $F\in\Fild\scrC$ such that

$$F^i\in \scrC_{\leq -i} \text{ for all } i.$$

(Note the asymmetry in the definition). This t-structure appeared first,
in a slightly different setting, in [4].
Its existence in this generality will be a consequence
of <cit.> together with Theorem <ref>.
We will denote the truncation functors for this t-structure by
$\tau^B_{\geq n}$ and $\tau^B_{\leq n}$, and the
homotopy objects by $\pi^B_n$ for all $n\in\bbZ$.

The hypothesis of $\scrC$ having
all sequential limits in Definition <ref> is
there just because we will use Theorem <ref> to prove
its existence. We believe that the t-structure can exist even
without this assumption on $\scrC$, but as all the examples that arise in
practice satisfy this extra hypothesis, we didn't bother finding a proof that
does not use it.

We can infer a few properties of the Beilinson t-structure from its

* Even if we are assuming $\scrC$ to be right separated, $\Fild\scrC$
need not be so; in fact, the full subcategory of
$\infty$-coconnective objects $\cap_n (\Fild\scrC)_{\leq n}$ consists
of all filtered objects whose associated graded is trivial, hence of
all the essentially constant objects.

* Since the full subcategory of $\infty$-connective objects consists of
the levelwise $\infty$-connective ones, $\Fild\scrC$ is left separated
if and only if $\scrC$ is so.

We learned the following fact from <cit.>; although the result in
loc. cit. is stated in less generality, the proof carries verbatim
to the general case. We report the argument here for the reader's convenience.

Let $\scrC$ be a stable having all sequential limits,
equipped with a right separated t-structure
$(\scrC_{\geq 0}, \scrC_{\leq 0})$, and let $\wtrunc{n}$ denote its
Whitehead truncation functors.
Let $\Fild(\scrC)$ be equipped with the Beilinson t-structure, and
let $\wtrunc{n}^B$ denote its Whitehead truncation functors.
Then, there is a natural equivalence of functors $\Fild\scrC\to\scrC$

\gr^i\circ\wtrunc{n}^B \simeq \wtrunc{n-i}\circ\gr^i

for all $i\in\bbZ$.

Notice that for any $i$, the exact functor $\gr^i\colon\Fild\scrC\to\scrC$
carries $(\Fild\scrC)_{\geq 0}$ to $\scrC_{\geq -i}$. Moreover, as by
<cit.> each $\scrC_{\geq -i}$ is closed under extensions,
the fiber sequence

$$F^i\to\gr^i F\to F^{i+1}[1]$$

proves that $\gr^i F\in\scrC_{\leq -i}$, and thus $\gr^i$ carries
also $(\Fild\scrC)_{\leq 0}$ to $\scrC_{\leq -i}$. That is, each $\gr^i$
is t-exact with respect to the Beilinson t-structure on $\Fild\scrC$
and the shifted t-structure $(\scrC_{\geq -i},\scrC_{\leq -i})$ on $\scrC$.
As any exact and t-exact functor between stable equipped with
t-structures commutes with the truncation functors associated to the
t-structures, the result follows.

In particular, in the hypotheses of Proposition <ref>,
if we denote by $\pi_n$ the t-structure homotopy object functors of $\scrC$
and by $\pi_n^B$ the t-structure homotopy object functors of $\Fild\scrC$,
we have the equivalence (natural in $F$)

\gr^i\pi_n^B F \simeq (\pi_{n-i}(\gr^i F))[-i]

for all $i,n\in\bbZ$.

We recall the following theorem from [12] (which is an
$\infty$-categorical generalization of <cit.>).
Given a recollement $\scrC_0\hookrightarrow\scrC\hookleftarrow\scrC_1$,
suppose both $\scrC_0$ and $\scrC_1$ are equipped with t-structures, then
there exists a t-structure on $\scrC$, called the glued t-structure,
given by

_≥0    = {X∈ |  j_L X ∈(_1)_≥0
and  i_L X ∈(_0)_≥0}

_≤0    = {X∈ |  j_L X ∈(_1)_≤0
and  i_R X ∈(_0)_≤0}

(with notations as in Remark <ref>).

Let $\scrC$ be a stable with all sequential limits equipped with a
right separated t-structure $(\scrC_{\geq 0},\scrC_{\leq 0})$.
Then the glued t-structure on $\Fild\scrC$
(via the recollement of Remark <ref>)
obtained by considering

* the trivial t-structure[that
is, the one given by $(\scrC,\{0\})$, where all objects are connective,
and only the zero object is coconnective]
on $\scrC$,

* the transferred t-structure on $\cFild\scrC$

is the Beilinson t-structure (with respect to
$(\scrC_{\geq 0},\scrC_{\leq 0})$) on $\Fild\scrC$, as per Definition

Before proving Theorem <ref>, we state one immediate
consequence of it (and the definition of glued t-structure given in Theorem

The Beilinson t-structure on $\Fild\scrC$ induces a t-structure on the
full subcategory of complete filtered objects $\cFild\scrC$ that is
equivalent to the transferred t-structure of Definition

Let us start by identifying the subcategory of connective objects for
the transferred t-structure on $\cFild\scrC$.
We have that $F\in(\cFild\scrC)_{\geq 0}$ if and only if
$\caush F \in \left(\coCh\scrC\right)_{\geq 0}$.
By Definition <ref>,
this is the case if and only if

(\caush F)^n \simeq \gr^n F[n] \in \scrC_{\geq 0}
\text{ for all } n

(\cFild\scrC)_{\geq 0} = \left\{F\in\cFild\scrC \ | \ \forall n \ \gr^n F \in
\scrC_{\geq -n}\right\}.

Now, according to Theorem <ref>, the connective
objects in the glued t-structure on $\Fild\scrC$ are given by all the
$G\in\Fild\scrC$ such that

$$LG\in\left(\cFild\scrC\right)_{\geq 0}$$

(the condition on $G^{-\infty}$ being empty, as we are considering the
trivial t-structure on $\scrC$);
but, as for all $n$ we have $\gr^n LG \simeq \gr^n G$, the above is
equivalent to the condition

\gr^n G \in \scrC_{\geq -n} \text{ for all } n

which in turn determines exactly the class of connective objects for the
Beilinson t-structure of Definition <ref>;
as, by <cit.>, the class of connective objects completely
determines the t-structure, provided its existence, the only thing left
is to chech is that the description of coconnective objects given in
Theorem <ref> coincides with the one given in Definition
<ref>; that is, we have to prove that

\left(\forall n \ \gr^n F \in \scrC_{\leq -n} \text{ and } F^{+\infty} \simeq 0\right)
\Longleftrightarrow
\left(\forall n \ F^n \in \scrC_{\leq -n}\right).

For the “only if” direction: <cit.>
implies that $\scrC_{\leq -n}$ is closed under
extensions, hence if $F^{n+1} \in \scrC_{\leq -n-1}$ and
$F^n \in \scrC_{\leq -n}$ the fiber sequence

$$F^n\to\gr^{n}F\to F^{n+1}[1]$$

proves that $\gr^n F\in\scrC_{\leq -n}$.
To check that $F^{+\infty}\simeq 0$, observe that any subset of the form
$\bbZ\op_{\geq n}$ is an initial subcategory of $\bbZ\op$, hence for any
$n\in\bbZ$ we have

F^{+\infty}\coloneqq\lim\left(\cdots\to F^{n+1}\to F^n \to \cdots\right)
\simeq\lim\left(\cdots\to F^{n+1}\to F^n\right);

as (by <cit.>), each $\scrC_{\leq n}$ is closed under all limits
existing in $\scrC$,

$$F^{+\infty}\in\scrC_{\leq n} \ \forall n\in\bbZ$$

(recall that $\scrC_{\leq m}\subseteq\scrC_{\leq n}$ for all pairs of integers
$m\leq n$) and thus $F^{+\infty}\simeq 0$ by the right separatedness

For the “if” direction, let us start by noticing that as
$\gr^{n+1}F[1]\in\scrC_{\leq -n}$, the fiber

$$F^n/F^{n+2}\to\gr^n F\to\gr^{n+1}F$$

proves $F^n/F^{n+2}\in\scrC_{\leq -n}$ (again, as the latter is closed
under limits in $\scrC$). We can now proceed inductively for $m\geq 2$ to
show that (as $\gr^{n+m}F[1]\in\scrC_{\leq -n-m+1}\subseteq\scrC_{\leq -n})$
the fiber sequence

$$F^n/F^{n+m+1}\to F^n/F^{n+m}\to \gr^{n+m}F[1]$$

implies all objects $F^n/F^k$ for $k>n$ lie in $\scrC_{\leq -n}$.
Since (again, by <cit.>) we know we can compute colimits in
$\scrC_{\leq -n}$ just by computing them in $\scrC$ and then reflecting
along the left adjoint to the inclusion (in particular, the colimit
is the same in both categories if the object already happened to land
in $\scrC_{\leq -n}$ when computed in $\scrC$), we have that

lim_k F^n/F^k
≃lim_k (F^k→F^n)

≃(lim_k F^k→F^n)

lies in $\scrC_{\leq -n}$. But, as by hypothesis $F^{+\infty}\simeq 0$,
we have that $F^n\in\scrC_{\leq -n}$ as desired.

Motivated by the previous results, we will refer to the transferred
t-structure on $\cFild\scrC$ also as the Beilinson t-structure.

In the situation of Theorem <ref>, passing to hearts one
gets a “recollement” of Abelian categories:

\begin{tikzcd}[column sep=huge]
\scrC_0^\heartsuit \ar[r, hook, "i"' description] &
\scrC^\heartsuit \ar[l, shift right=0.6ex, bend right, "i_L"']
\ar[from=r, hook, shift right=0.6ex, bend right, "(j_L)_!"']
\ar[r, "j_L" description]
\ar[from=r, hook', shift left=0.6ex, bend left, "j"]
\ar[l, shift left=0.6ex, bend left, "i_R"]
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
"\text{\scalebox{1}{\rotatebox{-90}{$\dashv$}}}"] &
\scrC_1^\heartsuit
\ar[l,phantom, shift left=2ex,
\ar[l,phantom, shift right=2ex,
\end{tikzcd}

where $i$, $j$ and $(j_L)_!$ are fully faithful. As shown already in
<cit.>, in this situation $j_L$ characterizes
$\scrC_1^\heartsuit$ as the quotient category

In particular, in the hypothesis of Theorem <ref>, we have
that (as $\scrC$ is endowed with the trivial t-structure)
$\scrC^\heartsuit\simeq 0$ and thus (using Remark <ref>)

\simeq\coCh\left(\scrC^\heartsuit\right).$$

It follows from Definition <ref> together with Corollary
<ref> that for any $F\in\cFild\scrC$ and $C\in
\coCh\scrC$ we have

\caush \left(\tau^B_{\leq n} F\right) \simeq
\tau^\text{lvl}_{\leq n} (\caush F) \quad
\text{and} \quad
\imp \left(\tau^\text{lvl}_{\leq n} C \right)\simeq
\tau^B_{\leq n} (\imp C)

for all $n\in\bbZ$, and similar formulae for the truncations above $n$.
Moreover, by Remark <ref>, we also have

$$\pi^B_n F \simeq \pi^\text{lvl}_n \caush F \quad
\text{and} \quad
\pi^\text{lvl}_n C \simeq \pi^B_n \imp C

for all $n\in\bbZ$.

One can construct on $\Fild\scrC$ the glued t-structure as in
Theorem <ref> regardless of the right separatedness
if $\scrC$ is not right separated, the class of coconnective objects will be
the full subcategory

\{F\in\Fild_{\leq 0} \ | \ \aush F \in (\coCh\scrC)_{\leq 0}
\text{ and } F^{-\infty} \simeq 0\}

but this class will in general not coincide with the one described in
Definition <ref>
(and, as by <cit.> the class of connective objects completely
determines the t-structure, there cannot exist
a t-structure exactly as in Definition <ref> if
the two classes of “candidate coconnective objects” do not coincide).
We are not aware of any application for this t-structure.

§ SYMMETRIC MONOIDAL STRUCTURES

The $\infty$-categorical Day convolution (introduced in [15] and
further developed in <cit.>) provides a way
to equip with a symmetric monoidal structure any functor category
$\Fun(\scrC,\scrD)$, provided that both $\scrC$ and $\scrD$ are symmetric
monoidal, and $\scrD$ is presentably so.
In particular (see Remark <ref>),
we can endow $\Fild\scrC$ with a symmetric monoidal structure whenever
$\scrC$ is presentably symmetric monoidal.
As it turns out, such monoidal structure induces one on $\cFild\scrC$, and
thus on $\coCh\scrC$, whenever Theorem <ref> applies.
In this section, we analyze these induced symmetric monoidal structures,
and their interaction with the t-structures introduced
in the previous section. In particular, we prove that the Day convolution
structure on both $\Fild\scrC$ and $\cFild\scrC$ is compatible with
Beilinson t-structures, and that the symmetric monoidal structure on
$\coCh\scrC$ provides a homotopy coherent refinement of the usual tensor
product of cochain complexes.

Let $\scrC$ be a presentably symmetric monoidal .
By <cit.> (see also <cit.>, with $\kappa$
chosen to be the strongly inaccessible cardinal determining the size of our
universe of small sets) we can endow $\Fild(\scrC)$ with the structure of a
symmetric monoidal , given by Day convolution (where the symmetric
monoidal structure on $\bbZ\op$ is given by addition).
Again, by <cit.>, if $F$ and $G$ are filtered objects in
$\scrC$, their Day convolution product is given by Kan extension of
$\tensor\circ(F,G)$ along $+$, hence by <cit.> and
<cit.> it is pointwise given by

\begin{equation}\label{eq-day-formula}
(F\tensor_{\text{Day}}G)^n \simeq \colim_{
\substack{(s,t)\in\bbZ\op\times\bbZ\op \\
s+t\geq n}}
F^s \tensor G^t
\end{equation}

where the shape of the colimit follows from inspection
of the comma category $(+\downarrow n)$ (see Definition <ref>).
In particular, $\Fild(\scrC)$ is presentably symmetric monoidal.
Similarly, we can endow $\prod_\bbZ\scrC\simeq\Fun(\bbZ^\delta,\scrC)$ with
a presentably symmetric monoidal structure given by Day convolution, whose
product is pointwise given by

\left((X_u)_{u\in\bbZ}\tensor_{\text{Day}}(Y_v)_{v\in\bbZ}\right)_n \simeq
\bigoplus_{s+t=n} X_s \tensor Y_t

where once again the shape of the colimit follows from inspection
of the comma category $(+\downarrow n)$.

Let $\scrC$ be a presentably symmetric monoidal , whose unit we'll denote
by $\mathbbm{1}$. Then, the unit object for $(\Fild\scrC,
\tensor_{\text{Day}})$ is given by the filtered object

$$\mathbbm{1}_{\langle\leq 0\rangle}\coloneqq
\cdots \to 0 \to 0 \to \mathbbm{1}\xto{\id}\mathbbm{1}\xto{\id}\cdots$$

consisting of copies of $\mathbbm{1}$ and identity morphisms for $n\leq 0$,
and of copies of $0$ for $n>0$.

First, let us note that it is enough to prove the following claim:

($\spadesuit$) For any $F\in\Fild\scrC$ and any $n\in\bbZ$
we have

$$(F\tensor_{\text{Day}}\mathbbm{1}_{\langle\leq 0\rangle})^n\simeq F^n.$$

In fact, the existence of the unitor equivalence
$\mathbbm{1}_{\text{Day}}\tensor\mathbbm{1}_{\langle\leq 0\rangle}\xto{\sim}
\mathbbm{1}_{\langle\leq 0\rangle}$
together with the claim implies the existence of equivalences

\left(\mathbbm{1}_{\text{Day}}\tensor
\mathbbm{1}_{\langle\leq 0\rangle}\right)^n\simeq
\mathbbm{1}_{\langle\leq 0\rangle}^n$$

for all $n\in\bbZ$.
As equivalences in $\Fild\scrC$ can be cheched pointwise, this is enough to
$\mathbbm{1}_{\text{Day}}\simeq \mathbbm{1}_{\langle\leq 0\rangle}$.

We now turn to the proof of claim ($\spadesuit$).
By (<ref>), this boils down to proving that the colimit
of the following diagram

\begin{tikzcd}
&\cdots \ar[d] & \cdots \ar[d] & \cdots \ar[d] & \cdots \ar[d] &
\cdots \ar[d] & \cdots \ar[d]\\
\cdots \ar[r] & 0 \ar[r]\ar[d] & 0 \ar[r]\ar[d]&
0 \ar[r]\ar[d]& 0 \ar[r]\ar[d]&
0 \ar[r]\ar[d]& 0 \\
\cdots \ar[r] & 0 \ar[r]\ar[d] & 0 \ar[r]\ar[d]&
0 \ar[r]\ar[d]& 0 \ar[r]\ar[d]& 0 & \\
\cdots \ar[r] & F^{n+3} \ar[r]\ar[d, equal] & F^{n+2} \ar[r]\ar[d, equal]&
F^{n+1} \ar[r]\ar[d, equal]& F^n && \\
\cdots \ar[r] & F^{n+3} \ar[r]\ar[d, equal] & F^{n+2} \ar[r]\ar[d, equal]&
F^{n+1} &&& \\
\cdots \ar[r] & F^{n+3} \ar[r]\ar[d, equal] & F^{n+2} &&&& \\
&\cdots &&&&& \\
\end{tikzcd}

is equivalent to $F^n$. By finality, it is enough to check that the colimit
of the following diagram

\begin{equation}\label{eq-zigzag}
\begin{tikzcd}
&&&&& \cdots \ar[d]\\
&&&& 0 \ar[r]\ar[d]& 0 & \\
&&& 0 \ar[r]\ar[d]& 0 && \\
&& F^{n+1} \ar[r]\ar[d, equal]& F^{n} &&& \\
& F^{n+2} \ar[r]\ar[d, equal]& F^{n+1} &&&& \\
F^{n+3} \ar[r]\ar[d, equal] & F^{n+2} &&&&& \\
\cdots &&&&& \\
\end{tikzcd}
\end{equation}

is equivalent to $F^n$.
If we denote by $A$ the colimit of the diagram

\begin{equation}\label{eq-zero-zig}
\begin{tikzcd}
&&&&& \cdots \ar[d]\\
&&&& 0 \ar[r]\ar[d]& 0 & \\
&&& 0 \ar[r]& 0 && \\
\end{tikzcd}
\end{equation}

and by $B$ the colimit of the diagram

\begin{equation}\label{eq-zag}
\begin{tikzcd}
&&& 0 \ar[d]&&& \\
&& F^{n+1} \ar[r]\ar[d, equal]& F^{n} &&& \\
& F^{n+2} \ar[r]\ar[d, equal]& F^{n+1} &&&& \\
F^{n+3} \ar[r]\ar[d, equal] & F^{n+2} &&&&& \\
\cdots &&&&& \\
\end{tikzcd}
\end{equation}

we have that, by <cit.>, we can decompose the colimit of
(<ref>) as the coproduct $A\oplus B$. As (<ref>)
consists only of zero objects, its colimit is zero. Thus, the colimit of
(<ref>) is equivalent to the colimit of (<ref>).
By finality, we can omit the top right arrow $0\to F^n$ from the diagram in
order to compute its colimit.
By applying inductively <cit.>, we see that $B$ can be computed
as the iterated pushout

As each of the pushouts is taken along an equivalence, we have $B\simeq F^n$,
as desired.

Recall the following definition.

Let $\scrC$ be a symmmetric monoidal , and let $L\colon\scrC\to\scrC$
be a localization functor. The functor $L$ is compatible with the
symmetric monoidal structure if for every $L$-equivalence
$X\to Y$, and every $Z\in\scrC$, the induced $X\tensor Z\to Y\tensor Z$
is an $L$-equivalence.

The following proposition already appeared as <cit.>,
we present an alternative proof.

Let $\scrC$ be a presentably symmetric monoidal . Then,
the localization functor $L\colon\Fild\scrC\to\cFild\scrC$ is compatible with
Day convolution.

Note that, by <cit.>, $\Fild\scrC$ admits an internal
mapping object given by

(where $\map_{\scrC}$ denotes the internal mapping object of $\scrC$)
By <cit.>, it suffices to prove that, given any
$F\in\Fild\scrC$ and any $G\in\cFild\scrC$, the internal mapping object is
complete; by the following computation

lim_n∈  (F,G)(n)

this is in fact the case.

It follows from Proposition <ref> and
<cit.> that if $\scrC$ is presentably symmetric monoidal,
we have an induced presentably symmetric monoidal structure on
$\cFild(\scrC)$, which we'll refer to as the completed Day convolution
monoidal structure and
will denote by $\widehat{\tensor}$. In particular, we have that

$$F \ \widehat{\tensor} \ G \simeq L \Big(F\tensor_{\text{Day}}G\Big).$$

From Proposition <ref> and
Remark <ref>, we have
that the unit for $\widehat\tensor$ is given by

$$L\mathbbm{1}_{\langle\leq 0\rangle}\simeq\mathbbm{1}_{\langle\leq 0\rangle}.$$

Let $\scrC$ be a stable presentably symmetric monoidal .
We refer to the symmetric monoidal structure induced on $\coCh(\scrC)$ by the
equivalence of Theorem <ref> as the coherent
cochains tensor product, and we will denote it just by $\tensor$.

It follows from Theorem <ref> together with
Remark <ref>
that the functor $\undch$ defined in Lemma
fits into the following commutative diagram

\begin{tikzcd}
\Fild\scrC\ar[r,"\aush"] \ar[d, "\gr"'] &[15pt] \coCh(\scrC) \ar[d,"\undch"] \\
\prod_\bbZ\scrC\ar[r,"(\Sigma^n)_{n\in\bbZ}"] &[15pt] \prod_\bbZ\scrC.
\end{tikzcd}

that is, $\undch \circ \aush F \simeq \left(\gr^n F [n]\right)_{n\in\bbZ}$
naturally in $F\in\Fild\scrC$.

The symmetric monoidal stucture given in Definition
<ref> is really a homotopy coherent generalization
of the usual tensor product of cochain complexes, in a sense made precise
by the following results.

The functor $\undch \circ \aush$ is symmetric monoidal.

This is a direct consequence of <cit.> together
with Remark <ref> and $(\Sigma^n)_{n\in\bbZ}$
being an equivalence.

Let $\scrC$ be a stable presentably symmetric monoidal .
Let $C$ and $D$ be elements of $\coCh(\scrC)$. Then, we have that

$$(C \tensor D)^n \simeq \bigoplus_{s+t=n} C^s \tensor_\scrC D^t .$$

By definition of $\undch$ (see Lemma
$(C\tensor D)^n \simeq \undch(C\tensor D)^n$. The result follows from
the following computation

(C D)^n

(C    D)^n

L (C _Day D)^n

(C _Day D)^n

(C _Day D)^n

(uC _Day uD)^n

⊕_s+t=n C^s _ D^t.

We now analyze the interaction between the symmetric monoidal structures
introduced in this section, and the t-structures introduced in Section
<ref>. We start by recalling the following definition
(see <cit.> and <cit.> for more details about the
general theory of the interaction between t-structures and symmetric monoidal

Let $\scrC$ be a stably symmetric monoidal equipped with a t-structure
$(\scrC_{\leq 0}, \scrC_{\geq 0})$.
The t-structure is said to be compatible with the symmetric monoidal
structure if the following conditions hold:

* The unit object for $\tensor$ lies in $\scrC_{\geq 0}$;

* Given any pair of connective objects $X,Y\in\scrC_{\geq 0}$,
their product $X\tensor Y$ lies in $\scrC_{\geq 0}$ as well.

Conditions (<ref>) and (<ref>) guarantee that
$\scrC_{\geq 0}$ inherits a symmetric monoidal structure from $\scrC$
such that the fully faithful inclusion
$\scrC_{\geq 0}\hookrightarrow\scrC$ is a (strong) symmetric monidal functor.

Let $\scrC$ be a presentably symmetric monoidal equipped with
a t-structure that is compatible with the monoidal structure. Then:

* The Beilinson t-structure on $\Fild\scrC$ is compatible with
the Day convolution product $\tensor_{\text{Day}}$;

* The Beilinson t-structure on $\cFild\scrC$ is compatible
with the completed Day convolution product $\widehat\tensor$;

* If $\scrC$ is moreover stable, the pointwise
t-structure on $\coCh\scrC$ is compatible with the coherent cochains
tensor product $\tensor$

By Proposition <ref>, we have that

\gr^i\mathbbm{1}_{\langle\leq 0\rangle}\simeq
\begin{cases}
0 &\text{ for } i\ne 1;\\
\mathbbm{1} &\text{ for } i=1.
\end{cases}

and thus the unit is Beilinson connective.
Let us now consider $F, G\in\Fild\scrC_{\geq 0}$.
By <cit.> we have that

\gr^i(F \tensor_{\text{Day}} G)\simeq \bigoplus_{s+t=i} \gr^s F\tensor\gr^t G

from which we immediately have that $F\tensor_{\text{Day}}G$ lies in
$(\Fild\scrC)_{\geq 0}$. This completes the proof of (<ref>).
By Proposition <ref>, applying the localization
functor $L$ has no effect on associated gradeds, hence (<ref>)
immediately implies (<ref>). Finally,
(<ref>) is a trivial consequence of
(<ref>) and the definition of $\tensor$ on $\coCh\scrC$.

It follows from <cit.> that Day convolution, and hence also the
tensor product of coherent chain complexes, induce a symmetric monoidal
structure on the hearts of the respective Beilinson t-structures.
From Corollary <ref>, we have
that the induced symmetric monoidal structure on

is the usual one.

§ COHERENT COCHAIN COMPLEXES AND TODA BRACKETS

In this section, we will have a closer look at the relation between
Toda brackets and coherent cochain complexes.
Our main result will be characterizing coherent cochain complexes in suitable
pointed as being precisely the sequences of objects

$$\cdots\to X^n \xto{f^n} X^{n+1}\xto{f^{n+1}} X^{n+2}\cdots$$

such that all pairwise composable morphisms are nullhomotopic (i.e. for all
$n\in\bbZ$ we have $f^{n+1}\circ f^n\simeq 0$) and all possible Toda brackets
are compatibly trivial (see Remark <ref>).

Let us first start with a short recollection about Toda brackets.
Given a pointed $\scrC$, let
$f^0\colon X^0 \to X^1$, $f^1\colon X^1 \to X^2$ and $f^2\colon X^2 \to X^3$
be morphisms in $\scrC$, such that $f^1 f^0\simeq f^2 f^1 \simeq 0$, and let
$\alpha\colon 0\Rightarrow f^1 f^0$ and $\beta\colon 0 \Rightarrow f^1 f^2$ be
two choices of nullhomotopies:

\begin{tikzcd}
X^0 \ar[r, "f^0"] \ar[rr, bend right=40, "0"']
\ar[rr, phantom, bend right=20, "\rotatebox{90}{$\Rightarrow$} \alpha"]
& X^1 \ar[r, "f^1"]
& X^2 \ar[r, "f^2"]
& X^3. \ar[from=ll, bend left=40, "0"]
\ar[from=ll, phantom, bend left=20, "\rotatebox{90}{$\Leftarrow$} \beta"]
\end{tikzcd}

In such a situation, the two whiskerings $f^2 \alpha$ and $\beta f^0$ determine
two paths $0\Rightarrow f^2 f^1 f^0$ in $\Map_*(X^0,X^3)$.
By gluing them along their endpoints, we obtain a map (pointed at zero)
$T\colon S^{1}\to\Map_*(X^0,X^3)$, which in turns determines a class
$\langle f^2,f^1,f^0 \rangle_{(\alpha, \beta)} \in \pi_1(\Map(X^0,X^3))$.
The homotopy class $\langle f^2,f^1,f^0 \rangle_{(\alpha, \beta)}$ is known as
the Toda bracket determined by $\alpha$ and $\beta$.

The customary approach to
Toda brackets would be not fix $\alpha$ and $\beta$ as part of the datum,
and to instead define the Toda bracket to be a subset of $\pi_1(\Map(X^0,X^3))$,
whose elements are given by all the possible choices of homotopy classes of
paths $([\alpha], [\beta])$. According to this definition, the object
referred above as $\langle h,g,f \rangle_{(\alpha, \beta)}$ is a specified
element of this subset.

The notion can be generalized to longer sequences of maps, provided that the
Toda brackets of the shorter sub-sequences are nullhomotopic; to illustrate how
the generalization works, let us consider the case of a sequence of length 4

\begin{tikzcd}
X^0 \ar[r, "f^0"]
& X^1 \ar[r, "f^1"]
& X^2 \ar[r, "f^2"]
& X^3 \ar[r, "f^3"]
& X^4.
\end{tikzcd}

and choices of nullhomotopies $\alpha$, $\beta$ and $\gamma$ for all consecutive
pairs of maps, for which $\langle f^2,f^1,f^0 \rangle_{(\alpha, \beta)}=0$
and $\langle f^3,f^2,f^1 \rangle_{(\beta, \gamma)}=0$; in such a situation,
we obtain a pair of nullhomotopies for a pair of maps $X^0\to\Sigma X^4$, one
for the composite

\begin{tikzcd}[column sep=6em]
X^0 \ar[r, "f^0"]
& X^1 \ar[r, "{\langle f^3,f^2,f^1 \rangle_{(\beta, \gamma)}}"]
& \Sigma X^4
\end{tikzcd}

and the other one for composite

\begin{tikzcd}[column sep=6em]
\ar[r, "{\langle f^2,f^1,f^0 \rangle_{(\alpha, \beta)}}"]
& \Sigma X^3 \ar[r, "\Sigma f^3"]
& \Sigma X^4;
\end{tikzcd}

putting together these two as before, we obtain a pointed map
$S^1\to\Map_*(X^0,\Sigma X^4)$, and thus a class
$\langle f^3,f^2,f^1,f^0 \rangle\in\pi_2(\Map_*(X^0,X^3))$,
the 4-fold Toda bracket of the
sequence[again, we are working with a coherent notion, depending also
on the choices of all the five nullhomotopies involved].
This idea clearly generalizes to longer sequences of maps,
provided that the shorter subsequences have nullhomotopic brackets.

Although it is possible to work with these notions using the approach sketched
above, in the rest of the section we will use a
slightly different perspective to define and generalize Toda brackets.
Our approach will be to define these classes by means
of the actions of certain algebra objects in graded pointed spaces;
the main advantage for us will be the ease of working
at once with all the possible $n$-fold Toda brackets in a $\bbZ$-indexed sequence
(provided that all the possible $(n-1)$-fold Toda brackets are
of course, one can recover the case of finite sequences
by considering a $\bbZ$-indexed sequence where only finitely many
objects are non-zero (see Definition <ref> for the details).
One other pleasant feature
of our approach will be to obtain an inherently coherent notion of Toda brackets
(that is, encoding also choices for all the involved homotopies, rather than
defining the notion up to a coset of “indeterminacies”).

Our strategy for proving that coherent chain complexes are precisely those
$\bbZ$-indexed sequences of morphisms where consecutive maps compose to zero
and all Toda brackets are coherently nullhomotopic requires a few steps:
in <ref>,
we will prove that for a pointed $\scrC$
(which we will for simplicity assume to be also presentable) both
$\Fun(\bbZ\op,\scrC)$[in the present section we will use the notation
$\Fun(\bbZ\op,\scrC)$ instead of the previously introduced $\Fild(\scrC)$
to stress that here we want to think not of structured
objects of independent interest, but of sequences
of morphisms upon which we intend to put relations;
in particular, the reader should
keep in mind that the appearence of this here has nothing to do with
the equivalence of Theorem <ref>]
and $\coCh(\scrC)$ are
of modules for suitable $\bbE_1$-algebra objects in $\Gr\spaces_*$.
Then, in <ref>
we will construct a sequence of $\bbE_1$-algebras in $\Gr\spaces_*$

\begin{equation}\label{eq-sequence-of-algebras}
R_1\to R_2\to R_3 \to \cdots
\end{equation}

such that the of modules over $R_1$ will be equivalent to
$\Fun(\bbZ\op,\scrC)$, the of modules over $\colim_n R_n$ will be
equivalent to $\coCh(\scrC)$, and such that the sequence induced
by (<ref>) on
homology will be equivalent to the resolution of $\Lambda(e)$ described in
Theorem <ref>, with $e$ of degree 1.
Modules over $R_n$ for a fixed $n\geq 2$ will
be sequences where all pairwise composable morphisms will be nullhomotopic,
all possible $m$-fold Toda brackets for $m\leq n$ will be defined and trivial,
and all the relevant nullhomotopies will be encoded in the module structure
(see Remark <ref> for the details).
In order to prove the existence of the sequence (<ref>),
we will use the results of Appendix <ref>.
As it turns out, the construction of the above sequence up to $R_3$ is
slightly more subtle than its extension to all the other $R_n$'s,
hence we will first construct the sequence inductively for $n\geq 4$,
and defer the construction of the beginning of the sequence
to <ref>.

§.§ Algebras in graded pointed spaces

The main goal of this section is to prove that, given a pointed $\scrC$,
both $\Fun(\bbZ\op,\scrC)$ and $\coCh(\scrC)$ can be expressed as of
modules over suitably defined $\bbE_1$-algebras in graded pointed spaces.
For simplicity, we will assume that $\scrC$ is also presentable, but with
some extra effort the results can be generalized further, using methods
along the lines of those employed in Section <ref>.

The stable analogue of the following proposition appeared as

Let $\scrC$ be a pointed . The following data are equivalent:

* self-equivalences $\scrC\to\scrC$;

* (left or right) actions of $\bbZ^\delta$ on $\scrC$;

* monoidal functors $\bbZ^\delta\to\Fun(\scrC,\scrC)$;

* monoidal functors $\bbZ^\delta\to\Fun^{\text{Rex}}(\scrC,\scrC)$;

* right exact monoidal functors

* (left or right) actions of $\Prefinpt{\bbZ^\delta}$ on $\scrC$
such that the action map commutes with finite colimits in each variable.

If $\scrC$ is pointed presentable, then the above are equivalent to:

* left adjoint monoidal functors

* actions of $\Prept{\bbZ^\delta}$ on $\scrC$ such that the action
map commutes with small colimits in each variable.

Here $\Prept{\bbZ^\delta}$ and $\Prefinpt{\bbZ^\delta}$ are defined as in <ref>.

The equivalence (<ref>) $\Leftrightarrow$
(<ref>) follows
from <cit.>: loc. cit. gives an equivalence

\simeq\Aut(\scrC)$$

where the monoidal structure on $\Fun(\scrC,\scrC)$ is given by composition,
and $\Aut(\scrC)\subset\Fun(\scrC,\scrC)$ denotes the full subcategory spanned
by self-equivalences.
(<ref>)$\Leftrightarrow$(<ref>) and
(<ref>)$\Leftrightarrow$(<ref>) are just
reformulations of the definition of action of a
symmetric monoidal .
(<ref>)$\Leftrightarrow$(<ref>) holds trivially, as
$\bbZ^\delta$ is a discrete category.
(<ref>)$\Leftrightarrow$(<ref>) follows from
Proposition <ref> and the monoidality of the pointed
Yoneda embedding (see <cit.>).
(<ref>)$\Leftrightarrow$(<ref>) and
follow at once from Proposition <ref>.

In what follows, unless otherwise specified, we will implicitly work with
left actions whenever we apply Proposition <ref>.
Recall that, in the terminology of [24], a left action of a monoidal $\scrC$ on some $\scrM$ is equivalent to exhibiting $\scrM$ as
left-tensored over $\scrC$, and to exhibiting $\scrM$ as a left $\scrC$-module
in (a suitably sized) of (see <cit.>).

Let $\scrC$ be a pointed . Then, precomposition with
$-1\colon\bbZ\to\bbZ$ induces an automorphism

$$(-)\{1\}\colon\coCh\scrC\to\coCh\scrC \ \colon
C^\bullet\mapsto C\{1\}^\bullet \simeq C^{\bullet -1};$$

by virtue of Proposition
<ref>, this endowes $\coCh\scrC$ with the structure of
an left-tensored over $\Prefinpt{\bbZ^\delta}\simeq\Gr(\spaces_*^{\text{fin}})$
(and when $\scrC$ is also presentable, the left-tensoring extends over
we will use the notation:

$$(-)\{n\}\colon\coCh\scrC\to\coCh\scrC \quad \forall n\in\bbZ$$

to denote the (right exact) endofunctor induced by $n$.

Let $\scrC$ be a pointed .
Similarly to the previous example, precomposition with
$-1\colon\bbZ\op\to\bbZ\op$ induces an automorphism

hence, $\Fild\scrC$ is also left-tensored over
(and when $\scrC$ is also presentable, the left-tensoring extends over
also in this case, we will use the notation:

$$(-)\{n\}\colon\Fild\scrC\to\Fild\scrC \quad \forall n\in\bbZ$$

to denote the automorphism induced by $n$.

Given any $n\in\bbN$ and any $t\in\bbZ$, we will denote by
$S^{n,t}\in\coCh\spaces_*$ the graded pointed space consisting of a copy of
$S^n$ in degree $t$, and copies of $\pt$ in all other degrees.

Let $C\in\coCh\scrC$ be a coherent cochain complex in a complete[or,
more generally, in a pointed where all the relevant right Kan extensions
exist and are pointwise], pointed .
The functor $-\tensor C \colon \Gr\spaces_* \to \coCh\scrC$ induced by
the left-tensoring of Example <ref> admits a right adjoint,
denoted $\innmap{}_*(C,-)$,
whose value on any $D\in\coCh\scrC$ is pointwise given by

$$\innmap{}_*(C,D)^n \simeq\ptMap\left(C\{n\},D\right).$$

By unraveling the definitions, the $\Gr\spaces_*$-action
on $\Fun^{\text{Rex}}(\coCh\scrC,\coCh\scrC)$ is obtained by Kan
extension of the functor
n\mapsto\{n\}$ along the functor $y\colon\bbZ^\delta\to\Gr\spaces_*\colon
n\mapsto S^{0,n}$. As the Kan extension is pointwise,

$$-\tensor C\simeq \Lan_y(\ev_C \circ\{\})$$

where $\ev_C\colon\Fun^{\text{Rex}}(\coCh\scrC,\coCh\scrC)\to\coCh\scrC$
denotes the evaluation at $C$ functor.
In particular, if we put $F\coloneqq\ev_C\circ\{\}$, we have that
$-\tensor C$ is, in the language of Appendix <ref>,
the pointed $F$-realization functor. Thus, it admits as a right adjoint
given by the pointed $F$-nerve functor $\nerve_F^{\text{pt}}$,
which by Proposition <ref> is given by

$$\nerve_F^{\text{pt}}(D)\simeq \ptMap(F(-),D)\simeq
\ptMap\left(C\{-\},D\right)$$

as desired.

We will refer to the right adjoint functor defined in Proposition
as the graded pointed mapping space functor.

Let $\scrC$ be a pointed presentable . Then,
there exists an $\bbE_1$-algebra in graded pointed spaces
such that the of coherent cochain
complexes in $\scrC$ is equivalent to the of right $A$-modules
in graded objects of $\scrC$:

Moreover, the underlying graded pointed space of $A$ is given by
$S^{0,0}\amalg S^{0,1}$.

Let us first notice that the claim for a generic pointed presentable
$\scrC$ can be deduced from the case $\scrC\simeq\spaces_*$,
by means of Lurie's tensor product (see <cit.>):
by <cit.>,

since preserving the zero object is a property, the equivalence above induces
an equivalence $\coCh(\scrC)\simeq\coCh(\spaces_*)\tensor\scrC$.
Now, by <cit.>,

\tensor_{\Gr\spaces_*}\scrC
\simeq \operatorname{RMod}_A(\Gr\scrC);$$

it thus suffices to prove our claim for the case of pointed spaces.

By Example <ref>, together with Proposition
<ref>, we have that $\coCh\spaces_*$ is left-tensored
over $\Gr\spaces_*$ (see Remark <ref>).
By <cit.>, it is thus enough to prove that:

* $\coCh(\spaces_*)$ admits geometric realizations of simplicial

* The action map $\Gr\spaces_* \times \coCh(\spaces_*) \to
\coCh(\spaces_*)$ preserves geometric realizations of simplicial objects.

* There exists an $M\in\coCh(\spaces_*)$ such that
the functor $-\tensor M \colon \Gr\spaces_* \to \coCh(\spaces_*)$
admits a right adjoint, denoted $\underline{\Map}{}_*(M,-)$.

* The functor $\underline{\Map}{}_*(M,-)$ preserves geometric realizations
of simplicial objects.

* The functor $\underline{\Map}{}_*(M,-)$ is conservative.

* For every coherent chain complex $C$ and every graded pointed space
$X$, the map

$$X\tensor \underline{\Map}{}_*(M,C)\tensor M \xto{X\tensor\eps_C}
X\tensor C$$

is adjoint to an equivalence

$$X\tensor \underline{\Map}{}_*(M,C)\to\underline{\Map}{}_*(M,X\tensor C).$$

We take $M$ to be the coherent cochain complex consisting of a copy of
$S^0$ sitting in degree $0$, a copy of $S^0$ sitting in degree $1$,
copies of $\pt$ elsewhere, and
the identity as its only possibly nontrivial differential.
(<ref>) follows from the cocompleteness of
(<ref>) follows from presentability of
$\spaces_*$ and Proposition <ref>.
(<ref>) and
(<ref>) hold for any choice of $M$, and
follow from Proposition <ref>.
Using the explicit description for the right adjoint given in Proposition
<ref>, we get that for our choice of
$M$, $\underline{\Map}{}_*(M,C)^n\simeq\ptMap(M\{n\},C)$
is equivalent to the space of choices of $f$ and $g$'s making the square

\begin{tikzcd}
S^0 \ar[r, "\id"]\ar[d, "f"'] & S^0 \ar[d, "g"] \\
C^n \ar[r, "\partial^n"]& C^{n+1}
\end{tikzcd}

commute; this space is in turn equivalent to $\ptMap(S^0,C^n)\simeq C^n$,
and thus the functor $\underline{\Map}{}_*(M,-)$ is equivalent to the
forgetful functor
$\undch\colon\coCh\spaces_*\to\Gr\spaces_*$ introduced in Lemma
implying (<ref>).

For (<ref>), by definition of adjoint map,
the adjoint map of interest can be factored as

$$ X\tensor\underline{\Map}{}_*(M,C)
\xto{\eta_{X\tensor\underline{\Map}{}_*(M,C)}}
\underline{\Map}{}_*(M,X\tensor \underline{\Map}{}_*(M,C)\tensor M)
\xto{\underline{\Map}{}_*(M,X\tensor\eps_C)}$$

\underline{\Map}{}_*(M,X\tensor C).$$

By a pointwise check, we have that
\underline{\Map}{}_*(M,\eta_{X\tensor C})$,
and $X\tensor\eps_c\simeq\eps_{X\tensor C}$;
by the conservativity of $\underline{\Map}{}_*(M,-)$,
it is sufficient to show that the composite

$$X\tensor C\xto{\eta_{X\tensor C}}
\underline{\Map}{}_*(M,X\tensor C)\tensor M
\xto{\eps_{X\tensor C}} X\tensor C$$

is an equivalence;
but, as the latter is one of the triangle identities, the desired condition

Finally, in the proof of <cit.> is shown that $A$ can be
identified with the internal hom object
thence we get the desired characterization for the graded pointed
space underlying $A$.

As, by <cit.>, the stabilization functor is symmetric
monoidal on presentable , we can stabilize Theorem
<ref>, to obtain an equivalence

for any stable presentable $\scrD$.

Let $\scrC$ be a pointed presentable . Then,
the of filtered objects of $\scrC$ is equivalent to the of right
$\rr{Free}_{\bbE_1}(S^{0,1})$-modules in graded objects:

\operatorname{RMod}_{\rr{Free}_{\bbE_1}(S^{0,1})}(\Gr\scrC).$$

Let us first prove the claim for the case $\scrC\simeq\spaces_*$.
This is basically an adaptation of <cit.> to
the case of pointed spaces.

First of all, notice that, as the functor
$u\colon\Fun(\bbZ\op,\spaces_*)\to\Gr\spaces_* $
given by precomposition with $\bbZ^\delta\to\bbZ\op\colon n\mapsto -n$
is symmetric monoidal, it induces a functor

\theta\colon
\Fun(\bbZ\op,\spaces_*)\simeq\Mod_{\mathbbm{1}_{\text{Day}}}

by Proposition <ref> together with <cit.>, we have
that $u\mathbbm{1}_{\langle\leq 0\rangle}$ is an $\bbE_\infty$-algebra
whose underlying $\bbE_1$-algebra is equivalent to

It is thus sufficient to show that the functor $\theta$ is an equivalence.
Let us start with fully faithfulness.
That is, we want to prove that the maps

$$\phi_{X,Y}\colon\ptMap(X,Y)\to\ptMap(\theta X, \theta Y)$$

are all equivalences.
As the functor $\phi_{-,Y}\colon\Fun(\bbZ\op,\spaces_*)\to
\Fun(\Delta^1,\spaces_*)$ sending a graded pointed space $X$ to the map
$\phi_{X,Y}$ sends colimits to limits, the full subcategory spanned by those
$X$ such that $\phi_{X,Y}$ is an equivalence is a pointed subcategory of
$\Fun(\bbZ\op,\spaces_*)$ closed under colimits.
Thus, as $\Fun(\bbZ\op,\spaces_*)$ is generated under colimits by objects
of the form $\ptYo_n$ for $n\in\bbZ$ (see Proposition
<ref>), it suffices to show that
$\phi_{-,Y}$ is an equivalence for such generators;
this is indeed the case, as for $X\simeq\ptYo_n$ we have
Essential surjectivity now follows from the observation that the essential
image of $\theta$ is closed under colimits, as $\theta$ is fully faithful
and commutes with colimits.

Similarly to the proof of Theorem <ref>,
the claim for general pointed presentable follows from
the properties of Lurie's tensor product.
By <cit.>,

and by <cit.>,

\tensor_{\Gr\spaces_*}\scrC
\simeq \operatorname{RMod}_{\rr{Free}_{\bbE_1}(S^{0,1})}(\Gr\scrC)$$

we thence have the desired claim.

We can describe more explicitly the equivalence of Theorem
Let us first observe that the action of
the free generator of $R_1 \coloneqq \rr{Free}_{\bbE_1}(S^{0,1})$ determines
the action of all other spaces of $R_1$.
By unraveling the definitions, specifying a map
$t\colon S^{0,1}\tensor X\to X$ on a graded object $X$ is equivalent to
specifying maps $X^n\to X^{n+1}$ for all $n\in\bbZ$.
Hence, given an $R_1$-module, the associated filtered object is the one
having as maps between consecutive objects the maps determined by $t$,
and the actions of the $s$-th powers of the generator
$t^s\colon S^{0,s}\tensor X\to X$ correspond to
choices for all the possible $s$-fold composites for the structure
maps of the resulting filtered object (i.e. maps
$X^n\to X^{n+s}$ for all $n>1$). This description makes
also clear the behavior of the functor going in the opposite direction.

§.§ The resolution sequence

We will now move to the construction of the sequence
(<ref>). By Theorem
<ref>, we can take
We will define the first map of the sequence to be the one obtained by
killing the square of the free generator of $R_1$.

By <cit.>, we have
$\left(\rr{Free}_{\bbE_1}(S^{0,1})\right)^n\simeq S^n$ for all $n\geq 0$.
In particular, the inclusion of pointed spaces
$S^{0,2}\to\rr{Free}(S^{0,1})$ induces a map of $\bbE_1$-algebras
We define $R_2$ to be the $\bbE_1$-algebra in graded pointed spaces
obtained as the pushout of $t^2$ along the augmentation map for the
free algebra $\rr{Free}_{\bbE_1}(S^{0,2})$:

\begin{equation}\label{eq-r2}
\begin{tikzcd}
\rr{Free}_{\bbE_1}(S^{0,2})\ar[r]\ar[d,"t^2"']& S^{0,0} \ar[d] \\
\rr{Free}_{\bbE_1}(S^{0,1})\ar[r]& R_2
\end{tikzcd}
\end{equation}

We pick the bottom horizontal map $\rr{Free}_{\bbE_1}(S^{0,1})
\simeq R_1 \to R_2$ to be the first map in the sequence

Notice that, as passing to homology preserves colimits, the pushout
square (<ref>) induces on homology the square of Theorem
<ref> for the case $n=2$.

In Lemma <ref>, we will show that there exists a map
$S^{1,3}\to R_2$, inducing on homology the map
$\mathrm{Free}_{\bbE_1}(r_3)\to \Lambda^{(2)}$ of Theorem <ref>.
Given such a map, we can consider the pushout diagram of $\bbE_1$-algebras

\begin{tikzcd}
\rr{Free}_{\bbE_1}(S^{1,3})\ar[r]\ar[d]& S^{0,0} \ar[d] \\
R_2\ar[r]& R_3
\end{tikzcd}

and pick the bottom horizontal map to be the second map in the sequence

Similarly, in Lemma <ref>,
we will show that there exists a map $S^{2,4}\to R_2$, inducing on homology
the map $\mathrm{Free}_{\bbE_1}(r_4)\to \Lambda^{(3)}$ of Theorem

Given such maps, we can construct the rest of the sequence by induction.

Let us assume we are given a sequence
$R_1\to R_2\to\cdots\to R_{n}$ with $n\geq 3$, such that every
$R_j$ comes equipped with a map $S^{j-1,j+1}\to R_j$
inducing on homology the map
$\mathrm{Free}_{\bbE_1}(r_{j+1})\to \Lambda^{(j)}$ of
Theorem <ref>,
and that each map $R_{j-1}\to R_{j}$ for $2\leq j\leq n$
fits in a commutative square of $\bbE_1$-algebras

\begin{tikzcd}
\rr{Free}_{\bbE_1}(S^{j-2,j})\ar[r]\ar[d]& S^{0,0} \ar[d] \\
R_{j-1}\ar[r]& R_{j}.
\end{tikzcd}

Our goal is to produce inductively
an $\bbE_1$-algebra $R_{n+1}$ and a map
$S^{n,n+2}\to R_{n+1}$ inducing on homology the map
$\mathrm{Free}_{\bbE_1}(r_{n+2})\to \Lambda^{(n+1)}$ of
Theorem <ref>.

Let us denote by $R_{n+1}$ the object obtained pushing out
the map $\rr{Free}_{\bbE_1}(S^{n-1,n+1})\to R_n$ we have by inductive
hypothesis along the augmentation map of the free algebra:

\begin{tikzcd}
\rr{Free}_{\bbE_1}(S^{n-1,n+1})\ar[r]\ar[d]& S^{0,0} \ar[d] \\
R_n\ar[r]& R_{n+1}.
\end{tikzcd}

We pick the bottom map of the above square to be the one for the
sequence (<ref>).
By passing to homology, we get precisely the pushout square
of Theorem <ref>.
By Lemma <ref>, we have an element $r_{n+1}$
of bidegree $(n+1,n-1)$ in
By Hurewicz's Theorem, the cohomology class $r_{n+1}$ determines
in an essentially unique way a map $S^{n,n+2}\to R_{n+1}$, which we will
denote $\langle t\rangle^{n+2}$ and call the
$(n+2)$-fold Toda power of $t$.
This completes the induction step needed to extend the sequence for
$m\geq 4$.

We can now understand the relations between $R_n$-modules for varying $n$:

* It follows from Construction <ref> together with
Remark <ref> that the $R_1$-module associated to
a filtered space $X$ can be given the structure of an $R_2$-module
if and only if all the composites of pairs of consecutive maps of $X$
are nullhomotopic, and specifying the structure of an
$R_2$-module is equivalent to choosing a nullhomotopy for each
such pair.

* It follows from Construction <ref> together with
Remark <ref> that given any $R_{n-1}$-module $C$,
it can be given the structure of an $R_n$-module if and only if
the map $\langle t\rangle^n\colon S^{n-2,n}\to\innmap{}_*(C,C)$ of
Construction <ref> [or,
in the case of $n=3$ (resp. $n=4$), the map
$\langle t \rangle^3$ defined in Construction <ref>
(resp. the map $\langle t\rangle^4$ defined in Construction
is trivial.
By unraveling the definitions, we see that if we denote
by $X$ the filtered space associated to $C$ (obtained by restriction of
scalars along $R_1\to R_{n-1}$), the components of $\langle t\rangle^n$
are maps

for all $m\in\bbZ$; specifying the structure of an $R_n$-module on $C$
is equivalent to choosing nullhomotopies for each such map.

Let $X^0\to X^1\to\cdots\to X^{n}$ be a sequence of pointed spaces and
maps between them, and assume that the filtered pointed space $X$ obtained
by extending the given sequence by zeroes
can be given the structure of an $R_{n-1}$-module.
Let us moreover fix one such structure and denote the resulting object by
We define the n-fold Toda bracket of $C_X$ to be the class

given by the only nontrivial component of the map
$\langle t\rangle^n\colon S^{n-2,n}\to\innmap{}_*(C_X,C_X)$ of Construction
<ref>, or,
in the case of $n=3$ (resp. $n=4$), the map
$\langle t \rangle^3$ defined in Construction <ref>
(resp. the map $\langle t\rangle^4$ defined in Construction

Given any filtered pointed space $X\in\Fun(\bbZ\op,\spaces_*)$, we say
it is a naive cochain complex if its corresponding $R_1$-module
can be given a structure of $R_2$-module. Given a naive cochain complex
$X$, we say that it has uniformly trivial n-fold Toda
brackets if it can be recursively given the structure of an $R_n$-module
after choosing $R_m$-module structures for all $m<n$.

As all pointed are canonically enriched over pointed spaces, the
definition of Toda brackets generalizes in an obvious way to sequences of
maps in an arbitrary pointed .
In particular, it extends to sequences of maps in any stable .

The $\bbE_1$-algebra $A$ of Theorem <ref>
is equivalent to the colimit
$\colim_n R_n$ of the sequence (<ref>);
i.e. for $\scrC$ pointed presentable with a special extremal separator,

$$\coCh(\scrC)\simeq\operatorname{RMod}_{\colim_{\bbZ\op} R_n}

It follows from the resolution sequence (<ref>),
together with the associated sequence in homology, that the underlying
graded space of $\colim_n R_n$ is equivalent to $S^{0,0}\amalg S^{0,1}$, as
is also the underlying space of the $\bbE_1$-algebra $A$ of Theorem
<ref>. As $S^{0,0}\amalg S^{0,1}$ is 0-truncated,
the $\bbE_1$-algebra structures on it are determined by the associative
ring structures on the discrete object $S^{0,0} \amalg S^{0,1}$
(living in the 1-category $\coCh(\mathbf{Top}_*)$).
But, by a direct check, there exists only
one nontrivial associative ring structure on $S^{0,0} \amalg S^{0,1}$.
As $\colim_n R_n$ is augmented over $S^{0,0}$, it cannot have trivial
multiplication; likewise, $A$ cannot be trivial
by Theorem <ref>. Therefore $A$ and $\colim_n R_n$
must be equivalent as $\bbE_1$ algebras.

The above proposition can be informally rephrased by saying that the datum
of a coherent cochain complex is equivalent to the datum of a naive
chain complex together with choices for all the nullhomotopies of pairs
of consecutive maps, and recursively defined choices of nullhomotopies for
all possible $n$-fold Toda brackets, for all $n\geq 3$.

§.§ The resolution sequence for $n\leq 3$

Let $\scrC$ be a pointed . Let $X=(X^n)_{n\in\bbZ}\in\Gr\scrC$.
Given any $s\in\bbZ$ and any $t\in\bbN$ will denote by
$\Omega^{t,s}$ the composite functor
$\Omega_\scrC^t\circ\{-s\} \simeq \{-s\}\circ\Omega_\scrC^t.$
That is,

Notice that, for $S^{0,0}\in\Gr\spaces_*$, we have
$\Omega^{0,s}\left(S^{0,0}\right)\simeq S^{0,-s}$.

Let $X\in\coCh\spaces_*$; we have that:

* For all $t\in\bbN$, $s\in\bbZ$,

$$\innmap{}_*(S^{t,s},X)\simeq(\Omega^t X)\{-s\}\simeq \Omega^{t,s}X;$$

* For all $t\in\bbN$, $s\in\bbZ$,

$$S^{t,s}\tensor_{\text{Day}}X \simeq (\Sigma^t X)\{s\}.$$

The structure map $\rr{Free}_{\bbE_1}(S^{0,1})\to R_2$ defined in
Construction <ref> determines a
degree $1$ square-zero element
$t\in(\pi_0(R_2))^1$, whose action
induces the following diagram of right $R_2$-modules:

\begin{tikzcd}
R_2 \ar[r, "t"] \ar[rr, bend right=40, "0"']
\ar[rr, phantom, bend right=20, "\rotatebox{90}{$\Rightarrow$}"]
& \Omega^{0,1}R_2 \ar[r, "t"]
& \Omega^{0,2}R_2 \ar[r, "t"]
& \Omega^{0,3}R_2. \ar[from=ll, bend left=40, "0"]
\ar[from=ll, phantom, bend left=20, "\rotatebox{90}{$\Leftarrow$}"]
\end{tikzcd}

In the situation of Remark <ref>,
let us denote by $F_2$ the fiber (in $R_2$-modules) of multiplication
by $t$, considered as a map $R_2\to\Omega^{0,1}R_2$.
Choosing a nullhomotopy $\alpha\colon 0\Rightarrow t^2$
is equivalent to chosing a factorization of $t$
through $\Omega^{0,1}F_2$.
By the following diagram (where all squares are Cartesian)

\begin{tikzcd}
R_2 \ar[r] \ar[rd, dotted, "\langle t\rangle^3"']
& \Omega^{0,1}F_2 \ar[r]\ar[d]
& \Omega^{0,1}R_2 \ar[d]\ar[from=ll, bend left=40, "t"]& \\
& \Omega^{0,3} R_2\ar[r]\ar[d]
& \Omega^{0,2}F_2\ar[d]\ar[r] & \pt\ar[d] \\
& \pt \ar[r]& \Omega^{0,2}R_2\ar[r]& \Omega^{0,3}R_2 \\
\end{tikzcd}

we see that there exists a map
$R_2\to\Omega^{1,3}R_2$ induced by such a factorization.
We call this map the 3-fold Toda power of $t$,
and denote it by $\langle t \rangle^3$.
With a little abuse of terminology, we will reserve the same name
for the map $S^{1,3}\to\innmap{}_*(R_2,R_2)$ obtained by adjoining over

\cfrac{R_2\to\innmap{}_*(S^{1,3},R_2)}
{\cfrac{R_2\tensor S^{1,3}\to R_2}

As $R_2$ is a free $R_2$-module of rank $1$, we have that
$\innmap{}_*(R_2,R_2)\simeq R_2$, hence the triple Toda power
can be equivalently expressed as a map $S^{1,3}\to R_2$.

The map $S^{1,3}\to R_2$ constructed above induces on homology the map
$\rr{Free}_{\bbE_1}(r_3)\to \Lambda^{(2)}$ defined in
Theorem <ref>.

Upon passing to homology degree-wise, using that
$H(R_2)\simeq \Lambda^{(2)}$ in the notation of Theorem
and that, by Remark <ref>, we can use
$\wt{P}_2 \coloneqq
\bbZ\langle e_1,e_2\rangle/(\partial e_2 = e_1^2)\simeq \Lambda^{(2)}$
as a representative
for the homology in the $1$-category of dg-modules,
the defining diagram can be represented by the strictly commuting
diagram of dg-modules

\begin{tikzcd}[ampersand replacement =\&]
\wt P_2 \ar[r, "\binom{e_1}{- e_2}"] \ar[rd]
\& \Sigma^{0,-1}\wt P_2 \oplus \Sigma^{-1,-2}\wt P_2
\ar{r}{\begin{pmatrix} 1 & 0 \end{pmatrix}}
\ar{d}{\begin{pmatrix}e_2 & e_1\end{pmatrix}}
\& \Sigma^{0,-1}\wt P_2 \ar[d, "\binom{e_1}{- e_2}"]\& \\[3.5em]
\& \Sigma^{-1,-3}\wt P_2\ar[r,"\binom{0}{1}"]\ar[d]
\& \Sigma^{0,-2}\wt P_2\oplus\Sigma^{-1,-3}\wt P_2
\ar{d}{\begin{pmatrix}1 & 0\end{pmatrix}}\ar[r] \& 0\ar[d] \\[3.5em]
\& 0 \ar[r]\& \Sigma^{0,-2}\wt P_2\ar[r,"e_1"]\& \Sigma^{0,-3}\wt P_2 \\
\end{tikzcd}

(where the negative signs are due to the direction of the homotopy,
going from $0$ to $e_2$).
Thus, the diagonal map is given by the action of $e_1e_2-e_2e_1$,
which by Remark <ref>
represents exactly the homology class
pointed by $r_3$ in the defining diagram for $\Lambda^{(3)}$.

Let us now consider the pushout square

\begin{tikzcd}
\rr{Free}_{\bbE_1}(S^{1,3})\ar[r]\ar[d]& S^{0,0} \ar[d] \\
R_2\ar[r]& R_3
\end{tikzcd}

where the vertical map is the one induced by the 3-fold Toda power
costructed above.
We pick the bottom horizontal map to be the second map in the sequence

By the commutativity of the following diagram of $R_3$-modules
(where we know that $\langle t\rangle^3\simeq 0$ by the defining pushout for

\begin{tikzcd}
R_3 \ar[r] \ar[rd, "t"']
& \Omega^{0,1}F_3 \ar[r]\ar[d]
& \Omega^{1,3}R_3 \ar[d]\ar[from=ll, bend left=40, "0"] \\
& \Omega^{0,1}R_3\ar[r]
& \Omega^{0,2}F_3 \\
\end{tikzcd}

we see that $t$ factors through
If we denote by $G_3\coloneqq\fib(\Omega^{0,1}F_3\to \Omega^{1,3} R_3)$,
we have the following diagram (where all squares are Cartesian)

\begin{tikzcd}
R_3 \ar[r] \ar[rd, dotted, "\langle t\rangle^4"']
& L_3 \ar[r]\ar[d]
& \Omega^{0,1}R_3 \ar[d]\ar[from=ll, bend left=40, "t"]& \\
& \Omega^{2,4} R_3\ar[r]\ar[d]
& \Omega^{0,1} G_3\ar[d]\ar[r] & \pt\ar[d] \\
& \pt \ar[r]& \Omega^{0,2}F_3\ar[r]& \Omega^{1,4} R_3 \\
\end{tikzcd}

and we define the composite

$$R_3\to L_3\to \Omega^{2,4}R_3$$
to be the
4-fold Toda power of $t$, denoted $\langle t\rangle^4$.
As for the 3-fold case, the datum of a 4-fold Toda bracket
uniquely determines a map $S^{2,4}\to R_3$ that we will refer to by the
same name.

The map $S^{2,4}\to R_3$ constructed above induces on homology the map
$\rr{Free}_{\bbE_1}(r_4)\to \Lambda^{(3)}$ defined in
Theorem <ref>.

As in Lemma <ref>, we can consider the diagram
of dg-modules obtained from the defining diagram of the $4$-fold Toda
power by passing to homology:

\begin{tikzcd}[ampersand replacement =\&, row sep=4em, column sep=1.3em]
\wt{P}_3 \ar{d}{\begin{pmatrix}e_1 \\ - e_2 \\ e_3\end{pmatrix}}
\ar[dd, bend right=80] \&\& \\
\Sigma^{0,-1}\wt P_3 \oplus \Sigma^{0,-2}\wt P_2\oplus\Sigma^{-1,-3}\wt P_2
\ar{r}{\begin{pmatrix}1 & 0 & 0\end{pmatrix}}
\ar{d}{\begin{pmatrix}e_3 & e_2 & e_1\end{pmatrix}}
\& \Sigma^{0,-1}\wt P_3
\ar{d}{\begin{pmatrix}e_1 \\ - e_2 \\ -e_3\end{pmatrix}}\& \\
\Sigma^{-2,-4} \wt{P}_3\ar{r}[swap]{\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}}\ar[d]
\& \Sigma^{0,-2}\wt P_2\oplus\Sigma^{-1,-3}\wt P_2 \oplus \Sigma^{-2,-4} \wt P_3
\ar{d}{\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0\end{pmatrix}}\ar[r] \& 0\ar[d] \\
0 \ar[r]
\&\Sigma^{0,-2}\wt P_2\oplus\Sigma^{-1,-3}\wt P_2
\ar{r}{\begin{pmatrix}e_2 & e_1\end{pmatrix}}\& \Sigma^{-1,-4} \wt P_3 \\
\end{tikzcd}

(again, the signs are consistent with the choice of having the nullhomotopies
starting at $0$)
and we see that the $4$-fold Toda power induces on homology the map
given by the action of

$$e_1e_3 - e_2^2 + e_3e_1$$

which by Remark <ref>
represents exactly the homology class
pointed by $r_4$ in the defining diagram for $\Lambda^{(4)}$.

§ TOTAL HOMOLOGY AND K-DECOMPOSITIONS

The Postnikov tower construction makes clear how a spectrum is
uniquely determined by its homotopy groups together with its k-invariants.
In particular, any spectrum $X$ uniquely determines co/fiber sequences of the

for each $n\in\bbZ$, and it's easy to verify that the $k_n$'s are such that

$$k_{n}\circ k_{n-1}[-1]\simeq 0$$

for all $n\in\bbZ$.
It is widely known among the experts
that the above observation admits a converse precisely when
the nullhomotopies for such pairwise compositions of
“truncated k-invariants” are suitably compatible
(see e.g. <cit.>, where an instance of this fact
is discussed for objects with finite Postnikov filtrations in the setting of
triangulated categories);
to be precise, a spectrum is uniquely determined by the datum of its homotopy
groups together with a collection of maps
$\{k_{n}\colon\pi_{n}X[n]\to\pi_{n+1}X[n+2]\}_{n\in\bbZ}$ such
that all pairs of composable maps compose to a nullhomotopic one,
and all the possible higher Toda brackets are trivial.

In this section we provide a rigorous formulation of the idea discussed above
using the language of coherent cochain complexes developed in the previous
sections, generalizing it to the context of stable satisfying some
mild hypothesis[that is, admitting sequential limits, sequential
colimits and equipped with a right complete t-structure]
and general homotopy objects in (ordinary) Abelian categories.
The process of recostructing an object from its decomposition in homotopy
objects and maps between their shifts will be given by the total homology
construction (see <ref>), a concept further investigated in the next
section, in relation to the spectral sequence generated by a coherent cochain

Let us begin by constructing an that encapsulates
the data needed to reconstruct the objects from their homotopy groups and their

Let $\scrC$ be a stable equipped with a
We will denote by $\Ch_{\scrC}^{+}(\scrC^\heartsuit)$ the full
subcategory of $\coCh\scrC$ spanned by those objects $C$ such that
$C^n\in\scrC^\heartsuit[2n]$ for all $n\in\bbZ$, and refer to it as the
of degenerate cochain complexes of $\scrC$.

The notation $\Ch_\scrC^+(\scrC^\heartsuit)$ is slightly abusive, as the
object depends also on the t-structure $\scrC$ is equipped with.

We can now construct a functor that incarnates the idea discussed in the
introduction of this section.

Let $\scrC$ be a stable equipped with a right complete t-structure.
By the definition of right completeness, there exists an equivalence

\scrC\simeq\lim\left(\cdots\to
\left(\scrC\right)_{\geq n}\xto{\tau_{\geq n+1}}
\left(\scrC\right)_{\geq n+1}\to\cdots\right).

By the right completeness of $\cFild\scrC$ (see Remark
<ref>, together with Theorem
<ref>), the fully faithful
inclusion $\scrC\hookrightarrow\Fild(\scrC)$ factors through
the subcategory of complete objects:

\wtrunc{\istar}\colon\scrC\hookrightarrow\cFild(\scrC).

Let $\scrC$ be a stable presentable equipped with a right complete
Notice that, by Remark <ref>,
the composite $\caush\circ\wtrunc\istar\colon\scrC\to\coCh(\scrC)$
factors through the inclusion
We define the k-decomposition functor

C   ↦(⋯→π_n-1X[2n-2]→π_nX[2n]→π_n+1[2n+2]→⋯)

to be such factorization of
$\caush\circ\wtrunc{\istar}$ through $\Ch_\scrC^+(\scrC^\heartsuit)$.
We refer to the differentials of the complex $\kinv C$ as the
k-invariants of $C$.

Let $R$ be an ordinary commutative ring, and let $\rrD(R)$ denote its
derived category (considered as a stable equipped with the usual
Given any object $X\in\rrD(R)$, its k-decomposition $\kinv X$ is given
by the coherent cochain complex

$$\cdots\to (H^{-n+1}X)[2n-2]\to(H^{-n} X)[2n]\to(H^{-n-1})X[2n+2]\to\cdots$$

with $H^{-n}X \in \Ab$.

Given any $X\in\Sp$, its k-decomposition $\kinv X$ is given by the
coherent cochain complex

\to(\pi_{n+1}X)[2n+2]\to\cdots$$

with $\pi_n X \in \Ab$.
The equivalence $\Ch_\Sp^+(\Ab)\simeq\Sp$ can be interpreted as saying
that any spectrum can be expressed as a coherent chain complex
of (suitably shifted) Abelian groups, and, viceversa, one can construct
a spectrum from the datum of objects in $\bigoplus_\bbZ \Ab[2n]$ and maps
between consecutive objects $A_n[2n]\to A_{n+1}[2n+2]$ such that all possible
Toda brackets between them are trivial.

The previous two examples make clear how, from the point of view of
coherent chain complexes, the only difference between an object in
$\rrD(\bbZ)$ and a spectrum lies in the $\bbZ$-linearity of the k-invariants:

Of course, this is just a consequence of a theorem of Shipley's
(see [30]) showing that the derived $\rrD(R)$ of a discrete
commutative ring $R$ is equivalent to the of $HR$-modules

We now turn our attention to the functor inverse to $\kinv$. It turns out that
the process reconstructing an object of $\scrC$ from its k-decomposition
is a particular case of the more general construction assigning to every
coherent cochain complex $C$ the object underlying its piled-up filtration
$\imp C$.

Let $\scrC$ be a stable with sequential limits and sequential colimits.
We define the total homology functor to be the composite functor

C   ↦(C)^-∞.

Let $\scrC$ be a stable with sequential limits and sequential colimits,
equipped with a right complete t-structure.
The restriction of $\tothom$ to $\Ch_\scrC^+(\scrC^\heartsuit)$
induces an equivalence

whose inverse is given by the k-decomposition functor of Definition

By the definition of right completeness, there exists an equivalence

\scrC\simeq\lim\left(\cdots\to
\left(\scrC\right)_{\geq n}\xto{\tau_{\geq n+1}}
\left(\scrC\right)_{\geq n+1}\to\cdots\right).

By the discussion in [24] right before Proposition 1.2.1.17, we can
give an alternative description of the right hand side as the full
subcategory of $\Fild(\scrC)$ spanned by those filtered objects for

* For each $n\in\bbZ$, $F^n\in\scrC_{\geq n}$;

* For each $m\geq n$, the associated map $F^m\to F^n$ induces an
equivalence $F^m\xto{\sim}\wtrunc{m}F^n$.

We first notice that, as long as (<ref>) holds,
we can replace (<ref>)
with the weaker assumption that

($\spadesuit$) For each $n\in\bbZ$, the associated map
$F^{n+1}\to F^n$ induces an equivalence

By induction, assume that for a fixed $n$ and some $m\geq n$
$F^m\to F^n$ induces an equivalence $F^m\to \wtrunc{m} F^n$. We want to prove
that $F^{m+1}\to F^n$ induces an equivalence $F^{m+1}\to\wtrunc{m+1}F^n$.
By ($\spadesuit$), $F^{m+1}\to F^m$
induces an equivalence $F^{m+1}\to\wtrunc{m+1}F^m$, and by our inductive

is an equivalence, hence the composite map $F^{m+1}\to\wtrunc{m+1}F^m\to
\wtrunc{m+1}F^n$
(which, by (<ref>), is precisely the image
under $\wtrunc{m+1}$ of the composite $F^{m+1}\to F^m\to F^n$)
is an equivalence as desired.

As $\wtrunc{\istar}$ is fully faithful, it induces an equivalence with its
essential image in $\cFild(\scrC)$ (see Remark <ref>).
As $\caush$ is an equivalence, it is sufficient to check that
By the definition of $\Ch_\scrC^+\scrC^\heartsuit$ and Lemma
<ref>, all objects in
$\imp\left(\Ch_\scrC^+\scrC^\heartsuit\right)$ satisfy
(<ref>) and
On the other hand, again by Lemma <ref>,
any filtered object satisfying conditions
(<ref>) and
is such that its shelled complex lies
in $\Ch_\scrC^+\scrC^\heartsuit$. Hence,
$\imp\left(\Ch_\scrC^+\scrC^\heartsuit\right)$ and
$\wtrunc{\istar}\scrC$ are two full subcategories of $\cFild\scrC$ spanned
by the same objects.
To identify its inverse, it now suffices to notice that

∘   ≃∘.25em∘∘

as desired.

We now turn our attention to the behavior of $\tothom$ on objects coming from
the (ordinary) category of chain complexes in the heart.

Let $\scrC$ be as in Theorem <ref>.
Then the t-structure homotopy objects of the total homology of an ordinary
chain complex in $\oC \in \coCh(\scrC^\heartsuit)$ (considered as an
element of $(\coCh\scrC)^\heartsuit$) are given by the cohomology of

$$\pi_{-n}\tothom\oC\simeq \rrH^n\oC.$$

By Proposition <ref>,
together with the hypothesis that $\oC$ lies in
$\coCh(\scrC^\heartsuit)$, we have that
$\gr^n\imp\oC\simeq \oC^n[-n]$; in particular, (as $\imp\oC$
is complete) we see that for any integer $n$,
$\imp\oC^n$ is $(-n)$-coconnective, and that

\begin{equation}\label{eq-pi-tothom}
\pi_m\left(\imp\oC^n\right) \cong
\pi_m\left(\imp\oC^{n-1}\right) \text{ for all } m\leq-n.
\end{equation}

By Remark <ref>, we have a co/fiber sequence

$$\oC^{n+1}[-n-1]\to \imp \oC^n/\imp \oC^{n+2} \to \oC^n[-n]$$

whose associated long exact sequence on homotopy objects lets us identify

\pi_m\left(\imp \oC^n/\imp \oC^{n+2}\right)\cong
\begin{cases}
\ker(d^n_{\oC}) \quad &\text{ for } m=-n;\\
\coker(d^n_{\oC}) \quad &\text{ for } m=-n-1;\\
0 \quad &\text{ else.}
\end{cases}

Again by Remark <ref>,
we have a co/fiber sequence

\imp \oC^n/\imp \oC^{n+2}\to\imp \oC^{n-1}/\imp \oC^{n+2}\to \oC^{n-1}[-n+1]

whose associated long exact sequence starting at $-n+1$ looks as follows

\cdots 0 \to \pi_{-n+1}\left(\imp \oC^{n-1}/\imp \oC^{n+2}\right)
\to \oC^{n-1} \to \ker(d^n) \to
\pi_{-n}\left(\imp \oC^{n-1}/\imp \oC^{n+2}\right) \to 0 \cdots

letting us identify $\pi_{-n}\left(\imp \oC^{n-1}/\imp \oC^{n+2}\right)$ as

$$\coker \left(\oC^{n-1}\xto{d^{n-1}} \ker(d^n)\right) \cong \rrH^n\oC.$$

As $\imp \oC^{n+2}$ is $(-n-2)$-coconnective, the long exact sequence
associated to the co/fiber sequence

$$\imp \oC^{n+2}\to\imp\oC^{n-1}\to\imp \oC^{n-1}/\imp \oC^{n+2}$$

shows that

\text{ for }m\geq-n-1$$

and thus in particular that $\pi_{-n}\left(\imp\oC^{n-1}\right)
\cong\rrH^n\oC$.
Finally, (<ref>) implies the desired result.

In the special case of $\scrC=\Sp$, the proof of Proposition
<ref> shows that $\imp\oC$ for an ordinary chain
complex of Abelian groups $\oC$ gives precisely the tower obtained through
the “brutal truncations”[sometimes referred to also as the
“stupid truncation” in the literature] of the complex $\oC$. That is,

$$\imp\oC^n\simeq H(\tau^{\geq n}\oC),$$

where $\tau^{\geq n}$ here denotes the brutal truncation (1-)functor and $H$
denotes the Eilenberg–MacLane ($\infty$-)functor $\coCh(\Ab)\to\Sp$.

In particular, we have that in the case of spectra the
total homology functor $\tothom$ restricted to $\coCh(\Ab)\subset\coCh(\Sp)$
coincides with the Eilenberg–MacLane functor.

We conclude the section with the following variant of Definition

Let $\scrC$ be a stable presentable equipped with a right separated
t-structure, and let $q\in\bbN$ be fixed.
Let $\Ch_{\scrC}^{+q}(\scrC^\heartsuit)$ denote the full
subcategory of $\coCh\scrC$ spanned by those objects $C$ such that
$C^n\in\scrC^\heartsuit[n(q+1)]$ for all $n\in\bbZ$.
Consider the full subcategory $\scrC_{q\text{-periodic}}
\subset\scrC$
spanned by the objects $X$ such that the t-structure homotopy objects
$\pi_n X$ are isomorphic to $0$ for $n$ not a multiple
of $q$:

\scrC_{q\text{-periodic}}\coloneqq\{X \ | \ \pi_n X \cong 0 \text { for }
n\not\equiv 0 \mod q\}.

We define $\kinv_q$ to be the factorization of
(where $\wtrunc{q\istar}$ denotes the sub-filtration of $\wtrunc{\istar}$
obtained by skipping all the stages of the Whitehead tower that are
not multiples of $q$)
through the inclusion
and refer to it as the
$q$-periodic k-decomposition functor.
Similarly to what happens for $\kinv$, the functor $\kinv_q$ induces
an equivalence $\scrC_{q\text{-periodic}}\simeq\Ch_\scrC^{+q}

We learned about the following example from Achim Krause.

As an instance of Variant <ref>, we can consider
the $2(p^n-1)$-periodic k-decomposition of the $n$-th Morava
K-theory ($n\geq1$) spectrum, for some fixed prime $p$.

the coherent cochain complex $\kinv_{2(p^n-1)}K(n)$ looks as follows

\bbF_p[(m+1)(p^n-1)]\to\cdots$$

(where $\bbF_p[m(p^n-1)]$ sits in degree $m$), and the differentials are
given by suitable shifts of the $n$-th Milnor primitive for the mod $p$
Steenrod algebra:

$$\partial^m\simeq Q_n[m]\in\calA_p.$$

§ THE SPECTRAL SEQUENCE ASSOCIATED TO A COHERENT COCHAIN COMPLEX

In this section, we discuss how coherent cochain complexes give rise to
spectral sequences.
We have the following result.

Let $\scrC$ be a stable with sequential limits and sequential colimits,
equipped with a right complete t-structure.
Then, every coherent cochain complex $C\in\coCh\scrC$ generates a spectral

whose $E_1$ page is given by the homotopy groups of the components of
$C$ and whose $E_1$ differentials

are obtained from the coherent differentials of $C$ by passing to
When the spectral sequence collapses at a finite stage, it converges
strongly to the homotopy groups of the total homology of $C$

$$E_1^{i,j}\cong\pi_{-j}C^i \Longrightarrow \pi_{-i-j}\tothom C.$$

We defer the proof to later in this section.
Of course, the above theorem follows at once from Theorem
<ref> together with Theorem <ref>