nLab spectral sequence of a filtered stable homotopy type

Idea

Each filtering on an object $X$ in a suitable stable (∞,1)-category $\mathcal{C}$ (a stable homotopy type $X$ such as a spectrum object, in particular possibly a chain complex) induces a spectral sequence whose first page consists of the homotopy groups of the homotopy cofibers of the filtering and which under suitable conditions converges to the homotopy groups of the total object $X$.

This is a generalization of the traditional spectral sequence of a filtered complex to which it reduces for $\mathcal{C} = Ch_\bullet(\mathcal{A})$ an (∞,1)-category of chain complexes presented in the projective model structure on chain complexes. Therefore the whole zoo of traditional spectral sequences that in turn are special cases of that of a filtered complex (see at spectral sequence – Examples) is all subsumed by the concept of spectral sequence of a filtered stable homotopy type.

Moreover, by applying general (∞,1)-categorical notion to naturally arising towers (such as the Whitehead tower, the chromatic tower) it naturally produces more specialized spectral sequences (such as the Atiyah-Hirzebruch spectral sequence, the chromatic spectral sequence, etc.). Specifically, applied to a coskeleton tower of a dual Cech nerve of an E-∞ algebra $E$ it naturally produces the $E$-Adams spectral sequence. See the discussion of the Examples below.

Therefore in (∞,1)-category theory one finds a lucky coincidence of historical terminology: spectral sequences are essentially sequences of spectra, when considered on homotopy groups.

The general construction can be summarized as follows:

Any homological functor

$$\mathcal{C}\to\mathcal{A}$$

from a stable (∞,1)-category to an abelian category induces a functor

$$Filt(\mathcal{C}) \to SpSeq(\mathcal{A})$$

from the stable (∞,1)-category of filtered objects in $\mathcal{C}$ to the abelian category of bigraded spectral sequences in $\mathcal{A}$.

Definition

Let throughout $\mathcal{C}$ be a stable (∞,1)-category, $\mathcal{A}$ an abelian category, and $\pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A}$ a homological functor on $\mathcal{C}$, i.e., a functor that transforms every cofiber sequence

$$X\to Y\to Z\to \Sigma X$$

in $\mathcal{C}$ into a long exact sequence

$$\dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots$$

in $\mathcal{A}$. We write $\pi_n=\pi\circ \Sigma^{-n}$.

Example

• $\mathcal{C}$ is arbitrary, $\mathcal{A}$ is the category of abelian groups and $\pi$ is taking the 0th homotopy group $\pi_0 \mathcal{C}(S,-)$ of the mapping spectrum out of some object $S\in\mathcal{C}$

• $\mathcal{C}$ is equipped with a t-structure, $\mathcal{A}$ is the heart of the t-structure, and $\pi$ is the canonical functor.

• $\mathcal{C} = D(\mathcal{A})$ is the derived category of the abelian category $\mathcal{A}$ and $\pi=H_0$ is the degree-0 chain homology functor.

• Any of the above with $\mathcal{C}$ and $\mathcal{A}$ replaced by their opposite categories.

(Co-)Filtered objects and their (co-)chain complexes

Definition

A filtered object in an (∞,1)-category in $\mathcal{C}$ is an object $X \in \mathcal{C}$ together with a sequential diagram $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$

$$\cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots$$

and an equivalence

$$X \coloneqq \underset{\rightarrow}{\lim}_n X_n$$

between $X$ and the (∞,1)-colimit of this sequence. (The sequence itself is the filtering on $X$.)

Dually, a co-filtered object in an (∞,1)-category in $\mathcal{C}$ is an object $X \in \mathcal{C}$ together with a sequential diagram $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$

$$\cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots$$

and an equivalence

$$X \coloneqq \underset{\leftarrow}{\lim}_n X_n$$

between $X$ and the (∞,1)-limit of this sequence. (The sequence itself is the co-filtering on $X$.)

This appears as (Higher Algebra, def. 1.2.2.9). The notions are equivalent under replacing $\mathcal{C}$ by its opposite category $\mathcal{C}^{op}$.

Remark

If $\mathcal{C}$ is presented by a sufficiently nice model category $C$ (for instance a combinatorial model category), then (∞,1)-colimits in $\mathcal{C}$ are computed by homotopy colimits in $C$. These in turn are computed as ordinary colimits in $C$ over a cofibrant diagram in the projective model structure on functors.

Specifically, as discussed at homotopy limit – Examples – Over sequential diagrams a cofibrant resolution of a sequential diagram $(\mathbb{N}, \leq) \to C$ is a sequential diagram all whose whose objects are cofibrant and all whose morphisms are cofibrations in $C$

$$\emptyset \stackrel{cof}{\to} X_0 \stackrel{cof}{\to} \cdots \to X^C_{n-1} \stackrel{cof}{\to} X_{n}^C \stackrel{cof}{\to} X_{n+1}^C \to \cdots \,,$$

where $X_n^C \in C$ denotes an object in the model category presenting the given object $X_n \in \mathcal{C}$.

Moreover, in many model categories that appear in practice the cofibrations are in particular monomorphisms, this is the case in particular in a projective model structure on chain complexes. In these cases then a filtering on an object $X \in \mathcal{C}$ in the abstract sense of (∞,1)-categories is presented by a filtered object $X^C \in C$ in the sense of plain category theory.

The intrinsic definition makes manifest however that the monomorphism-aspect here is just a means of a presentation of the filtering and not an intrinsic aspect of the homotopy theory.

Definition

Let $I$ be a linearly ordered set. An $I$-chain complex in a stable (∞,1)-category $\mathcal{C}$ is an (∞,1)-functor

$$F \;\colon\; I^{\Delta[1]} \longrightarrow \mathcal{C}$$

from the subposet of $I \times I$ on pairs of elements $i \leq j$, such that

1. for each $n \in I$, $F(n,n) \simeq 0$ is the zero object;

2. for all $i \leq j \leq k$ the induced diagram

   $$\array{ & F(i,j) &\longrightarrow& F(i,k) \\ & \downarrow && \downarrow \\ 0 \simeq & F(j,j) &\longrightarrow& F(j,k) }$$

   is a homotopy pushout square (hence equivalently, by stability, a homotopy pullback).

Write

$$Gap(I,\mathcal{C}) \hookrightarrow Func(I^{\Delta[1]}, \mathcal{C})$$

for the full sub-(∞,1)-category of diagrams satisfying these conditions.

This is Higher Algebra, def. 1.2.2.2.

Remark

The conditions in def. imply by the pasting law that also all squares

$$\array{ F(i,k) &\longrightarrow& F(i,l) \\ \downarrow && \downarrow \\ F(j,k) &\longrightarrow& F(j,l) }$$

for all $i \leq k$ and $k \leq l$ are homotopy pushout squares.

Definition

Given a $\mathbb{Z}$-chain complex $F$ in $\mathcal{C}$ as in def. , define a sequential diagram in the (triangulated) homotopy category $Ho(\mathcal{C})$ of $\mathcal{C}$

$$C_\bullet \;\colon\; (\mathbb{Z}, \leq)^{op} \longrightarrow Ho(\mathcal{C})$$

by setting

$$C_n \coloneqq \Sigma^{-n} F(n-1,n) \in Ho(\mathcal{C})$$

and taking

$$d_n \coloneqq \Sigma^{-n} \delta_n \;\colon\; C_n \longrightarrow C_{n-1}$$

to be the $n$-fold de-suspension of the connecting homomorphisms of the defining homotopy fiber sequences

$$F(n-1,n) \to F(n-1, n+1) \to F(n,n+1) \,,$$

hence the $(n+1)$-fold de-suspension of the morphism $\delta_n$ in the following pasting of homotopy pushouts

$$\array{ F(n-1,n) &\longrightarrow& F(n-1,n+1) &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\longrightarrow& F(n,n+1) &\stackrel{\delta_n}{\longrightarrow}& \Sigma F(n-1,n) }$$

where the total outer homotopy pushout exhibits the suspension of $F(n-1,n)$, by the pasting law.

Proposition

The sequence $C_\bullet$ in def. is a chain complex in that the $d_\bullet$ are differentials, hence in that for all $n \in \mathbb{Z}$ we have that the composite

$$d_n \circ d_{n+1} = 0$$

is the zero morphism in the triangulated category $Ho(\mathcal{C})$.

(Higher Algebra, remark 1.2.2.3)

Proof

Consider the pasting diagram

$$\array{ F(n-2,n) &\longrightarrow& F(n-2,n+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow && \downarrow \\ F(n-1,n) &\longrightarrow& F(n-1,n+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow &(c)& \downarrow \\ 0 &\longrightarrow& F(n,n+1) &\underset{\delta_n}{\longrightarrow}& \Sigma F(n-1,n) }$$

where the squares labeled “c” are (co-)cartesian (homotopy pushouts) ( by def. and by remark and ). Notice that the homotopy pushout of the outermost span gives the suspension

$$\array{ F(n-2,n) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow \\ 0 &\longrightarrow& \Sigma F(n-2,n) } \,.$$

Therefore we have two paths of morphisms of span diagrams, the first is

$$\left( \array{ F(n-2,n) &\to& F(n-2,n+1) \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-2,n) &\to& 0 \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-1,n) &\to& 0 \\ \downarrow \\ 0 } \right)$$

which gives on homotopy pushouts

$$F(n,n+1) \longrightarrow \Sigma F(n-2,n) \longrightarrow \Sigma F(n-1,n)$$

and the second is

$$\left( \array{ F(n-2,n) &\to& F(n-2,n+1) \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-1,n) &\to& F(n-1,n+1) \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-1,n) &\to& 0 \\ \downarrow \\ 0 } \right)$$

which on homotopy pushouts is

$$F(n,n+1) \stackrel{\simeq}{\longrightarrow} F(n,n+1) \stackrel{\delta_n}{\longrightarrow} \Sigma F(n-1,n)$$

(all by the pasting law). By the commutativity of the original pasting diagram these two paths are equivalent. Therefore on homotopy pushouts this exhibits a factorization of $\delta_n$ through $\Sigma F(n-2,n)$:

$$\array{ F(n,n+1) &\longrightarrow& \Sigma F(n-2,n) \\ & {}_{\mathllap{\delta_n}}\searrow & \downarrow \\ && \Sigma F(n-1,n) } \,.$$

Pasting this to the homotopy pushout that defines $\Sigma \delta_{n-1}$

$$\array{ F(n,n+1) &\longrightarrow& \Sigma F(n-2,n) &\longrightarrow& 0 \\ & {}_{\mathllap{\delta_n}}\searrow & \downarrow &(c)& \downarrow \\ && \Sigma F(n-1,n) &\underset{\Sigma \delta_{n-1}}{\longrightarrow}& \Sigma^2 F(n-2,n-1) }$$

and then suspending the result $n$ times yields a diagram that exhibits a null-homotopy

$$d_{n-1} \circ d_n \simeq 0$$

in $\mathcal{C}$.

The following proposition observes that the $\mathbb{Z}$-chain complexes of def. are, despite the explict appearance of square diagrams, equivalently already determined by a sequential diagram.

Proposition

Consider the inclusion of posets

$$(\mathbb{Z}, \leq) \to (\mathbb{Z}, \leq)^{\Delta[1]}$$

given by

$$n \mapsto (-\infty, n) \,.$$

The induced (∞,1)-functor

$$Func((\mathbb{Z}\cup \{-\infty\}, \leq)^{\Delta[1]} , \mathcal{C}) \longrightarrow Func((\mathbb{Z}, \leq), \mathcal{C})$$

restricts to an equivalence between the (∞,1)-category $Gap(\mathbb{Z},\mathcal{C})$ of $\mathbb{Z}\cup \{\infty\}$-chain complexes in $\mathcal{C}$ (def. ) and that of filtered objects in $\mathcal{C}$ (def. ). The equivalence is given by left and right (∞,1)-Kan extension.

This is Higher Algebra, lemma 1.2.2.4.

Remark

The inverse functor can be described informally as follows:

given a filtered object $X_\bullet$, the associated chain complex $X(\bullet,\bullet)$ is given by taking each entry $X(n,n+r)$ to be given by the homotopy cofiber of $X_n \to X_{n+r}$

$$X(n, n+r) = \operatorname{cofib}(X_n\to X_{n+r})$$

because that makes the squares

$$\array{ & X(-\infty,n) &\longrightarrow& X(-\infty,n+r) \\ & \downarrow && \downarrow \\ 0 \simeq & X(n, n) &\longrightarrow& X(n,n+r) }$$

be homotopy pushout squares.

Spectral sequence of a filtered object

We discuss now how in the presence of sequential colimits, every filtered object induces a spectral sequence which converges to its homotopy groups, equipped with the induced filtering. The discussion for co-filtered objects is formally dual, but also spelled out below, for reference.

Remark

Let $X_\bullet$ be a filtered object in the sense of def. . Write $X(\bullet,\bullet)$ for the corresponding $\mathbb{Z}$-complex, according to prop. . Then for all $i \leq j \leq k$ there is a long exact sequence of homotopy groups in $\mathcal{A}$ of the form

$$\cdots \to \pi_n X(i,j) \to \pi_n X(i,k) \to \pi_n X(j,k) \to \pi_{n-1}X(i,j) \to \cdots \,.$$

Definition

Define for $p,q \in \mathbb{Z}$ and $r \geq 1$ an object $E_r^{p,q} \in \mathcal{A}$ by the canonical epi-mono factorization

$$\pi_{p+q} X(p-r,p) \twoheadrightarrow E_r^{p,q} \hookrightarrow \pi_{p+q} X(p-1, p+r-1)$$

in the abelian category $\mathcal{A}$, of the morphism $X((p-r,p) \leq (p-1,p+r-1))$, so that $E_r^{p,q}$ is the image of this morphism. Moreover, define morphisms

$$d_r \;\colon\; E_r^{p,q} \to E_r^{p-r, q+r-1}$$

to be the restriction (the image on morphisms) of the connecting homomorphism

$$\array{ \pi_{p+q} X(p-r, p) &\longrightarrow& E_r^{p,q} &\longrightarrow& \pi_{p+q} X(p-1, p+r-1) \\ {}^{\mathllap{\delta}}\downarrow && \downarrow^{\mathrlap{d_r}} && \downarrow^{\mathrlap{\delta}} \\ \pi_{p+q-1} X(p-2r, p-r) &\longrightarrow& E_r^{p-r,q+r-1} &\longrightarrow& \pi_{p+q-1} X(p-r-1, p-1) }$$

in the long exact sequence of homotopy groups of remark ,

• on the left for the case $(i \leq j \leq k) = (p-2r \leq p - r \leq p)$

• on the right for the case $(i \leq j \leq k) = (p - r - 1 \leq p - 1 \leq p + r - 1)$.

Remark

For $r = 1$ def. reduces to

$$\begin{aligned} E_1^{p,q} & \simeq \pi_{p+q} X(p-1,p) \\ & \simeq \pi_{q} (\Sigma^{-p} X(p-1,p)) \\ & \simeq \pi_{q} (C_{p}) \end{aligned}$$

where $C_p$ is the $p$th element in the chain complex associated with $X(\bullet,\bullet)$ according to def. .

(Higher Algebra, construction 1.2.2.6)

Proposition

In def. we have $d^r\circ d^r = 0$ for all $r \geq 1$ and all $p,q \in \mathbb{Z}$.

Moreover, there are natural isomorphisms (natural in $X_\bullet$)

$$E_{r+1}^{p,q} \simeq \frac{ ker(d_r \colon E_r^{p,q} \to E_r^{p-r, q+r-1}) }{ im(d_r \colon E_r^{p+r, q-r+1} \to E_r^{p,q}) } \,.$$

Thus, $\{E_r^{\bullet,\bullet}\}_{r\geq 1}$ is a homology spectral sequence in the abelian category $\mathcal{A}$, functorial in the filtered object $X_\bullet$, with first page

$$\begin{aligned} E_1^{p,q} &= \pi_{p+q} \operatorname{cofib}(X_{p-1}\to X_{p}) \\ & \simeq \pi_q (C_p) \end{aligned} \,.$$

(Higher Algebra, prop. 1.2.2.7)

Proof

Since $d_r$ is by definition the image morphism of a connecting homomorphism, for showing $d_r \circ d_r = 0$ it suffices to show that the connecting homomorphisms compose to the zero morphism, $\delta_r \circ \delta_r \simeq 0$. This is the same argument as in the proof of prop. , generalized from vertical steps of length 1 to vertical steps of length $r$.

Explicitly, we have the pasting diagram

$$\array{ F(p-2r,p) &\longrightarrow& F(p-2r,p+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow && \downarrow \\ F(p-r,n) &\longrightarrow& F(p-r,p+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow &(c)& \downarrow \\ 0 &\longrightarrow& F(p,p+r) &\underset{\delta_r}{\longrightarrow}& \Sigma F(p-r,p) }$$

where the squares labeled “c” are (co-)cartesian (homotopy pushouts). By the universal property of the pushout, this induces a factorization

$$\array{ F(p,p+1) &\longrightarrow& \Sigma F(p-2r,p) \\ & {}_{\mathllap{\delta_r}}\searrow & \downarrow \\ && \Sigma F(p-r,p) } \,.$$

Pasting this in turn to the homotopy pushout that defines $\Sigma \delta_{p-r}$

$$\array{ F(p,p+1) &\longrightarrow& \Sigma F(p-2r,p) &\longrightarrow& 0 \\ & {}_{\mathllap{\delta_r}}\searrow & \downarrow &(c)& \downarrow \\ && \Sigma F(p-r,p) &\underset{\Sigma \delta_{r}}{\longrightarrow}& \Sigma^2 F(p-2r,p-1) }$$

and then suspending the result $n$ times yields a diagram that exhibits a null-homotopy

$$\delta_{r} \circ \delta_r \simeq 0$$

in $\mathcal{C}$.

Next, to show the homology isomorphisms; consider for fixed $p,q,r$ the usual abbreviation

$$C \coloneqq E_r^{p,q}$$

for the $r$-relative chains,

$$Z \coloneqq ker(d_r \colon E_r^{p,q} \to E_r^{p-r, q+r-1})$$

for the $r$-relative cycles and

$$B \coloneqq im(d_r \colon E_r^{p+r, q-r+1} \to E_r^{p,q})$$

for the $r$-relative boundaries, all in bidegree $p,q$.

We claim that the canonical maps induce a sequence of morphisms in $\mathcal{A}$ of the form

$$\pi_{p+q} X(p-r-1, p) \stackrel{\phi}{\to} Z \stackrel{\phi'}{\to} Z/B \stackrel{\psi'}{\to} C/B \stackrel{\psi}{\to} \pi_{p+q} X(p-1, p+r)$$

and that $\phi'\circ \phi$ is an epimorphism and $\psi \circ \phi'$ is a monomorphism. By the uniqueness of the image factorization in the abelian category $\mathcal{A}$, this will prove the proposition.

To see that $\pi_{p+q} X(p-r-1,p)$ is indeed in the kernel of $d_r$ consider the commuting diagram

$$\array{ \pi_{p+q} X(p-r-1,p) &\longrightarrow& \pi_{p+q-1} X(p-2r, p- r-1) \\ \downarrow && \downarrow & \searrow \\ \pi_{p+q} X(p-r, p) &\longrightarrow& \pi_{p+q-1}X(p-2r, p-r) \\ \downarrow && \downarrow \\ E_r^{p,q} &\stackrel{d_r}{\longrightarrow}& E_r^{p-r, q+r-1} && \pi_{p+q-1} X(p - 2r, p-r) \\ && \downarrow & \swarrow \\ && \pi_{p+q-1} X(p - r - 1, p - 1) } \,.$$

Since the bottom right morphism is a monomorphism by construction, the claim is equivalently that the total composite from top-left to bottom right is zero. By commutativity of the diagram this factors through the composite from top-right to bottom-right. As indicated, this in turn factors through two consecutive morphisms of an $(i \leq j \leq k)$-square, which by definition of $\mathbb{Z}$-chain complex is null-homotopic.

By a dual argument one has that $\pi_{p+q}X(p-1, p+r)$ is in the coimage of $d_r$. This shows that we indeed have the above sequence of morphisms $\stackrel{\phi}{\to}\stackrel{\phi'}{\to}\stackrel{\psi'}{\to}\stackrel{\psi}{\to}$.

It now remains to show that $\phi$ is an epimorphism (dually $\psi$ will be a monomorphism.) (…Higher Algebra, p. 41…)

We can now consider the convergence of the spectral sequence of prop. . To state that efficiently, first consider the following definition

Definition

Given a filtered object, def. , $X \simeq \underset{\longrightarrow}{\lim}_n X_n \in \mathcal{C}$, say that the induced filtering on its homotopy groups $F^\bullet \pi_\bullet X$ is given by the images of the homotopy groups of the strata of $X$

$$F^p \pi_{p+q}X \coloneqq im\left( \pi_{p+q} X_{p} \to \pi_{p+q} X \right) \,\,\, \in \mathcal{A} \,.$$

(Higher Algebra, p. 43)

Proposition

Assume that $\mathcal{C}$ admits all sequential colimits and that $\pi$ preserves these. Let $X \simeq \underset{\longrightarrow}{\lim}_n X_n$ be a filtered object, def. , for filtering with $X_{n \lt 0} \simeq 0$. Then the spectral sequence of prop. , converges to the homotopy groups of $X$

$$E_1^{p,q} = \pi_{p+q} \operatorname{cofib}(X_{p-1}\to X_{p}) \simeq \pi_q (C_p) \;\;\Rightarrow\;\; \pi_{p+q} X \,,$$

where the first page is identified following remark .

In detail, for all $p,q \in \mathbb{Z}$ the differentials $d_r \colon E_r^{p,q} \to E_r^{p-r, q+r-1}$ vanish for $r \gt p$, and the colimit (in $\mathcal{A}$)

$$E^{p,q}_\infty \coloneqq \underset{\longrightarrow}{\lim}_{r \gt p} E_r$$

is isomorphic to the associated graded object of the filtered homotopy groups of def. :

$$E^{p,q}_\infty \simeq F^p \pi_{p+q}(X) / F^{p-1} \pi_{p+q}(X) \,.$$

This is due to (Higher Algebra, prop. 1.2.2.14). A quick review is in (Wilson 13, theorem 1.2.1).

Proof

The assumption $X_{n \lt 0} \simeq 0$ implies that for $i,j \lt 0$ we have, by remark ,

$$X(i,j) \simeq cofib(X_i \to X_j) \simeq 0 \;\;\;\; for\, i,j \lt 0$$

and therefore it follows that $E_r^{p-r,q+r-1}$, being a quotient of $\pi_{p+q} X(p-2r, p-r)$, vanishes for $r \gt p$.

The same assumption implies that

$$X(p-r,p) \simeq X_p \;\;\;\; for\, p \gt r$$

and so $E_\infty^{p,q}$ is

$$E_\infty^{p,q} \simeq im\left( \pi_{p+q} X_p \to \pi_{p+q} Y \right)$$

for

$$Y \coloneqq \underset{\longrightarrow}{\lim}_r X(p-1,p+r) \,.$$

We need to show that this image is the associated graded object of the filtered homotopy groups.

To that end, observe that the homotopy fiber sequences

$$X_{p-1} \to X_{p+r} \to X(p-1,p+r)$$

for all $r$ give a homotopy fiber sequence under the colimit over $r$ of the form

$$X_{p-1} \to X \to Y \,.$$

The corresponding long exact sequence of homotopy groups truncates on the left to read

$$0 \to F^{p-1} \pi_{p+q}(X) \stackrel{ker(f')}{\hookrightarrow} \pi_{p+q} X \stackrel{f'}{\to} \pi_{p+q}Y \,.$$

By construction the morphism $f'$ appearing here factors the morphism $f$ whose image we need to compute as

$$\array{ && \pi_{p+q}X \\ & {}^{\mathllap{g}}\nearrow && \searrow^{\mathrlap{f'}} \\ \pi_{p+q} X(p) && \stackrel{f}{\longrightarrow} && \pi_{p+q} Y }$$

Using these relation we can now express $E_\infty^{p,q} \simeq im(f)$ as:

$$\begin{aligned} E_\infty^{p,q} & \simeq im(f) \\ & \simeq im(f'|_{im(g)}) \\ & \simeq im(f'|_{F^p \pi_{p+q} X}) \\ & \simeq F^p \pi_{p+q}X/ker(f') \\ & \simeq F^p \pi_{p+q} X/F^{p-1} \pi_{p+q}X \end{aligned} \,.$$

Remark

While historically the appearances of the root “spectr-” in “spectral sequence” and in “spectrum” (stable homotopy types) are unrelated, prop. and prop. say that there is a lucky coincidence of terminology:

Every sequence of spectra manifests itself on homotopy groups in a spectral sequence.

Moreover, the discussion below in Examples shows that also conversely, essentially every spectral sequence that appears in practice comes from a sequence of spectra this way.

(See also the title of (Wilson 13)).

Remark

The spectral sequence above itself only actually depends to the triangulated homotopy category $Ho(\mathcal{C})$. But its $\infty$-functorial dependence on the filtered object needs the full structure of the (∞,1)-category $\mathcal{C}$

Spectral sequence of a cofiltered object

We discuss here the dual notion to the spectral sequence of a filtered object above, now for a cofiltered object.

The following does not just dualize but also change the indexing convention on top of dualization. Needs further discussion/harmonization.

Proposition

Consider the inclusion of posets

$$(\mathbb{Z}, \leq) \to (\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}$$

given by

$$n \mapsto (n,\infty) \,.$$

The induced (∞,1)-functor

$$Func((\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}, \mathcal{C}) \longrightarrow Func((\mathbb{Z}, \leq), \mathcal{C})$$

restricts to an equivalence between the (∞,1)-category of $\mathbb{Z}\cup \{\infty\}$-chain complexes in $\mathcal{C}$ (def. ) and that of generalized filtered objects in $\mathcal{C}$ (def. ).

Given a filtered object $X_\bullet$, the associated chain complex $X(\bullet,\bullet)$ is given by the homotopy fiber

$$X(n, n+r) = \operatorname{fib}(X_n\to X_{n+r}).$$

Definition

For a cofiltered object $X_\bullet$, def. , write

$$K_n \coloneqq fib(X_n \to X_{n+1})$$

for the homotopy fiber of the $n$th structure map, for all $n \in \mathbb{Z}$, and define an exact couple

$$\array{ && \pi_\bullet(K_\bullet) \\ & \swarrow && \nwarrow \\ \pi_\bullet(X_\bullet) && \stackrel{}{\longrightarrow} && \pi_\bullet(X_\bullet) }$$

where the maps are given by the long exact sequences

$$\cdots \to \pi_\bullet(X_{n+1}) \to \pi_\bullet(K_n) \to \pi_\bullet(X_n) \to \pi_\bullet(X_{n+1}) \to \pi_{\bullet+1}(K_n) \to \cdots$$

This exact couple gives rise in the usual way to a spectral sequence.

Let $X_\bullet$ be a cofiltered object.

Definition

Define for $p,q \in \mathbb{Z}$ and $r \geq 1$ the object $E^r_{p,q}$ by the canonical epi-mono factorization

$$\pi_{p} X(q-r+1,q+1) \twoheadrightarrow E^r_{p,q} \hookrightarrow \pi_{p} X(q, q+r)$$

in the abelian category $\mathcal{A}$, and define the differential

$$d^r \;\colon\; E_{p,q}^r \to E_{p-1, q-r}^r$$

to be the restriction of the connecting homomorphism

$$\pi_{p} X(q,q+r) \to \pi_{p-1} X(q-r, q)$$

from the long exact sequence of remark ,
for the case $i=q-r$, $j=q$, and $k=q+r$.

Proposition

$d^r\circ d^r = 0$ and there are natural (in $X_\bullet$) isomorphisms

$$E^{r+1}\cong \operatorname{ker}(d^r)/\operatorname{im}(d^r).$$

Thus, $\{E^r_{*,*}\}_{r\geq 1}$ is a bigraded spectral sequence in the abelian category $\mathcal{A}$, functorial in the filtered object $X_\bullet$, with

$$E^1_{p,q} = \pi_p \operatorname{fib}(X_q\to X_{q+1}), \qquad d^r: E^r_{p,q}\to E^r_{p-1,q-r}.$$

If sequential limits and sequential colimits exist in $\mathcal{A}$, we can form the limiting term $E^\infty_{*,*}$ of this spectral sequence.

On the other hand, the graded object $\pi_\bullet (X)$ admits a filtration by

$$F_q \pi_p (X) = \operatorname{ker}(\pi_p (X)\to \pi_p(X_q))$$

and we would like to compare $E^\infty_{*,*}$ with the associated graded of this filtration. We say that

Definition

The spectral sequence converges weakly if there is a canonical isomorphism

$$E^\infty_{p,q} \cong F_q\pi_p(X)/ F_{q-1}\pi_p(X)$$

for every $p,q\in\mathbb{Z}$.

We say that the spectral sequence converges strongly if it converges weakly and if, in addition, the filtration $F_\bullet\pi_p(X)$ is complete on both sides, that is:

$$\underset{\rightarrow}{\lim}_q F_q\pi_p (X) \simeq \pi_p(X) \simeq \underset{\leftarrow}{\lim}_q F^q\pi_p (X),$$

where $F^\bullet$ is the cofiltration.

Remark

The meaning of the word canonical in def. is somewhat subtle since, in general, there is no map from one side to the other. However, there always exists a canonical relation between the two, and we ask that this relation be an isomorphism (see Hilton-Stammbach, VIII.7).

Proposition

Let $\mathcal{C}$ be a stable (∞,1)-category and let $\pi:\mathcal{C}\to\mathcal{A}$ be a homological functor where $\mathcal{A}$ is an abelian category which admits sequential limits. Let $X_\bullet$ be a filtered object in $\mathcal{C}$ such that $\underset{\leftarrow}{\lim} X_\bullet$ exists. Suppose further that:

1. For every $n$, the diagram $r\mapsto \operatorname{fib}(X_{n-r}\to X_n)$ has a limit in $\mathcal{C}$ and that limit is preserved by $\pi$.
2. For every $n$, $\pi_n(X_r)=0$ for $r\gg 0$.

Then the spectral sequence $\{E^r_{*,*}\}_{r\geq 1}$ in $\mathcal{A}$ converges strongly (def. ). We write:

$$E_{p,q}^1 = \pi_{p} \operatorname{fib}(X_q\to X_{q+1}) \Rightarrow \pi_{p} (\underset{\leftarrow}{\lim} X_\bullet)$$

There is also a dual statement in which limits are replaced by colimits, but it is in fact a special case of the proposition with $\pi$ replaced by $\pi^{op}$. A proof of this proposition (in dual form) is given in (Higher Algebra, prop. 1.2.2.14). Review is in (Wilson 13, theorem 1.2.1).

Examples

General

tower diagram/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

Example

For $\mathcal{A}$ a good abelian category and $\mathcal{C} = Ch_\bullet(\mathcal{A})$ the (∞,1)-category of chain complexes in $\mathcal{A}$, we recover, by , the traditional notion of a spectral sequence of a filtered complex.

(Higher Algebra, example 1.2.2.11).

Example

Let $\mathcal{C} = Spec^{op}$ be the opposite (∞,1)-category of spectra, let $\mathcal{A}$ be the opposite category of abelian groups, and let $\pi$ be the functor $[K,-]$ where $K$ is spectrum. Then condition (1) in Proposition holds for all filtered objects if and only if $K$ is a finite spectrum. When the filtered object is the Whitehead tower of a spectrum $E$, the associated spectral sequence is the Atiyah-Hirzebruch spectral sequence with target $E^*(K)$. It is thus strongly convergent if $K$ is a finite spectrum.

Example

For $\mathcal{C}$ a stable (∞,1)-category and $X_\bullet$ a simplicial object in an (∞,1)-category in $\mathcal{C}$, then the simplicial skeleta of $X$ give it the structure of a filtered object in an (∞,1)-category. The corresponding spectral sequence of a filtered stable homotopy type has as its first page the Moore complexes of the corresponding simplicial objects of homotopy groups.

See at spectral sequence of a simplicial stable homotopy type.

As a special case of example we have:

Example

The $E$-based Adams spectral sequence that approximates homotopy classes of maps between two spectra $X$ and $Y$ using a ring spectrum $E$ is a special case of the above spectral sequence, with $\mathcal{C}=Spec$, $\pi=[X,-]$, and the filtered object associated with the cosimplicial spectrum $E^{\wedge\bullet+1}\wedge Y$. Bousfield’s theorems on the convergence of the Adams spectral sequence can be rephrased as giving sufficient conditions on $X$, $Y$, and $E$ for condition (1) in Proposition to hold (see Bousfield, Theorems 6.6 and 6.10).

See J-homomorphism and chromatic homotopy for an exposition.

Canonical cosimplicial resolution of $E_\infty$-algebras

We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from E-∞ algebras (dual Cech nerves/Sweedler corings/Amitsur complexes).

In this form this appears as (Lurie 10, theorem 2). A review is in (Wilson 13, 1.3). For the analog of this in the traditional formulation see (Ravenel, ch. 3, prop. 3.1.2).

Definition

Given an cosimplicial object in an (∞,1)-category with (∞,1)-colimits

$$Y \;\colon\; \Delta \longrightarrow \mathcal{C}$$

its totalization $Tot Y \simeq \underset{\leftarrow}{\lim}_n Y_n$ is filtered, def. , by the totalizations of its coskeleta

$$Tot Y \to \cdots \to Tot (cosk_2 Y) \to Tot (cosk_1 Y) \to Tot (cosk_0 Y) \to 0 \,.$$

Definition

The filtration spectral sequence, prop. , applied to the filtration of a totalization by coskeleta as in def. , we call the spectral sequence of a simplicial stable homotopy type.

(Higher Algebra, prop. 1.2.4.5)

Proposition

The spectral sequence of a simplicial stable homotopy type has as first page/$E_1$-term the cohomology groups of the Moore complex associated with the cosimplicial objects of homotopy groups

$$E_2^{p,q} = H^p(\pi_q(Tot (cosk_\bullet(Y)))) \Rightarrow \pi_{p-q} Tot(Y) \,.$$

By the discussion at ∞-Dold-Kan correspondence and spectral sequence of a filtered stable homotopy type. This appears as (Higher Algebra, remark 1.2.4.4). Review is around (Wilson 13, theorem 1.2.4).

Definition

Let $S$ be an E-∞ ring and let $E$ be an E-∞ algebra over $S$, hence an E-∞ ring equipped with a homomorphism

$$S \longrightarrow E \,.$$

The canonical cosimplicial object associated to this (the $\infty$-Cech nerve/Sweedler coring/Amitsur complex) is that given by the iterated smash product/tensor product over $S$:

$$E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C} \,.$$

More generally, for $X$ an $S$-∞-module, the canonical cosimplicial object is

$$E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C} \,.$$

Proposition

If $E$ is such that the self-generalized homology $E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E)$ (the dual $E$-Steenrod operations) is such that as a module over $E_\bullet \coloneqq \pi_\bullet(E)$ it is a flat module, then there is a natural equivalence

$$\pi_\bullet \left( E^{\wedge^{n+1}_S} \wedge_S X \right) \simeq E_\bullet(E^{\wedge^n_S}) \otimes_{E_\bullet} E_\bullet(X) \,.$$

Reviewed for instance as (Wilson 13, prop. 1.3.1).

Remark

This makes $(E_\bullet, E_\bullet(E))$ be the commutative Hopf algebroid formed by the $E$-Steenrod algebra. See there for more on this.

Example

The condition in prop. is satisfied for

• $E = H \mathbb{F}_p$ an Eilenberg-MacLane spectrum with $mod\;p$ coefficients;

• $E = B P$ the Brown-Peterson spectrum;

• $E = MU$ the complec cobordism spectrum.

It is NOT satisfied for

• $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum for integers coefficients;

• $E = M S U$.

Remark

Under good conditions (…), $\pi_\bullet$ of the canonical cosimplicial object provides a resolution of comodule tensor product and hence computes the Ext-groups over the commutative Hopf algebroid:

$$H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X)))) \simeq Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X)) \,.$$

(…)

Remark

There is a canonical map

$$L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)$$

from the $E$-Bousfield localization of spectra of $X$ into the totalization.

(Lurie 10, lecture 30, prop. 1)

We consider now conditions for this morphism to be an equivalence.

Definition

For $R$ a ring, its core $c R$ is the equalizer in

$$c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.$$

Proposition

Let $E$ be a connective E-∞ ring such that the core or $\pi_0(E)$, def. is either of

• the localization of the integers at a set $J$ of primes, $c \pi_0(E) \simeq \matbb{Z}[J^{-1}]$;

• $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark is an equivalence

$$L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.$$

(Bousfield 79, Lurie 10, lecture 30, prop. 3, Lurie 10, lecture 31,).

Related concepts

filtered objects

• filtered topological space

  • spectral sequence of a tower of fibrations

• filtered vector space

• filtered chain complex

  • spectral sequence of a filtered complex

  • Hodge filtration

• filtered ring

• filtered object in an (∞,1)-category

• filtered stable homotopy type

  • spectral sequence of a filtered stable homotopy type

• filtered probability space

associated graded objects

• associated graded vector space

• associated graded ring

References

The general theory is set up in section 1.2.2 of

• Jacob Lurie, Higher Algebra.

A quick exposition of that is for instance in section 1.2 of

• Dylan WilsonSpectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra lecture at 2013 Pre-Talbot Seminar (pdf)

The case of the derived category of an arbitrary abelian category is discussed in details in Chapter VIII of

• P. Hilton, U. Stammbach, A Course in Homological Algebra, Graduate Texts in Mathematics 4

The traditional discussion of the Adams spectral sequence in this style originates in

• Aldridge Bousfield, The localization of spectra with respect to homology, Topology vol 18 (1979) (pdf)

see also at Bousfield localization of spectra. The formulation of this in modern chromatic homotopy theory is discussed in

• Jacob Lurie, Chromatic Homotopy Theory, lecture notes (2010)