(also nonabelian homological algebra)
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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:
from a stable (∞,1)-category to an abelian category induces a functor
from the stable (∞,1)-category of filtered objects in $\mathcal{C}$ to the abelian category of bigraded spectral sequences in $\mathcal{A}$.
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
in $\mathcal{C}$ into a long exact sequence
in $\mathcal{A}$. We write $\pi_n=\pi\circ \Sigma^{-n}$.
$\mathcal{C}$ 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.
A filtered object in an (∞,1)-category in $\mathcal{C}$ is an object $X \in \mathcal{C}$ together with a sequential diagram $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$
and an equivalence
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 $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$
and an equivalence
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 $\mathcal{C}^{op}$.
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 $C$
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.
Let $I$ be a linearly ordered set. An $I$-chain complex in a stable (∞,1)-category $\mathcal{C}$ is an (∞,1)-functor
from the subposet of $I \times I$ on pairs of elements $i \leq j$, such that
for each $n \in I$, $F(n,n) \simeq 0$ is the zero object;
for all $i \leq j \leq k$ the induced diagram
is a homotopy pushout square (hence equivalently, by stability, a homotopy pullback).
Write
for the full sub-(∞,1)-category of diagrams satisfying these conditions.
This is Higher Algebra, def. 1.2.2.2.
The conditions in def. imply by the pasting law that also all squares
for all $i \leq k$ and $k \leq l$ are homotopy pushout squares.
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 $\mathcal{C}$
by setting
and taking
to be the $n$-fold de-suspension of the connecting homomorphisms of the defining homotopy fiber sequences
hence the $(n+1)$-fold de-suspension of the morphism $\delta_n$ in the following pasting of homotopy pushouts
where the total outer homotopy pushout exhibits the suspension of $F(n-1,n)$, by the pasting law.
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
is the zero morphism in the triangulated category $Ho(\mathcal{C})$.
(Higher Algebra, remark 1.2.2.3)
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
Therefore we have two paths of morphisms of span diagrams, the first is
which gives on homotopy pushouts
and the second is
which on homotopy pushouts is
(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 $\Sigma F(n-2,n)$:
Pasting this to the homotopy pushout that defines $\Sigma \delta_{n-1}$
and then suspending the result $n$ times yields a diagram that exhibits a null-homotopy
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.
Consider the inclusion of posets
given by
The induced (∞,1)-functor
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.
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 $X_n \to X_{n+r}$
because that makes the squares
be homotopy pushout squares.
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.
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
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
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
to be the restriction (the image on morphisms) of the connecting homomorphism
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 $(i \leq j \leq k) = (p - r - 1 \leq p - 1 \leq p + r - 1)$.
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)
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 $X_\bullet$)
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
(Higher Algebra, prop. 1.2.2.7)
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
where the squares labeled “c” are (co-)cartesian (homotopy pushouts). By the universal property of the pushout, this induces a factorization
Pasting this in turn to the homotopy pushout that defines $\Sigma \delta_{p-r}$
and then suspending the result $n$ times yields a diagram that exhibits a null-homotopy
in $\mathcal{C}$.
Next, to show the homology isomorphisms; consider for fixed $p,q,r$ the usual abbreviation
for the $r$-relative chains,
for the $r$-relative cycles and
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
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
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
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 $X$
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 $X$
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 $\mathcal{A}$)
is isomorphic to the associated graded object of the filtered homotopy groups of def. :
This is due to (Higher Algebra, prop. 1.2.2.14). A quick review is in (Wilson 13, theorem 1.2.1).
The assumption $X_{n \lt 0} \simeq 0$ implies that for $i,j \lt 0$ we have, by remark ,
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
and so $E_\infty^{p,q}$ is
for
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
for all $r$ give a homotopy fiber sequence under the colimit over $r$ of the form
The corresponding long exact sequence of homotopy groups truncates on the left to read
By construction the morphism $f'$ appearing here factors the morphism $f$ whose image we need to compute as
Using these relation we can now express $E_\infty^{p,q} \simeq im(f)$ as:
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)).
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 $\mathcal{C}$
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.
Consider the inclusion of posets
given by
The induced (∞,1)-functor
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
For a cofiltered object $X_\bullet$, def. , write
for the homotopy fiber of the $n$th structure map, for all $n \in \mathbb{Z}$, and define an exact couple
where the maps are given by the long exact sequences
This exact couple gives rise in the usual way to a spectral sequence.
Let $X_\bullet$ be a cofiltered object.
Define for $p,q \in \mathbb{Z}$ and $r \geq 1$ the object $E^r_{p,q}$ by the canonical epi-mono factorization
in the abelian category $\mathcal{A}$, and define the differential
to be the restriction of the connecting homomorphism
from the long exact sequence of remark ,
for the case $i=q-r$, $j=q$, and $k=q+r$.
$d^r\circ d^r = 0$ and there are natural (in $X_\bullet$) isomorphisms
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
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
and we would like to compare $E^\infty_{*,*}$ with the associated graded of this filtration. We say that
The spectral sequence converges weakly if there is a canonical isomorphism
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:
where $F^\bullet$ is the cofiltration.
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).
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:
Then the spectral sequence $\{E^r_{*,*}\}_{r\geq 1}$ in $\mathcal{A}$ converges strongly (def. ). We write:
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).
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).
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.
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:
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.
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).
Given an cosimplicial object in an (∞,1)-category with (∞,1)-colimits
its totalization $Tot Y \simeq \underset{\leftarrow}{\lim}_n Y_n$ is filtered, def. , by the totalizations of its coskeleta
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)
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
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).
Let $S$ be an E-∞ ring and let $E$ be an E-∞ algebra over $S$, hence an E-∞ ring equipped with a homomorphism
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 $S$:
More generally, for $X$ an $S$-∞-module, the canonical cosimplicial object is
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
Reviewed for instance as (Wilson 13, prop. 1.3.1).
This makes $(E_\bullet, E_\bullet(E))$ be the commutative Hopf algebroid formed by the $E$-Steenrod algebra. See there for more on this.
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;
$E = M S U$.
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:
(…)
There is a canonical map
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.
For $R$ a ring, its core $c R$ is the equalizer in
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
(Bousfield 79, Lurie 10, lecture 30, prop. 3, Lurie 10, lecture 31,).
The general theory is set up in section 1.2.2 of
A quick exposition of that is for instance in section 1.2 of
The case of the derived category of an arbitrary abelian category is discussed in details in Chapter VIII of
The traditional discussion of the Adams spectral sequence in this style originates in
see also at Bousfield localization of spectra. The formulation of this in modern chromatic homotopy theory is discussed in
Last revised on November 25, 2019 at 14:31:34. See the history of this page for a list of all contributions to it.