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Introduction to Stable homotopy theory -- 1-1

Stable homotopy theory Sequential spectra

We give an introduction to the stable homotopy category and to its key computational tool, the Adams spectral sequence. To that end we introduce the modern tools, such as model categories and highly structured ring spectra. In the accompanying seminar we consider applications to cobordism theory and complex oriented cohomology such as to converge in the end to a glimpse of the modern picture of chromatic homotopy theory._


Lecture notes.

Main page: Introduction to Stable homotopy theory.

Previous section: Prelude – Classical homotopy theory

This section_ Part 1 – Stable homotopy theory

This subsection: Part 1.1 – Stable homotopy theory – Sequential spectra

Next subsection: Part 1.2 – Stable homotopy theory – Structured Spectra

Next section: Part 2 – Adams spectral sequences



Stable homotopy theory – Sequential spectra

\,

The Prelude on Classical homotopy theory ended with the following phenomenon:

Definition

The reduced suspension/looping operation on pointed (def.) compactly generated topological spaces (def.) is the smash-tensor/hom-adjunction (cor.) for the standard 1-sphere smash product from the left:

(ΣΩ):Top cg */Maps(S 1,) *S 1()Top cg */. (\Sigma \dashv \Omega) \;\colon\; Top_{cg}^{\ast/} \; \underoverset {\xrightarrow[Maps(S^1,-)_\ast]{}} {\xleftarrow{S^1 \wedge (-)}} {\bot} \; Top_{cg}^{\ast/} \,.
Proposition

With respect to the classical model structure on pointed compactly generated topological spaces (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} (thm., prop.)

  1. the adjunction in def. is a Quillen adjunction (def.)

    (ΣΩ):(Top cg */) QuillenMaps(S 1,) *S 1()(Top cg */) Quillen, (\Sigma \dashv \Omega) \;\colon\; (Top_{cg}^{\ast/})_{Quillen} \underoverset {\underset{Maps(S^1,-)_\ast}{\longrightarrow}} {\overset{S^1 \wedge (-)}{\longleftarrow}} {\bot} (Top_{cg}^{\ast/})_{Quillen} \,,
  2. its induced adjoint pair of derived functors on the classical pointed homotopy category (by this prop.) is the canonical suspension/looping adjunction (according to this prop.)

    (ΣΩ):Ho(Top */)ΩΣHo(Top */). (\Sigma \dashv \Omega) \;\colon\; Ho(Top^{\ast/}) \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} Ho(Top^{\ast/}) \,.

See (this prop.).

The stable homotopy category Ho(Spectra)Ho(Spectra) is to be the result of stabilizing the adjunction in prop. , in the sense of forcing it to become an equivalence of categories in a compatible way, i.e. such as to fit into a diagram of categories of the form

Ho(Top */) ΩΣ Ho(Top */) Σ Ω Σ Ω Ho(Spectra) ΩΣ Ho(Spectra). \begin{matrix} Ho(Top^{\ast/}) & \underoverset{\xrightarrow[\quad\Omega\quad]{}}{\xleftarrow{\quad\Sigma\quad}}{\scriptsize{\bot}} & Ho(Top^{\ast/}) \\ \left.\scriptsize{\mathllap{\Sigma^\infty}}\right\downarrow \dashv \left\uparrow\scriptsize{\mathrlap{\Omega^\infty}}\right. && \left.\scriptsize{\mathllap{\Sigma^\infty}}\right\downarrow \dashv \left\uparrow\scriptsize{\mathrlap{\Omega^\infty}}\right. \\ Ho(Spectra) & \underoverset{\xrightarrow[\quad\Omega\quad]{}}{\xleftarrow{\quad\Sigma\quad}}{\simeq} & Ho(Spectra) \end{matrix} \,.

Moreover, for stable homotopy theory proper we are to refine this situation from homotopy categories to model categories and ask it to be the diagram of derived functors (according to this prop.) of a diagram of Quillen adjunctions (def.)

(Top cg */) Quillen ΩΣ (Top cg */) Quillen Σ Ω Σ Ω SeqSpec(Top cg) stable QΩΣ SeqSpec(Top cg) stable, \array{ (Top_{cg}^{\ast/})_{Quillen} & \underoverset{\underoverset{\Omega}{\bot}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{} & (Top^{\ast/}_{cg})_{Quillen} \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ SeqSpec(Top_{cg})_{stable} & \underoverset{\underset{\Omega}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{\simeq_Q} & SeqSpec(Top_{cg})_{stable} } \,,

This we establish in theorem below.

The notation Σ \Sigma^\infty and Ω \Omega^\infty is meant to be suggestive of the intuition behind how this stabilization will work: The universal way of making a topological space XX become stable under suspension is to pass to its infinite suspension in a suitable sense. That suitable sense is going to be called the suspension spectrum of XX (def. below). Conversely, if an object does not change up to equivalence, by forming its loop spaces, it must give an infinite loop space.

In contrast to the classical homotopy category, the stable homotopy category is a triangulated category (a shadow of the fact that the (∞,1)-category of spectra is a stable (∞,1)-category). As such it may be thought of as a refinement of the derived category of chain complexes (of abelian groups): every chain complex gives rise to a spectrum and every chain map to a map between these spectra (the stable Dold-Kan correspondence), but there are many more spectra and maps between them than arise from chain complexes and chain maps.

There is a variety of different models for the stable homotopy theory of spectra, some of which fits into this hierarchy:

  1. sequential spectra with their model structure on sequential spectra

  2. symmetric spectra with their model structure on symmetric spectra

  3. orthogonal spectra with their model structure on orthogonal spectra

  4. excisive functors with their model structure for excisive functors

As one moves down this list, the objects modelling the spectra become richer. This means on the one hand that their abstract properties become better as one moves down the list, on the other hand it means that it is more immediate to construct and manipulate examples as one stays further up in the list.

We start with plain sequential spectra as a transparent means to construct the stable homotopy category. In order to discuss ring spectra it is convenient to first pass to the richer model of highly structured spectra, this we do in Part II

The most lighweight model for spectra are sequential spectra. They support most of stable homotopy theory in a straightforward way, and have the advantage that examples tend to be immediate (for instance the proof of the Brown representability theorem spits out sequential spectra).

The key disadvantage of sequential spectra is that they do not support a functorial smash product of spectra before passing to the stable homotopy category, much less a symmetric smash product of spectra. This is the structure needed for a decent discussion of the higher algebra of ring spectra. To accomodate this, further below we enhance sequential spectra to the more highly structured models given by symmetric spectra and orthogonal spectra. But all these models are connected by a free-forgetful adjunction and for working with either it is useful to have the means to pass back and forth between them.

Sequential pre-spectra

The following def. is the traditional component-wise definition of sequential spectra. It was first stated in (Lima 58) and became widely appreciated with (Boardman 65).

It is generally supposed that G. W. Whitehead also had something to do with it, but the latter takes a modest attitude about that. (Adams 74, p. 131)

Below in prop. we discuss an equivalent definition of sequential spectra as “topological diagram spectra” (Mandell-May-Schwede-Shipley 00), namely as topologically enriched functors (defn.) on a topologically enriched category of n-spheres, which is useful for establishing the stable model category structure (below) and for establishing the symmetric monoidal smash product of spectra (in 1.2).

Throughout, our ambient category of topological spaces is Top cgTop_{cg}, the category of compactly generated topological space (defn.).

Definition

A sequential prespectrum in topological spaces, or just sequential spectrum for short (or even just spectrum), is

  1. an \mathbb{N}-graded pointed compactly generated topological space

    X =(X nTop cg */) n X_\bullet = (X_n \in Top_{cg}^{\ast/})_{n \in \mathbb{N}}

    (the component spaces);

  2. pointed continuous functions

    σ n:S 1X nX n+1 \sigma_n \colon S^1 \wedge X_n \to X_{n+1}

    for all nn \in \mathbb{N} (the structure maps) from the smash product (defn.) of one component space with the standard 1-sphere to the next component space.

A homomorphism f:XYf \colon X \to Y of sequential spectra is a sequence f :X Y f_\bullet \colon X_\bullet \to Y_\bullet of base point-preserving continuous functions between component spaces, such that these respect the structure maps in that all diagrams of the form

S 1X n S 1f n S 1Y n σ n X σ n Y X n+1 f n+1 Y n+1 \begin{matrix} S^1 \wedge X_n &\xrightarrow{\quad S^1 \wedge f_n\quad}& S^1 \wedge Y_n \\ \left\downarrow\scriptsize{\mathrlap{\sigma_n^X}}\right. && \left\downarrow\scriptsize{\mathrlap{\sigma_n^Y}}\right. \\ X_{n+1} &\xrightarrow{\qquad f_{n+1}\qquad}& Y_{n+1} \end{matrix}

commute.

Write SeqSpec(Top cg)SeqSpec(Top_{cg}) for this category of topological sequential spectra.

Due to the classical adjunction

Top cg */Maps(S 1,) *S 1()Top cg */ Top_{cg}^{\ast/} \underoverset {\underset{Maps(S^1,-)_\ast}{\longrightarrow}} {\overset{S^1 \wedge (-)}{\longleftarrow}} {\bot} Top_{cg}^{\ast/}

from classical homotopy theory (this prop.), the definition of sequential spectra in def. is equivalent to the following definition

Definition

A sequential prespectrum in topological spaces, or just sequential spectrum for short (or even just spectrum), is

  1. an \mathbb{N}-graded pointed compactly generated topological space

    X =(X nTop cg */) n X_\bullet = (X_n \in Top_{cg}^{\ast/})_{n \in \mathbb{N}}

    (the component spaces);

  2. pointed continuous functions

    σ˜ n:X nMaps(S 1,X n+1) * \tilde \sigma_n \colon X_n \to Maps(S^1,X_{n+1})_\ast

    for all nn \in \mathbb{N} (the adjunct structure maps) from one component space to the pointed mapping space (def., exmpl.) out of S 1S^1 into the next component space.

A homomorphism f:XYf \colon X \to Y of sequential spectra is a sequence f ˜:X Y \widetilde{f_\bullet} \colon X_\bullet \to Y_\bullet of base point-preserving continuous function, such that all diagrams of the form

X n f n Y n σ˜ n X σ˜ n Y Maps(S 1,X n+1) * Maps(S 1,f n+1) * Maps(S 1,Y n+1) * \begin{matrix} X_n &\xrightarrow{\space{0}{0}{45}f_n\space{0}{0}{45}}& Y_n \\ \left.\scriptsize{\mathllap{\tilde \sigma^X_n}}\right\downarrow && \left\downarrow\scriptsize{\mathrlap{\tilde \sigma^Y_n}}\right. \\ Maps(S^1, X_{n+1})_\ast &\xrightarrow[\quad Maps(S^1,f_{n+1})_\ast\quad]{}& Maps(S^1, Y_{n+1})_\ast \end{matrix}

commute.

Example

For XTop */ cgX\in Top^{\ast/_{cg}} a pointed topological space, its suspension spectrum Σ X\Sigma^\infty X is the sequential spectrum , def. , with

  • (Σ X) nS nX(\Sigma^\infty X)_n \coloneqq S^n \wedge X (smash product of XX with the n-sphere);

  • σ n:S 1S nXS n+1X\sigma_n \colon S^1 \wedge S^n \wedge X \overset{\simeq}{\longrightarrow} S^{n+1}X (the canonical homeomorphism).

This construction extends to a functor

Σ :Top cg */SeqSpec(Top cg). \Sigma^\infty \;\colon\; Top^{\ast/}_{cg} \longrightarrow SeqSpec(Top_{cg}) \,.
Example

The suspension spectrum (example ) of the point is the standard sequential sphere spectrum

𝕊 seqΣ S 0. \mathbb{S}_{seq} \coloneqq \Sigma^\infty S^0 \,.

Its nnth component space is the standard n-sphere

(𝕊 seq) n=S n. (\mathbb{S}_{seq})_n = S^n \,.
Example

A fundamental example of a spectrum that is not just a suspension spectrum is the universal real Thom spectrum, denoted MO. For details on this see Part S – Thom spectra.

There are are also the universal complex Thom spectrum denoted MU, and the universal symplectic Thom spectrum denoted MSp. Their standard construction first yields an example of a “sequential S 2S^2-spectrum”; which we introduce below in def. ; and then there is an adjunction (prop. ) that canonically turns this into an ordinary sequential spectrum.

Definition

Let XSeqSpec(Top cg)X\in SeqSpec(Top_{cg}) be a sequential spectrum (def. ) and KTop cg */K \in Top^{\ast/}_{cg} a pointed compactly generated topological space. Then

  1. XKX \wedge K (the smash tensoring of XX with KK) is the sequential spectrum given by

    • (XK) nX nK(X \wedge K)_n \coloneqq X_n \wedge K (smash product on component spaces (defn.))

    • σ n XKσ n Xid K\sigma_n^{X \wedge K} \coloneqq \sigma_n^{X} \wedge id_{K}.

  2. Maps(K,X) *Maps(K,X)_\ast (the powering of KK into XX) is the sequential spectrum with

    • (Maps(K,X) *) nMaps(K,X n) *(Maps(K,X)_\ast)_n \;\coloneqq\; Maps(K,X_n)_\ast (compactly generated pointed mapping space (def., def.))

    • σ n Maps(K,X) *:S 1Maps(K,X n)(const,id)Maps(K,S 1X n) *Maps(K,σ n) *Maps(K,X n+1) *\sigma_n^{Maps(K,X)_\ast} \;\colon\; S^1 \wedge Maps(K,X_n) \overset{(const,id)}{\longrightarrow} Maps(K,S^1 \wedge X_n)_\ast \overset{Maps(K,\sigma_n)_\ast}{\longrightarrow} Maps(K,X_{n+1})_\ast,

    where (const,id):[s,ϕ][const s,ϕ](const, id) \;\colon\; [s,\phi] \mapsto [const_s,\phi].

These operations canonically extend to functors

()():SeqSpec(Top cg)×Top cg */SeqSpec(Top cg) (-)\wedge (-) \;\colon\; SeqSpec(Top_{cg}) \times Top^{\ast/}_{cg} \longrightarrow SeqSpec(Top_{cg})

and

Maps(,) *:(Top cg */) op×SeqSpec(Top cg)SeqSpec(Top cg). Maps(-,-)_\ast \;\colon\; (Top^{\ast/}_{cg})^{op} \times SeqSpec(Top_{cg}) \longrightarrow SeqSpec(Top_{cg}) \,.
Example

The tensoring (def. ) of the standard sphere spectrum 𝕊 std\mathbb{S}_{std} (def. ) with a space XTop cgX \in Top_{cg} is isomorphic to the suspension spectrum of XX (def. ):

𝕊 stdXΣ X. \mathbb{S}_{std} \wedge X \simeq \Sigma^\infty X \,.
Proposition

For any KTop cg */K \in Top^{\ast/}_{cg} the functors of smash tensoring and powering with KK, from def. , constitute a pair of adjoint functors

SeqSpec(Top cg)Maps(K,) *()KSeqSpec(Top cg). SeqSpec(Top_{cg}) \underoverset { \underset{Maps(K,-)_\ast}{\longrightarrow} } { \overset{(-) \wedge K}{\longleftarrow} } {\bot} SeqSpec(Top_{cg}) \,.
Proof

For X,YSeqSpec(Top cg)X, Y\in SeqSpec(Top_{cg}) and KTop cg */K \in Top_{cg}^{\ast/}, let

XKfY X \wedge K \overset{f}{\longrightarrow} Y

be a morphism, with component maps fitting into commuting squares of the form

S 1X nK S 1f n S 1Y n σ n XK σ n Y X n+1K f n+1 Y n+1. \array{ S^1 \wedge X_n \wedge K &\overset{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n \\ {}^{\mathllap{\sigma^X_n \wedge K}}\downarrow && \downarrow^{\sigma^Y_n} \\ X_{n+1} \wedge K &\overset{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.

Applying degreewise the adjunction

Top cg */Maps(K,) *()KTop cg */ Top_{cg}^{\ast/} \underoverset {\underset{Maps(K,-)_\ast}{\longrightarrow}} {\overset{(-) \wedge K}{\longleftarrow}} {\bot} Top_{cg}^{\ast/}

from classical homotopy theory (this prop.) gives that these squares are in natural bijection with squares of the form

S 1X n S 1f n˜ Maps(K,S 1Y n) * σ n X Maps(K,σ n Y) * X n+1 f n+1˜ Maps(K,Y n+1) *. \array{ S^1 \wedge X_n &\overset{\widetilde{S^1 \wedge f_n}}{\longrightarrow}& Maps(K,S^1 \wedge Y_n)_\ast \\ {}^{\mathllap{\sigma_n^X}}\downarrow && \downarrow^{\mathrlap{Maps(K,\sigma_n^Y)_\ast}} \\ X_{n+1} &\overset{\widetilde{f_{n+1}}}{\longrightarrow}& Maps(K, Y_{n+1})_\ast } \,.

But since the map S 1f nS^1 \wedge f_n is the smash product of two maps, only one of which involves the smash factor of KK, one sees that here the top map factors through the map (const,id)(const,id) from def. .

Hence the commuting square above factors as

S 1X n S 1f n˜ S 1Maps(K,Y n) * σ n X σ n Maps(K,Y) * X n+1 f n+1˜ Maps(K,Y n+1) *. \array{ S^1 \wedge X_n &\overset{S^1 \wedge \widetilde{f_n}}{\longrightarrow}& S^1 \wedge Maps(K, Y_n)_\ast \\ {}^{\mathllap{\sigma_n^X}}\downarrow && \downarrow^{\mathrlap{\sigma_n^{Maps(K,Y)_\ast}}} \\ X_{n+1} &\overset{\widetilde {f_{n+1}}}{\longrightarrow}& Maps(K, Y_{n+1})_\ast } \,.

This gives the structure maps for a homomorphism

f˜:XMaps(K,Y) *. \tilde f \;\colon\; X \longrightarrow Maps(K,Y)_\ast \,.

Running this argument backwards shows that the map ff˜f \mapsto \tilde f given thereby is a bijection.

Remark

For the adjunction of prop. it is crucial that the smash tensoring in def. is from the right, at least as long as the structure maps in def. are defined as they are, with the circle smash factor on the left. We could change both jointly: take the structure maps to be from smash products with the circle on the right, and take smash tensoring to be from the left. But having both on the right or both on the left does not work.

Proposition

The functor Σ \Sigma^\infty that forms suspension spectra (def. ) has a right adjoint functor Ω \Omega^\infty

(Σ Ω ):SeqSpec(Top cg)Ω Σ Top cg */, (\Sigma^\infty \dashv \Omega^\infty) \;\colon\; SeqSpec(Top_{cg}) \underoverset {\underset{\Omega^{\infty}}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {\bot} Top_{cg}^{\ast/} \,,

given by picking the 0-component space:

Ω (X)=X 0. \Omega^\infty(X) = X_0 \,.
Proof

By def. the components f nf_n of a homomorphism of sequential spectra of the form

Σ XfY \Sigma^\infty X \overset{f}{\longrightarrow} Y

have to make these diagrams commute

S 1S nX S 1f n S 1Y n σ n Y S n+1X f n+1 Y n+1 \array{ S^1 \wedge S^n X &\overset{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_{n} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\sigma^Y_n}} \\ S^{n+1} \wedge X &\overset{f_{n+1}}{\longrightarrow}& Y_{n+1} }

for all nn \in \mathbb{N}. Since here the left vertical map is an isomorphism by def. , this uniquely fixes f n+1f_{n+1} in terms of f nf_n. Hence the only freedom in specifying ff is in the choice of the component f 0:XY 0f_0 \colon X \longrightarrow Y_0, which is equivalently a morphism

Xf˜Ω Y. X \overset{\tilde f}{\longrightarrow} \Omega^\infty Y \,.

Stable homotopy groups

In analogy to how homotopy groups are the fundamental invariants in classial homotopy theory, the fundamental invariants of stable homtopy theory are stable homtopy groups:

Definition

The stable homotopy groups of a sequential prespectrum XX, def. , is the \mathbb{Z}-graded abelian group given by the colimit of homotopy groups of the component spaces (def.)

π (X)lim kπ +k(X k), \pi_\bullet(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{\bullet+k}(X_{k}) \,,

where the colimit is over the sequential diagram whose component morphisms are given in terms of the structure maps of def. by

π q+k(X k)[S q+k,X k] *(S 1()) S q+k,X k[S q+k+1,S 1X k] *[S q+k+1,σ k][S q+k+1,X k+1] *π q+k+1(X k+1) \pi_{q+k}(X_k) \overset{\simeq}{\to} [S^{q+k},X_k]_\ast \overset{(S^1\wedge(-))_{S^{q+k},X_k}}{\longrightarrow} [S^{q+k+1}, S^1 \wedge X_k]_\ast \overset{[S^{q+k+1}, \sigma_k]}{\longrightarrow} [S^{q+k+1}, X_{k+1}]_\ast \overset{\simeq}{\to} \pi_{q+k+1}(X_{k+1})

and equivalently are given in terms of the adjunct structure maps of def. by

π q+k(X k)[S q+k,X k] *[S q+k,σ˜ k][S q+k,Maps(S 1,X k+1) *] *[S 1S q+k,X k+1] *π q+k+1(X k+1). \pi_{q+k}(X_k) \overset{\simeq}{\longrightarrow} [S^{q+k}, X_k]_\ast \overset{[S^{q+k}, \tilde \sigma_k]}{\longrightarrow} [S^{q+k}, Maps(S^1,X_{k+1})_\ast]_\ast \simeq [S^1 \wedge S^{q+k}, X_{k+1}]_\ast \simeq \pi_{q+k+1}(X_{k+1}) \,.

The colimit starts at

k={0 ifq0 |q| ifq<0 k = \left\{ \array{ 0 & if \; q \geq 0 \\ {\vert q\vert} & if \; q \lt 0 } \right.

This canonically extends to a functor

π :SeqSpec(Top cg)Ab . \pi_\bullet \;\colon\; SeqSpec(Top_{cg}) \longrightarrow Ab^{\mathbb{Z}} \,.
Proposition

The two component morphisms given in def. indeed agree.

Proof

Consider the following instance of the defining naturality square of the (S 1())Maps(S 1,) *(S^1 \wedge (-)) \dashv Maps(S^1,-)_\ast-adjunction of prop. :

[S 1X k,S 1X k] * [X k,Maps(S 1,S 1X k) *] * [S 1α,σ k] [α,Maps(S 1,σ k) *] * [S 1S q+k,X k+1] * [S q+k,Maps(S 1,X k+1) *] *. \array{ [S^1 \wedge X_k, S^1 \wedge X_k]_\ast &\overset{\simeq}{\longrightarrow}& [X_k, Maps(S^1, S^1 \wedge X_k)_\ast]_\ast \\ {}^{\mathllap{[S^1 \wedge \alpha, \sigma_k]}}\downarrow && \downarrow^{\mathrlap{[\alpha, Maps(S^1,\sigma_k)_\ast]_\ast}} \\ [S^1 \wedge S^{q+k}, X_{k+1}]_\ast &\underset{\simeq}{\longrightarrow}& [S^{q+k}, Maps(S^1, X_{k+1})_\ast]_\ast } \,.

Then consider the identity element in the top left hom-set. Its image under the left vertical map is the first of the two given component morphisms. Its image under going around the other way is the second of the two component morphisms. By the commutativity of the diagram, these two images agree.

Example

Given XTop cg */X \in Top^{\ast/}_{cg}, then the stable homotopy groups (def. ) of its suspension spectrum (example ) are given by

π q S(X) π q(Σ X) =lim kπ q+k(S kX) lim kπ q(Ω k(Σ kX)). \begin{aligned} \pi_q^S(X) & \;\coloneqq\; \pi_q(\Sigma^\infty X) \\ & = \underset{\longrightarrow}{\lim}_{k} \pi_{q + k}(S^k \wedge X) \\ & \simeq \underset{\longrightarrow}{\lim}_{k} \pi_{q}(\Omega^k ( \Sigma^k X)) \end{aligned} \,.

Specifically for X=S 0X = S^0 the 0-sphere, with suspension spectrum the standard sphere spectrum (def. ), its stable homotopy groups are the stable homotopy groups of spheres:

π q S(S 0) π q(𝕊) =lim kπ q+k(S k). \begin{aligned} \pi_q^S(S^0) & \;\coloneqq\; \pi_q(\mathbb{S}) \\ & = \underset{\longrightarrow}{\lim}_{k} \pi_{q + k}(S^k) \end{aligned} \,.

Recall the Freudenthal suspension theorem, which states that if XX is an n-connected pointed CW-complex then the comparison map

π q(X)π q+1(ΣX) \pi_{q}(X) \longrightarrow \pi_{q+1}(\Sigma X)

is an isomorphism for q2nq \leq 2n. This implies first of all that every Σ kX\Sigma^k X is (k1)(k-1)-connected

π 0(ΣX) * π 1(Σ 2X) π 0(ΣX)* π 2(Σ 3X) π 1(Σ 2X)π 0(ΣX)* \begin{aligned} \pi_0(\Sigma X) & \simeq \ast \\ \pi_1(\Sigma^2 X) & \simeq \pi_0(\Sigma X) \simeq \ast \\ \pi_2(\Sigma^3 X) & \simeq \pi_1(\Sigma^2 X) \simeq \pi_0(\Sigma X) \simeq \ast \\ \cdots \end{aligned}

and then that the qqth stable homotopy group of XX is attained at stage k=q+2k = q+2 in the colimit:

π q S(X)π q+(q+2)(Σ q+2X). \pi^S_q(X) \simeq \pi_{q + (q+2)}(\Sigma^{q+2}X) \,.

Historically, this fact was one of the motivations for finding a stable homotopy category (def. below).

Definition

A morphism f:XYf \colon X \longrightarrow Y of sequential spectra, def. , is called a stable weak homotopy equivalence, if its image under the stable homotopy group-functor of def. is an isomorphism

π (f):π (X)π (Y). \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \overset{\simeq}{\longrightarrow} \pi_\bullet(Y) \,.

Omega-spectra

In order to motivate Omega-spectra consider the following shadow of the structure they will carry:

Example

A \mathbb{Z}-graded abelian group is equivalently a sequence {A n} n\{A_n\}_{n \mathbb{Z}} of \mathbb{N}-graded abelian groups A nA_n, together with isomorphisms

A nA n+1[1], A_n \simeq A_{n+1}[1] \,,

(where [1][1] denotes the operation of shifting all entries in a graded abelian group down in degree by -1). Because this means that the sequence of \mathbb{N}-graded abelian groups is of the following form

a 3 a 2 a 1 a 2 a 1 a 0 a 1 a 0 a 1 a 0A 0 a 1A 1 a 2A 2 . \array{ \vdots && \vdots \\ a_3 && a_2 && a_1 && \cdots \\ a_2 && a_1 && a_0 && \cdots \\ a_1 && a_0 && a_{-1} && \cdots \\ \underset{A_0}{\underbrace{a_0}} && \underset{A_1}{\underbrace{a_{-1}}} && \underset{A_2}{\underbrace{a_{-2}}} && \cdots } \,.

This allows to recover the \mathbb{Z}-graded abelian group {a n} n\{a_n\}_{n \in \mathbb{Z}} from an \mathbb{N}-sequence of \mathbb{N}-graded abelian groups.

Then consider the case that the \mathbb{N}-graded abelian groups here are homotopy groups of some topological space. Then shifting the degree of the component groups corresponds to forming loop spaces, because for any topological space XX then

π (ΩX)π +1(X). \pi_\bullet(\Omega X) \simeq \pi_{\bullet + 1}(X) \,.

(This may be seen concretely in point-set topology or abstractly by looking at the long exact sequence of homotopy groups for the fiber sequence ΩXPath *(X)X\Omega X \to Path_*(X) \to X.)

We find this kind of behaviour for the stable homotopy groups of Omega-spectra below in example .

Definition

An Omega-spectrum is a sequential spectrum XX of topological spaces, def. , such that the (smash product \dashv pointed mapping space)-adjuncts σ˜ n\tilde \sigma_n of the structure maps σ n:ΣX nX n+1\sigma_n \colon \Sigma X_n \to X_{n+1} of XX are weak homotopy equivalences (def.), hence classical weak equivalences (def.):

σ˜ n:X nW clMaps(S 1,X n+1) * \tilde \sigma_n \;\colon\; X_n \stackrel{\in W_{cl}}{\longrightarrow} Maps(S^1,X_{n+1})_\ast

for all nn \in \mathbb{N}.

Equivalently: an Omega-spectrum is a sequential spectrum in the incarnation of def. such that all adjunct structure maps are weak homotopy equivalences.

Example

The Brown representability theorem (thm.) implies (prop.) that every generalized (Eilenberg-Steenrod) cohomology theory (def.) is represented by an Omega-spectrum (def. ).

Applied to ordinary cohomology with coefficients some abelian group AA, this yields the Eilenberg-MacLane spectra HAH A (exmpl.). These are the Omega-spectra whose nnth component space is an Eilenberg-MacLane space

(HA) nK(A,n). (H A)_n \simeq K(A,n) \,.

A genuinely generalized (i.e. non-ordinary, hence “extra-ordinary”) cohomology theory is topological K-theory K ()K^\bullet(-). Applying the Brown representability theorem to topological K-theory yields the K-theory spectrum denoted KU.

Omega-spectra are singled out among all sequential pre-spectra as having good behaviour under forming stable homotopy groups.

Example

If a sequential spectrum XX is an Omega-spectrum, def. , then its colimiting stable homotopy groups reduce to the actual homotopy groups of the component spaces, in that:

XOmega-spectrumπ k(X){π kX 0 ifk0 π 0X |k| ifk<0. X \; \text{Omega-spectrum} \;\;\;\;\; \Rightarrow \;\;\;\;\; \pi_k(X) \simeq \left\{ \array{ \pi_k X_0 & if\; k \geq 0 \\ \pi_0 X_{\vert k \vert} & if \; k \lt 0 } \right. \,.

(Hence the stable homotopy groups of an Omega-spectrum realize the general pattern discussed in example .)

Proof

For an Omega-spectrum, the adjunct structure maps σ˜ X\tilde \sigma_X are weak homotopy equivalences, by definition, hence are classical weak equivalences. Hence [S 1,σ˜ n] *[S^1, \tilde \sigma_n]_\ast is an isomorphism (prop.). Therefore, by prop. , the sequential colimit in def. is entirely over isomorphisms and hence is given already by the first object of the sequence.

We now show that every sequential pre-spectrum may be completed to an Omega-spectrum, up to stable weak homotopy equivalence:

Definition

For XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}), define a spectrum QXSeqSpec(Top cg)Q X \in SeqSpec(Top_{cg}) and a morphism

η X:XQX \eta_X \;\colon\; X \longrightarrow Q X

(to be called the spectrification of XX) as follows.

First introduce for the given components X kX_k and adjunct structure maps σ˜ k\tilde \sigma_k of XX (from def. ) the notation

Z 0,kX k,σ˜ 0,kσ˜ k. Z_{0,k} \coloneqq X_{k} \,, \;\;\;\;\;\; \tilde \sigma_{0,k} \coloneqq \tilde \sigma_k \,.

Now assume, by induction, that sets of objects {Z i,k} k\{Z_{i,k}\}_{k \in \mathbb{N}} and maps {Z i,kσ˜ i,kΩZ i,k+1} k\{Z_{i,k} \overset{\tilde \sigma_{i,k}}{\to} \Omega Z_{i,k+1}\}_{k \in \mathbb{N}} have been constructed for some ii \in \mathbb{N}.

Then construct Z i+1,kTop cgZ_{i+1,k}\in Top_{cg} by factorizing σ˜ i,k\tilde \sigma_{i,k}, with respect to the model structure (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} (thm.) as a classical cofibration followed by a classical weak equivalence. More specifically, apply the small object argument (prop.) with respect to the set of generating cofibrations I TopI_{Top} (def.) to produce functorial factorizations (def.) into a relative cell complex followed by a weak homotopy equivalence (just as in the proof of this lemma):

σ˜ i,k:Z i,kI TopCellι i,kZ i+1,kW clϕ i,kΩZ i,k+1. \tilde \sigma_{i,k} \;\colon\; Z_{i,k} \underoverset{\in I_{Top} Cell}{\iota_{i,k}}{\longrightarrow} Z_{i+1,k} \underoverset{\in W_{cl}}{\phi_{i,k}}{\longrightarrow} \Omega Z_{i,k+1} \,.

Then define σ˜ i+1,k\tilde \sigma_{i+1,k} as the composite

σ˜ i+1,k:Z i+1,kϕ i,kΩZ i,k+1Ω(ι i,k+1)ΩZ i+1,k+1. \tilde \sigma_{i+1,k} \;\colon\; Z_{i+1,k} \overset{\phi_{i,k}}{\longrightarrow} \Omega Z_{i,k+1} \overset{\Omega (\iota_{i,k+1})}{\longrightarrow} \Omega Z_{i+1,k+1} \,.

This produces for each ii \in \mathbb{N} a commuting diagram of the form

X k=Z 0,k I TopCellι 0,k Z 1,k I TopCellι 1,k Z 2,k I TopCellι 2,k σ˜ k=σ˜ 0,k σ˜ 1,k σ˜ 2,k ΩX k+1=ΩZ 0,k+1 Ω(ι 0,k+1) ΩZ 1,k+1 Ω(ι 1,k+1) ΩZ 2,k+1 Ω(ι 2,k+1) . \array{ X_k = Z_{0,k} &\underoverset{\in I_{Top} Cell}{\iota_{0,k}}{\longrightarrow}& Z_{1,k} &\underoverset{\in I_{Top} Cell}{\iota_{1,k}}{\longrightarrow}& Z_{2,k} &\underoverset{\in I_{Top} Cell}{\iota_{2,k}}{\longrightarrow}& \cdots \\ {}^{\mathllap{\tilde \sigma_k = \tilde \sigma_{0,k}}}\downarrow && {}^{\mathllap{\tilde \sigma_{1,k}}}\downarrow && {}^{\mathllap{\tilde \sigma_{2,k}}}\downarrow && \cdots \\ \Omega X_{k+1} = \Omega Z_{0,k+1} &\overset{\Omega (\iota_{0,k+1})}{\longrightarrow}& \Omega Z_{1,k+1} &\overset{\Omega (\iota_{1,k+1})}{\longrightarrow}& \Omega Z_{2,k+1} &\overset{\Omega (\iota_{2,k+1})}{\longrightarrow}& \cdots } \,.

That this indeed commutes is the identity

σ˜ i+1,kι i,k =(Ω(ι i,k+1)ϕ i,k)ι i,k =Ω(ι i,k+1)(ϕ i,kι i,k) =Ω(ι i,k+1)σ˜ i,k. \begin{aligned} \tilde \sigma_{i+1,k} \circ \iota_{i,k} & = (\Omega(\iota_{i,k+1}) \circ \phi_{i,k}) \circ \iota_{i,k} \\ & = \Omega(\iota_{i,k+1}) \circ (\phi_{i,k} \circ \iota_{i,k}) \\ & = \Omega(\iota_{i,k+1}) \circ \tilde \sigma_{i,k} \end{aligned} \,.

Now let QXQ X be the spectrum with component spaces the colimit

(QX) klim iZ i,k (Q X)_k \coloneqq \underset{\longrightarrow}{\lim}_i Z_{i,k}

and with adjunct structure maps (via def. ) given by the map induced under colimits by the above diagrams

σ˜ k QXlimσ˜ i,k:QXΩ(QX). \tilde \sigma^{Q X}_k \;\coloneqq\; \underset{\longrightarrow}{\lim} \tilde \sigma_{i,k} \;\colon\; Q X \longrightarrow \Omega(Q X) \,.

Notice that this is indeed well-defined: since each component map X i,kX i+1,kX_{i,k} \to X_{i+1,k} is a relative cell complex and since the 1-sphere S 1S^1 is compact, it follows (lemma) that

lim iΩZ i,k =lim iMaps(S 1,Z i,k) * Maps(S 1,lim iZ i,k) * =Ωlim iZ i,k (ΩQX). \begin{aligned} \underset{\longrightarrow}{\lim}_{i} \Omega Z_{i,k} & = \underset{\longrightarrow}{\lim}_{i} Maps(S^1, Z_{i,k})_\ast \\ & \simeq Maps(S^1, \underset{\longrightarrow}{\lim}_i Z_{i,k} )_\ast \\ & = \Omega \underset{\longrightarrow}{\lim}_i Z_{i,k} \\ & \simeq (\Omega Q X) \end{aligned} \,.

Finally, let

η X:XQX \eta_X \colon X \to Q X

be degreewise the inclusion of the first component (i=0i = 0) into the colimit. By construction, this is a homomorphism of sequential spectra (according to def. ).

Proposition

Let XSeqSpec(Top cg)X\in SeqSpec(Top_{cg}) be a sequential prespectrum with j X:XQXj_X \colon X \to Q X from def. . Then:

  1. QXQ X is an Omega-spectrum (def. );

  2. η X:XQX\eta_X \colon X \to Q X is a stable weak homotopy equivalence (def. ):

  3. η X\eta_X is a level weak equivalence (is in W strictW_{strict}, def. ) precisely if XX is an Omega-spectrum;

  4. a morphism f:XYf \colon X \to Y is a stable weak homotopy equivalence (def. ), precisely if Qf:QXQYQ f \colon Q X \to Q Y is a level weak equivalence (is in W strictW_{strict}, def. ).

(Schwede 97, lemma 2.1.3 and remark before section 2.2)

Proof

Since the colimit defining QXQ X is a transfinite composition of relative cell complexes, each component map X k(QX) kX_k \to (Q X)_k is itself a relative cell complex. Since n-spheres are compact topological spaces, it follows (lemma) that each element of a homotopy group in π ((QX) k)\pi_\bullet((Q X)_k) is in the image of a finite stage π (Z i,k)\pi_\bullet(Z_{i,k}) for some ii \in \mathbb{N}. From this, all statements follow by inspection at finite stages.

Regarding first statement:

Since each σ˜ i,k\tilde \sigma_{i,k} by construction is a weak homotopy equivalence followed by an inclusion of stages in the colimit, as any element of π q((QX) k)\pi_q((Q X)_k) is sent along σ˜ k QX\tilde \sigma^{Q X}_k it passes through one such π q(σ˜ i,k)\pi_q(\tilde \sigma_{i ,k}) at some stage ii, hence also through all the following, and is hence identically preserved in the colimit.

Regarding the second statement:

By the previous statement and by example , the map π (η X):π (X)π (QX)\pi_\bullet(\eta_X) \colon \pi_\bullet(X)\to \pi_\bullet(Q X) is given in degree q0q \geq 0 by

lim kπ q+k(X k)lim kπ q(Ω kX k)π q((QX) 0) \underset{ \simeq \underset{\longrightarrow}{\lim}_k \pi_q \left(\Omega^k X_k\right) } { \underbrace{ \underset{\longrightarrow}{\lim}_{k \in \mathbb{N}} \pi_{q+k}(X_k) }} \longrightarrow \pi_q \left( (Q X)_0 \right)

and similarly in degree q<0q\lt 0. Now using the compactness of the spheres and the definition of QQ we compute on the right:

π q((QX) 0) =π q(lim kZ k,0) lim kπ q(Z k,0) lim kπ q(Ω kX k), \begin{aligned} \pi_q( (Q X)_0 ) & = \pi_q (\underset{\longrightarrow}{\lim}_k Z_{k,0}) \\ & \simeq \underset{\longrightarrow}{\lim}_k \pi_q ( Z_{k,0}) \\ & \simeq \underset{\longrightarrow}{\lim}_k \pi_q ( \Omega^k X_k ) \end{aligned} \,,

where the last isomorphism is π q\pi_q applied to the composite of the weak homotopy equivalences

Z k,0W clϕ k1,0ΩZ k1,1Ω kZ 0,k=Ω kX k. Z_{k,0} \underoverset{\in W_{cl}}{\phi_{k-1,0}}{\longrightarrow} \Omega Z_{k-1,1} \to \cdots \to \Omega^k Z_{0,k} = \Omega^k X_k \,.

Regarding the third statement:

In one direction:

If XX is an Omega-spectrum in that all its adjunct structure maps σ˜ k\tilde \sigma_k are weak homotopy equivalences, then by two-out-of-three also the maps ι i,k\iota_{i,k} in def. are weak homotopy equivalences. Hence (j X) k:X k(QX) k(j_X)_k \colon X_k \to (Q X)_k is the map into a sequential colimit over acyclic relative cell complexes, and again by the compactness of the spheres, this means that it is itself a weak homotopy equivalence.

In the other direction:

If η X\eta_X is degrewise a weak homotopy equivalence, then by applying two-out-of-three (def.) to the compatibility squares for the adjunct structure morphisms (def. ), using that σ˜ n QX\tilde \sigma^{Q X}_n is a weak homotopy equivalence by the first point above

X n W cl(j X) n (QX) n σ˜ n X W cl σ˜ n QX Maps(S 1,X n+1) Maps(S 1,(j X) n+1)W cl Maps(S 1,(QX) n+1) \array{ X_n &\underoverset{\in W_{cl}}{(j_X)_n}{\longrightarrow}& (Q X)_n \\ {}^{\mathllap{\tilde \sigma^X_n}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma^{Q X}_n}}_{\mathrlap{\in W_{cl}}} \\ Maps(S^1,X_{n+1}) &\underoverset{\Maps(S^1,(j_X)_{n+1})}{\in W_{cl}}{\longrightarrow}& Maps(S^1, (Q X)_{n+1}) }

implies that also σ˜ n XW cl\tilde \sigma^X_n \in W_{cl}, hence that XX is an Omega-spectrum.

The fourth statement follows with similar reasoning.

Remark

In the case that XX is a CW-spectrum (def. ) then the sequence of resolutions in the definition of spectrification in def. is not necessary, and one may simply consider

(Q CWX) nlim kΩ kX n+k. (Q_{CW} X)_n \;\coloneqq\; \underset{\longrightarrow}{\lim}_k \Omega^k X_{n+k} \,.

See for instance (Lewis-May-Steinberger 86, p. 3) and (Weibel 94, 10.9.6 and topology exercise 10.9.2).

As topological diagrams

In order to conveniently understand the stable model category structure on spectra, we now consider an equivalent reformulation of the component-wise definition of sequential spectra, def. , as topologically enriched functors (defn.).

Definition

Write

ι:StdSpheresTop cg */ \iota \;\colon\; StdSpheres \longrightarrow Top_{cg}^{\ast/}

for the non-full topologically enriched subcategory (def.) of that of pointed compactly generated topological spaces (def.) where:

  • objects are the standard n-spheres S nS^n, for nn \in \mathbb{N}, identified as the smash product powers S n(S 1) nS^n \coloneqq (S^1)^{\wedge^n} of the standard circle;

  • hom-spaces are

    StdSpheres(S n,S k+n){* for k<0 S k otherwise StdSpheres(S^{n}, S^{k+n}) \coloneqq \left\{ \array{ \ast & for & k \lt 0 \\ S^k & otherwise } \right.
  • composition is induced from composition in Top cg */Top^{\ast/}_{cg} by regarding the hom-space S kS^k above as its image in Maps(S n,S k+n) *Maps({S^n},S^{k+n})_\ast under the adjunct

S kMaps(S n,S k+n) * S^{k} \stackrel{}{\to} Maps({S^n},S^{k+n})_\ast

of the canonical isomorphism

S kS nS k+n. S^k \wedge S^n \overset{\simeq}{\longrightarrow} S^{k+n} \,.

This induces the category

[StdSpheres,Top cg */] [StdSpheres, Top^{\ast/}_{cg}]

of topologically enriched functors on StdSpheresStdSpheres with values in Top cg */Top_{cg}^{\ast/} (exmpl.).

Proposition

There is an equivalence of categories

() seq:[StdSpheres,Top cg */]SeqSpec(Top cg) (-)^seq \;\colon\; [StdSpheres,\; Top_{cg}^{\ast/}] \overset{\simeq}{\longrightarrow} SeqSpec(Top_{cg})

from the category of topologically enriched functors on the category of standard spheres of def. to the category of topological sequential spectra, def. , which is given on objects by sending X[StdSpheres,Top cg */]X \in [StdSpheres,Top_{cg}^{\ast/}] to the sequential prespectrum X seqX^{seq} with components

X n seqX(S n) X^{seq}_n \coloneqq X(S^n)

and with structure maps

S 1X n seqσ nX n seqS 1Maps(X n seq,X n+1 seq) * \frac{ S^1 \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n }{ S^1 \longrightarrow Maps(X^{seq}_n, X^{seq}_{n+1})_\ast }

being the adjunct of the component map of XX on spheres of consecutive dimension.

Proof

First observe that from its components on consecutive spheres the functor XX is already uniquely determined. Indeed, by definition the hom-space between non-consecutive spheres StdSpheres(S n,S n+k)StdSpheres(S^n, S^{n+k}) is the smash product of the hom-spaces between the consecutive spheres, for instance:

S 1S 1 = StdSpheres(S n,S n+1)StdSpheres(S n+1,S n+2) S 2 = StdSpheres(S n,S n+2), \array{ S^1 \wedge S^1 & = & StdSpheres(S^n, S^{n+1}) \wedge StdSpheres(S^{n+1}, S^{n+2}) \\ {}^{\mathllap{\simeq}}\downarrow && {}^{\mathllap{\simeq}}\downarrow^{\circ} \\ S^2 & = & StdSpheres(S^n , S^{n+2}) } \,,

and so functoriality completely fixes the former by the latter.

This means that we actually have a bijection between classes of objects.

Now observe that a natural transformation f:XYf \colon X \to Y between two functors on StdSpheresStdSpheres is equivalently a collection of component maps f n:X nY nf_n \colon X_n \to Y_n, such that for each sS 1s \in S^1 then the following squares commute

X(S n) f n Y S n X S n,S n+1(s) Y S n,S n+1(s) X(S n+1) f n+1 Y(S n+1), \array{ X(S^n) &\overset{f_n}{\longrightarrow}& Y^{S^{n}} \\ {}^{\mathllap{X_{S^n,S^{n+1}}(s)}}\downarrow && \downarrow^{\mathrlap{Y_{S^n,S^{n+1}}(s)}} \\ X(S^{n+1}) &\underset{f_{n+1}}{\longrightarrow}& Y(S^{n+1}) } \,,

By the smash/hom adjunction, the square equivalently factors as

X(S n) f n Y S n (s,id) (s,id) S 1X(S n) id×f n S 1Y(S n) σ n X σ n Y X(S n+1) undersetf n+1 Y(S n+1). \array{ X(S^n) &\overset{f_n}{\longrightarrow}& Y^{S^{n}} \\ {}^{\mathllap{(s,id)}}\downarrow && \downarrow^{\mathrlap{(s,id)}} \\ S^1 \wedge X(S^n) &\underset{id \times f_n}{\longrightarrow}& S^1 \wedge Y(S^n) \\ {}^{\mathllap{\sigma^X_n}} \downarrow && \downarrow^{\mathrlap{\sigma^Y_n}} \\ X(S^{n+1}) &underset{f_{n+1}}{\longrightarrow}& Y(S^{n+1}) } \,.

Here the top square commutes in any case, and so the total rectangle commutes precisely if the lower square commutes, hence if under our identification the components {f n}\{f_n\} constitute a homomorphism of sequential spectra.

Hence we have an isomorphism on all hom-sets, and hence an equivalence of categories.

Further below we use prop. to naturally induce a model structure on the category of topological sequential spectra.

Remark

Under the equivalence of prop. , the general concept of tensoring of topologically enriched functors over topological spaces (according to this def.) restricts to the concept of tensoring of sequential spectral over topological spaces according to def. .

Proposition

The category SeqSpec(Top cq)SeqSpec(Top_{cq}) of sequential spectra (def. ) has all limits and colimits, and they are computed objectwise:

Given

X :ISeqSpec(Top cg) X_\bullet \;\colon\; I \longrightarrow SeqSpec(Top_{cg})

a diagram of sequential spectra, then:

  1. its colimiting spectrum has component spaces the colimit of the component spaces formed in Top cgTop_{cg} (via this prop. and this corollary):

    (lim iX(i)) nlim iX(i) n, (\underset{\longrightarrow}{\lim}_i X(i))_n \simeq \underset{\longrightarrow}{\lim}_i X(i)_n \,,
  2. its limiting spectrum has component spaces the limit of the component spaces formed in Top cgTop_{cg} (via this prop. and this corollary):

    (lim iX(i)) nlim iX(i) n; (\underset{\longleftarrow}{\lim}_i X(i))_n \simeq \underset{\longleftarrow}{\lim}_i X(i)_n \,;

moreover:

  1. the colimiting spectrum has structure maps in the sense of def. given by

    S 1(lim iX(i) n)lim i(S 1X(i) n)lim iσ n X(i)lim iX(i) n+1 S^1 \wedge (\underset{\longrightarrow}{\lim}_i X(i)_n) \simeq \underset{\longrightarrow}{\lim}_i ( S^1 \wedge X(i)_n ) \overset{\underset{\longrightarrow}{\lim}_i \sigma_n^{X(i)}}{\longrightarrow} \underset{\longrightarrow}{\lim}_i X(i)_{n+1}

    where the first isomorphism exhibits that S 1()S^1 \wedge(-) preserves all colimits, since it is a left adjoint by prop. ;

  2. the limiting spectrum has adjunct structure maps in the sense of def. given by

    lim iX(i) nlim iσ˜ n X(i)lim iMaps(S 1,X(i) n) *Maps(S 1,lim iX(i) n) * \underset{\longleftarrow}{\lim}_i X(i)_n \overset{\underset{\longleftarrow}{\lim}_i \tilde \sigma_n^{X(i)}}{\longrightarrow} \underset{\longleftarrow}{\lim}_i Maps(S^1, X(i)_n)_\ast \simeq Maps(S^1, \underset{\longleftarrow}{\lim}_i X(i)_n)_\ast

    where the last isomorphism exhibits that Maps(S 1,) *Maps(S^1,-)_\ast preserves all limits, since it is a right adjoint by prop. .

Proof

That the limits and colimits exist and are computed objectwise follows via prop. from the general statement for categories of topological functors (prop.). But it is also immediate to directly check the universal property.

Example

The initial object and the terminal object in SeqSpec(Top cg)SeqSpec(Top_{cg}) agree and are both given by the spectrum constant on the point, which is also the suspension spectrum Σ *\Sigma^\infty \ast (def. ) of the point). We will denote this spectrum *\ast or 00 (since it is hence a zero object ):

* n=* \ast_n = \ast
S 1* n**. S^1 \wedge \ast_n \simeq \ast \overset{\simeq}{\to} \ast \,.
Example

The coproduct of spectra X,YSeqSpec(Top cg)X, Y \in SeqSpec(Top_{cg}), called the wedge sum of spectra

XYXY X \vee Y \coloneqq X \sqcup Y

is componentwise the wedge sum of pointed topological spaces (exmpl.)

(XY) n=X nY n (X \vee Y)_n = X_n \vee Y_n

with structure maps

σ n XY:S 1(XY)S 1XS 1Y(σ n X,σ n Y)X n+1Y n+1. \sigma_n^{X \vee Y} \;\colon\; S^1 \wedge (X \vee Y) \simeq S^1 \wedge X \,\vee\, S^1 \wedge Y \overset{(\sigma_n^X, \sigma_n^Y)}{\longrightarrow} X_{n+1} \vee Y_{n+1} \,.
Example

For XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}) a sequential spectrum, def. , its standard cylinder spectrum is its smash tensoring X(I +)X \wedge (I_+), according to def. , with the standard interval (def.) with a basepoint freely adjoined (def.). The component spaces of the cylinder spectrum are the standard reduced cylinders (def.) of the component spaces of XX:

(X(I +)) n=X nI +. (X \wedge (I_+))_n = X_n \wedge I_+ \,.

By the functoriality of the smash tensoring, the factoring

S 0:S 0S 0I +S 0 \nabla_{S^0} \;\colon\; S^0 \vee S^0 \longrightarrow I_+ \longrightarrow S^0

of the codiagonal on the 0-sphere through the standard interval with a base point adjoined, gives a factoring of the codiagonal of XX through its standard cylinder spectrum

X:XXX(S 0S 0I +)X(I +)X(I +S 0)X \nabla_X \;\colon\; X \vee X \overset{X \wedge (S^0 \vee S^0 \to I_+)}{\longrightarrow} X \wedge (I_+) \overset{X \wedge (I_+ \to S^0) }{\longrightarrow} X

(where we are using that wedge sum is the coproduct in pointed topological spaces (exmpl.).)

Suspension and looping

We discuss models for the operation of reduced suspension and forming loop space objects of sequential spectra.

Definition

For XX a sequential spectrum, then

  1. the standard suspension of XX is the smash product-tensoring XS 1X \wedge S^1 according to def. ;

  2. the standard looping of XX is the smash powering Maps(S 1,X) *Maps(S^1,X)_\ast according to def. .

Proposition

For XSeqSpec(Top cg)X\in SeqSpec(Top_{cg}), the standard suspension XS 1X \wedge S^1 of def. is equivalently the cofiber (formed via prop. ) of the canonical inclusion of boundaries into the standard cylinder spectrum X(I +)X \wedge (I_+) of example :

XS 1cofib(XXX(I +)). X \wedge S^1 \simeq cofib\left( X \vee X \to X \wedge (I_+) \right) \,.
Proof

This is immediate from the componentwise construction of the smash tensoring and the componentwise computation of colimits of spectra via prop. .

This means that once we know that XXX(I +)X\vee X \to X \wedge (I_+) is suitably a cofibration (to which we turn below) then the standard suspension is a homotopy-correct model for the suspension operation. However, some properties of suspension are hard to prove directly with the standard suspension model. For such there are two other models for suspension and looping of spectra. These three models are not isomorphic to each other in SeqSpec(Top cg)SeqSpec(Top_{cg}), but (this is lemma below) they will become isomorphic in the stable homotopy category (def. ).

Definition

For XX a sequential spectrum (def. ) and kk \in \mathbb{Z}, the kk-fold shifted spectrum of XX is the sequential spectrum denoted X[k]X[k] given by

  • (X[k]) n{X n+k forn+k0 * otherwise(X[k])_n \coloneqq \left\{ \array{X_{n+k} & for \; n+k \geq 0 \\ \ast & otherwise } \right. ;

  • σ n X[k]{σ n+k X forn+k0 0 otherwise\sigma_n^{X[k]} \coloneqq \left\{ \array{ \sigma^X_{n+k} & for \; n+k \geq 0 \\ 0 & otherwise} \right. .

Definition

For XX a sequential spectrum, def. , then

  1. the alternative suspension of XX is the sequential spectrum ΣX\Sigma X with

    1. (ΣX) nS 1X n(\Sigma X)_n \coloneqq S^1 \wedge X_n (smash product on the left (defn.))

    2. σ n ΣXS 1(σ n X)\sigma_n^{\Sigma X} \coloneqq S^1 \wedge (\sigma^X_n).

    in the sense of def. ;

  2. the alternative looping of XX is the sequential spectrum ΩX\Omega X with

    1. (ΩX) nMaps(S 1,X n) *(\Omega X)_n \coloneqq Maps(S^1,X_n)_\ast;

    2. σ˜ n ΩXMaps(S 1,σ˜ n X) *\tilde \sigma_n^{\Omega X} \coloneqq Maps(S^1,\tilde \sigma^X_n)_\ast

    in the sense of def. .

Remark

In various references the “alternative suspension” from def. is called the “fake suspension” (e.g. Goerss-Jardine 96, p. 499, Jardine 15, section 10.4).

Remark

There is no direct natural isomorphism between the standard suspension (def. ) and the alternative suspension (def. ). This is due to the non-trivial graded commutativity (braiding) of smash products of spheres. (We discuss braiding of the smash product more in detail in Part 1.2, this example).

Namely a natural isomorphism ϕ:ΣXXS 1\phi \colon \Sigma X \longrightarrow X \wedge S^1 (or alternatively the other way around) would have to make the following diagrams commute:

S 1S 1X n id S 1ϕ n S 1X nS 1 S 1σ n (nc) σ nS 1 S 1X n+1 ϕ n+1 X n+1S 1 \array{ S^1 \wedge S^1 \wedge X_n &\overset{id_{S^1} \wedge \phi_n}{\longrightarrow}& S^1 \wedge X_n \wedge S^1 \\ {}^{\mathllap{S^1 \wedge \sigma_n}}\downarrow &(nc)& \downarrow^{\mathrlap{\sigma_n \wedge S^1}} \\ S^1 \wedge X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1} \wedge S^1 }

and naturally so in XX.

The only evident option is to have ϕ\phi be the braiding homomorphisms of the smash product

ϕ n=τ S 1,X n:S 1X nX nS 1. \phi_n = \tau_{S^1, X_n} \;\colon\; S^1 \wedge X_{n} \overset{\simeq}{\to} X_{n} \wedge S^1 \,.

It may superficially look like this makes the above diagram commute, but it does not. To make this explicit, consider labeling the two copies of the circle appearing here as S a 1S^1_a and S b 1S^1_b. Then the diagram we are dealing with looks like this:

S a 1S b 1X n S a 1X nS b 1 S a 1σ n (nc) σ nS b 1 S a 1X n+1 X n+1S b 1 \array{ S_a^1 \wedge S_b^1 \wedge X_n &\longrightarrow& S_a^1 \wedge X_n \wedge S_b^1 \\ {}^{\mathllap{S^1_a \wedge \sigma_n}}\downarrow &(nc)& \downarrow^{\mathrlap{\sigma_n \wedge S^1_b}} \\ S_a^1 \wedge X_{n+1} &\underset{}{\longrightarrow}& X_{n+1} \wedge S_b^1 }

If we had S a 1σ nS^1_a \wedge \sigma_n on the left and σ nS a 1\sigma_n \wedge S^1_a on the right, then the naturality of the braiding would indeed give a commuting diagram. But since this is not the case, the only way to achieve this would be by exchanging in the top left

S a 1S b 1S b 1S a 1. S^1_a \wedge S^1_b \longrightarrow S^1_b \wedge S^1_a \,.

However, this map is non-trivial. It represents 1-1 in [S 2,S 2] *=π 2(S 2)=[S^2, S^2]_\ast = \pi_2(S^2) = \mathbb{Z}. Hence inserting this map in the top of the previous diagram still does not make it commute.

But this technical problem points to its own solutions: if we were to restrict to the homotopy category of spectra which had structure maps only of the form S 2X nX n+2S^2 \wedge X_n \to X_{n+2}, then the braiding required to make the two models of suspension comparable would be

S a 2S b 1S b 1S a 2 S^2_a \wedge S^1_b \longrightarrow S^1_b \wedge S^2_a

and this map is indeed trivial, up to homotopy. This we make precise as lemma below.

More generally, the kind of issue encountered here is taken care of by the concept of symmetric spectra, to which we turn in Part 1.2.

Remark

The looping and suspension operations in def. and def. commute with shifting, def. . Therefore in expressions like Σ(X[1])\Sigma (X[1]) etc. we may omit the parenthesis.

Proposition

The constructions from def. , def. and def. form pairs of adjoint functors SeqSpecSeqSpecSeqSpec \to SeqSpec like so:

  1. ()[1]()[1](-)[-1] \dashv (-)[1];

  2. ()S 1Maps(S 1,) *(-)\wedge S^1 \dashv Maps(S^1,-)_\ast;

  3. ΣΩ\Sigma \dashv \Omega.

Proof

Regarding the first statement:

A morphism of the form f:X[1]Yf \;\colon\; X[-1] \longrightarrow Y has components of the form

X 2 f 2 Y 3 X 1 f 2 Y 2 X 0 f 1 Y 1 * f 0=0 Y 0 \array{ \vdots && \vdots \\ X_2 &\overset{f_2}{\longrightarrow}& Y_3 \\ X_1 &\overset{f_2}{\longrightarrow}& Y_2 \\ X_0 &\overset{f_1}{\longrightarrow}& Y_1 \\ \ast &\overset{f_0 = 0}{\longrightarrow}& Y_0 }

and the compatibility condition with the structure maps in lowest degree is automatically satisfied

* (S 1f 0)=0 S 1Y 0 σ 0 X[1]=0 σ 0 Y X 0 f 1 Y 1. \array{ \ast &\overset{(S^1 \wedge f_0) = 0}{\longrightarrow}& S^1 \wedge Y_0 \\ {}^{\mathllap{\sigma^{X[-1]}_0 = 0}}\downarrow && \downarrow^{\mathrlap{\sigma^Y_0}} \\ X_0 &\overset{f_1}{\longrightarrow}& Y_1 } \,.

Therefore this is equivalent to components

X 2 f 2 Y 3 X 1 f 2 Y 2 X 0 f 1 Y 1 \array{ \vdots && \vdots \\ X_2 &\overset{f_2}{\longrightarrow}& Y_3 \\ X_1 &\overset{f_2}{\longrightarrow}& Y_2 \\ X_0 &\overset{f_1}{\longrightarrow}& Y_1 }

hence to a morphism XY[1]X \longrightarrow Y[1].

The second statement is a special case of prop. .

Regarding the third statement:

This follows by applying the (smash product\dashvpointed mapping space)-adjunction isomorphism twice, like so:

Morphisms f:ΣXYf\colon \Sigma X \to Y in the sense of def. are in components given by commuting diagrams of this form:

S 1S 1X n S 1f n S 1Y n S 1σ n X σ n Y S 1X n+1 f n+1 Y n+1. \array{ S^1 \wedge S^1 \wedge X_{n} &\overset{S^1 \wedge f_{n}}{\longrightarrow}& S^1 \wedge Y_{n} \\ {}^{\mathllap{S^1 \wedge \sigma_n^X}}\downarrow && \downarrow^{\mathrlap{\sigma^Y_n}} \\ S^1 \wedge X_{n+1} &\underset{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.

Applying the adjunction isomorphism diagonally gives a natural bijection to diagrams of this form:

S 1X n f n Y n σ n X σ˜ n Y X n+1 f n+1˜ Maps(S 1,Y n+1) *. \array{ S^1 \wedge X_n &\overset{f_n}{\longrightarrow}& Y_n \\ {}^{\mathllap{\sigma^X_n}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma^Y_n}} \\ X_{n+1} &\underset{\widetilde {f_{n+1}}}{\longrightarrow}& Maps(S^1,Y_{n+1})_\ast } \,.

(To see this in full detail, for instance for the adjunct of the left and bottom morphism: chase the identity id S 1X n+1id_{S^1 \wedge X_{n+1}} in both ways

Hom(S 1X n+1,S 1X n+1) Hom(X n+1,Maps(S 1,S 1X n+1) *) Hom(S 1σ n X,f n+1) Hom(σ n X,Maps(S 1,f n+1) *) Hom(S 1S 1X n,Y n+1) Hom(S 1X n,Maps(S 1,Y n+1) *) \array{ Hom(S^1 \wedge X_{n+1}, S^1 \wedge X_{n+1}) &\overset{\simeq}{\longrightarrow}& Hom(X_{n+1}, Maps(S^1, S^1 \wedge X_{n+1})_\ast) \\ {}^{\mathllap{Hom(S^1 \wedge \sigma^X_n, f_{n+1})}}\downarrow && \downarrow^{\mathrlap{Hom(\sigma^X_n, Maps(S^1, f_{n+1})_\ast)}} \\ Hom(S^1 \wedge S^1 \wedge X_n, Y_{n+1}) &\overset{\simeq}{\longrightarrow}& Hom(S^1 \wedge X_n, Maps(S^1, Y_{n+1})_\ast) }

through the adjunction naturality square. The other cases follow analogously.)

Then applying the adjunction isomorphism diagonally once more gives a further bijection to commuting diagrams of this form:

X n f n˜ Maps(S 1,Y n) * σ˜ n Maps(S 1,σ˜ n Y) * Maps(S 1,X n+1) * Maps(S 1,f n+1˜) * Maps(S 1,Maps(S 1,Y n+1) *) *. \array{ X_n &\overset{\widetilde {f_n}}{\longrightarrow}& Maps(S^1,Y_n)_\ast \\ {}^{\mathllap{\tilde \sigma_n}}\downarrow && \downarrow^{\mathrlap{Maps(S^1,\tilde \sigma^Y_n)_\ast}} \\ Maps(S^1, X_{n+1})_\ast &\underset{Maps(S^1,\widetilde{f_{n+1}})_\ast}{\longrightarrow}& Maps\left(S^1, Maps(S^1,Y_{n+1})_\ast\right)_\ast } \,.

This, finally, equivalently exhibits homomorphisms of the form

XΩY X \longrightarrow \Omega Y

in the sense of def. .

Proposition

The following diagram of adjoint pairs of functors commutes:

Top cg */ ΩΣ Top cg */ Σ Ω Σ Ω SeqSpec(Top cg) ΩΣ SeqSpec(Top cg), \array{ Top_{cg}^{\ast/} & \underoverset{\underoverset{\Omega}{\bot}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{} & Top^{\ast/}_{cg} \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ SeqSpec(Top_{cg}) & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} & SeqSpec(Top_{cg}) } \,,

Here the top horizontal adjunction is from prop. , the vertical adjunction is from prop. and the bottom adjunction is from prop. .

Proof

It is sufficient to check

Σ ΣΣΣ . \Sigma^\infty \circ \Sigma \simeq \Sigma \circ \Sigma^\infty \,.

From this the statement

Ω ΩΩΩ \Omega^\infty \circ \Omega \simeq \Omega \circ \Omega^\infty

follows by uniqueness of adjoints.

So let XTop cg */X \in Top_{cg}^{\ast/}. Then

  • (ΣΣ X) n=S 1S nX(\Sigma \Sigma^\infty X)_n = S^1 \wedge S^n \wedge X,

  • σ n (ΣΣ X):S 1S 1S nXS 1idS 1S 1+nX\sigma^{(\Sigma \Sigma^\infty X)}_n \colon S^1 \wedge S^1 \wedge S^n \wedge X \overset{S^1 \wedge id}{\longrightarrow} S^1 \wedge S^{1+n} \wedge X,

while

  • (Σ ΣX) n=S nS 1X(\Sigma^\infty \Sigma X)_n = S^n \wedge S^1 \wedge X,

  • σ n (Σ ΣX):S 1S nS 1XidS 1XS 1+nS 1X\sigma_n^{(\Sigma^\infty \Sigma X)}\colon S^1\wedge S^n \wedge S^1 \wedge X \overset{id \wedge S^1 \wedge X}{\longrightarrow} S^{1+n} \wedge S^1 \wedge X,

where we write “id” for the canonical isomorphism. Clearly there is a natural isomorphism given by the canonical identifications

S 1S nX(S 1) n+1XS nS 1X. S^1 \wedge S^n \wedge X \overset{\simeq}{\longrightarrow} (S^1)^{\wedge^{n+1}}\wedge X \overset{\simeq}{\longrightarrow} S^n \wedge S^1 \wedge X \,.

(As long as we are not smash-permuting the S 1S^1 factor with the S nS^n factor – and here we are not – then the fact that they get mixed under this isomorphism is irrelevant. The point where this does become relevant is the content of remark below.)

The strict model structure on sequential spectra

The model category structure on sequential spectra which presents stable homotopy theory is the “stable model structure” discussed below. Its fibrant-cofibrant objects are (in particular) Omega-spectra, hence are the proper spectrum objects among the pre-spectrum objects.

But for technical purposes it is useful to also be able to speak of a model structure on pre-spectra, which sees their homotopy theory as sequences of simplicial sets equipped with suspension maps, but not their stable structure. This is called the “strict model structure” for sequential spectra. Its main point is that the stable model structure of interest arises from it via left Bousfield localization.

Definition

Say that a homomorphism f :X Y f_\bullet \colon X_\bullet \to Y_\bullet in the category SeqSpec(Top)SeqSpec(Top), def. is

We write W strictW_{strict}, Fib strictFib_{strict} and Cof strictCof_{strict} for these classes of morphisms, respectively.

Recall the sets

I Top */{S + n1(ι n) +D + n} n I_{Top^{\ast/}} \coloneqq \{S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+\}_{n \in \mathbb{N}}
J Top */{D n(j n) +D n×I} n J_{Top^{\ast/}} \coloneqq \{D^n \overset{(j_n)_+}{\longrightarrow} D^n \times I\}_{n \in \mathbb{N}}

of standard generating (acyclic) cofibrations (def.) of the classical model structure on pointed topological spaces (thm.).

Definition

Write

I seq strict{y(S n)i +} S nStdSpheres,i +I Top */[StdSpheres,Top */]SeqSpec(Top) I_{seq}^{strict} \coloneqq \left\{ y(S^n) \cdot i_+ \right\}_{{S^n \in StdSpheres,} \atop {i_+ \in I_{Top^{\ast/}}}} \;\; \in [StdSpheres, Top^{\ast/}] \simeq SeqSpec(Top)

and

J seq strict{y(S n)j +} S nStdSpheresj +J Top */[StdSpheres,Top */]SeqSpec(Top), J_{seq}^{strict} \coloneqq \left\{ y(S^n) \cdot j_+ \right\}_{{ S^n \in StdSpheres} \atop {j_+ \in J_{Top^{\ast/}}}} \;\; \in [StdSpheres, Top^{\ast/}] \simeq SeqSpec(Top) \,,

for the set of morphisms arising as the tensoring (remark ) of a representable (exmpl.) with a generating acyclic cofibration of the classical model structure on pointed topological spaces (def.).

Theorem

The classes of morphisms in def. give the structure of a model category (def.) to be denoted SeqSpec(Top) strictSeqSpec(Top)_{strict} and called the strict model structure on topological sequential spectra (or: level model structure).

Moreover, this is a cofibrantly generated model category with generating (acyclic) cofibrations the set I seq strictI_{seq}^{strict} (resp. J seq strictJ_{seq}^{strict}) from def. .

Proof

Prop. says that the category of sequential spectra is equivalently an enriched functor category

SeqSpec(Top)[StdSpheres,Top cg */]. SeqSpec(Top) \simeq [StdSpheres,\; Top_{cg}^{\ast/}] \,.

Accordingly, this carries the projective model structure on functors (thm.). This immediately gives the statement for the fibrations and the weak equivalences.

It only remains to check that the cofibrations are as claimed. To that end, consider a commuting square of sequential spectra

X h A f Y B. \array{ X &\stackrel{h}{\longrightarrow}& A \\ \downarrow^{\mathrlap{f}} && \downarrow \\ Y &\longrightarrow& B } \,.

By definition, this is equivalently an \mathbb{N}-collection of commuting diagrams in Top cgTop_{cg} of the form

X n h n A n f n Y n B n \array{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} && \downarrow \\ Y_n &\longrightarrow& B_n }

such that all structure maps are respected.

S 1X n σ n X X n+1 S 1f n f n+1 S 1Y n σ n Y Y n+1 S 1B n σ n B B n+1=S 1X n σ n X X n+1 S 1h n h n+1 S 1A n σ n A A n+1 S 1B n σ n B B n+1. \array{ S^1 \wedge X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ \downarrow^{\mathrlap{S^1 \wedge f_n}} && \downarrow^{\mathrlap{f_{n+1}}} \\ S^1 \wedge Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1} \\ & \searrow && \searrow \\ && S^1 \wedge B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1} } \;\;\; = \;\;\; \array{ S^1 \wedge X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ & \searrow^{\mathrlap{S^1 \wedge h_n}} && \searrow^{\mathrlap{h_{n+1}}} \\ && S^1 \wedge A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1} \\ && \downarrow && \downarrow \\ && S^1 \wedge B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1} } \,.

Hence a lifting in the original diagram is a lifting in each degree nn, such that the lifting in degree n+1n+1 makes these diagrams of structure maps commute.

Since components are parameterized over \mathbb{N}, this condition has solutions by induction:

First of all there must be an ordinary lifting in degree 0. Since the strict fibrations are degreewise classical fibrations, this gives the condition that for f f_\bullet to be a strict cofibration, then f 0f_0 is to be a classical cofibration.

Then assume that a lifting l nl_n in degree nn has been found

X n h n A n f n l n Y n B n. \array{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} &\nearrow_{\mathrlap{l_n}}& \downarrow \\ Y_n &\longrightarrow& B_n } \,.

Now the lifting l n+1l_{n+1} in the next degree has to also make the following diagram commute

S 1X n σ n X X n+1 S 1f n f n+1 h n+1 S 1Y n σ n Y Y n+1 S 1l n l n+1 S 1A n σ n A A n+1. \array{ S^1 \wedge X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ \downarrow^{\mathrlap{S^1 \wedge f_n}} && \downarrow^{\mathrlap{f_{n+1}}} & \searrow^{\mathrlap{h_{n+1}}} \\ S^1 \wedge Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1} && \\ & \searrow^{\mathrlap{S^1 \wedge l_n}} && \searrow^{\mathrlap{l_{n+1}}} & \downarrow \\ && S^1 \wedge A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1} } \,.

This is a cocone under the commuting square for the structure maps, and therefore the outer diagram is equivalently a morphism out of the domain of the pushout product f nσ n Xf_n \Box \sigma_n^X (def.), while the compatible lift l n+1l_{n+1} is equivalently a lift against this pushout product:

S 1Y nS 1X nX n+1 (σ n AS 1l n,h n+1) A n+1 f nσ n X l n+1 Y n+1 B n+1. \array{ S^1 \wedge Y_n \underset{S^1 \wedge X_n}{\sqcup} X_{n+1} &\stackrel{(\sigma_n^A \circ S^1 \wedge l_n,\;h_{n+1})}{\longrightarrow}& A_{n+1} \\ {}^{\mathllap{f_n \Box \sigma^X_n}}\downarrow &{}^{\mathllap{l_{n+1}}}\nearrow& \downarrow \\ Y_{n+1} &\stackrel{}{\longrightarrow}& B_{n+1} } \,.

This shows that f f_\bullet is a strict cofibration precisely if, in addition to f 0f_0 being a classical cofibration, all these pushout products are classical cofibrations.

Suspension and looping

Proposition

The (Σ Ω )(\Sigma^\infty \dashv \Omega^\infty)-adjunction from prop. is a Quillen adjunction (def.) between the classical model structure on pointed topological spaces (thm., prop.) and the strict model structure on topological sequential spectra of theorem :

(Σ Ω ):SeqSpec(Top cg) strictΩ Σ (Top cg */) Quillen. (\Sigma^\infty \dashv \Omega^\infty) \;\colon\; SeqSpec(Top_{cg})_{strict} \underoverset {\underset{\Omega^{\infty}}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {\bot} (Top_{cg}^{\ast/})_{Quillen} \,.
Proof

It is clear that Ω \Omega^\infty preserves fibrations and acyclic cofibrations. This is sufficient to deduce a Quillen adjunction.

Just for the record, we spell out a direct argument that also Σ \Sigma^\infty preserves cofibrations and acyclic cofibrations:

Let f:XYf \colon X\longrightarrow Y be a morphism in Top cg */Top^{\ast/}_{cg} and

Σ f:Σ XΣ Y \Sigma^\infty f \colon \Sigma^\infty X \longrightarrow \Sigma^\infty Y

its image.

Since the structure maps in a suspension spectrum, example , are all isomorphisms, we have for all nn \in \mathbb{N} an isomorphism

(Σ X) n+1S 1(Σ X) nS 1(Σ Y) nS 1(Σ Y) n. (\Sigma^\infty X)_{n+1} \underset{S^1 \wedge (\Sigma^\infty X)_n}{\coprod} S^1 \wedge (\Sigma^\infty Y)_n \simeq S^1 \wedge (\Sigma^\infty Y)_n \,.

Therefore Σ f\Sigma^\infty f is a strict cofibration, according to def. , precisely if (Σ f) 0=f(\Sigma^\infty f)_0 = f is a classical cofibration and all the structure maps of Σ Y\Sigma^\infty Y are classical cofibrations. But the latter are even isomorphisms, so that this is no extra condition (prop.). Hence Σ \Sigma^\infty sends classical cofibrations of spaces to strict cofibrations of sequential spectra.

Furthermore, since S n():(Top cg */) Quillen(Top cg */) QuillenS^n \wedge (-) \colon (Top_{cg}^{\ast/})_{Quillen} \to (Top_{cg}^{\ast/})_{Quillen} is a left Quillen functor for all nn \in \mathbb{N} by prop. it sends classical acyclic cofibrations to classical acyclic cofibrations. Hence Σ \Sigma^\infty, which is degreewise given by S n()S^n \wedge(-), sends classical acyclic cofibrations to degreewise acyclic cofibrations, hence in particular to degreewise weak equivalences, hence to weak equivalences in the strict model structure on sequential spectra.

This shows that Σ \Sigma^\infty is a left Quillen functor.

Proposition

The (ΣΩ)(\Sigma \dashv \Omega)-adjunction from prop. is a Quillen adjunction (def.) with respect to the strict model structure on sequential spectra of theorem .

SeqSpec(Top cg) strictΩΣSeqSpec(Top cg) strict. SeqSpec(Top_{cg})_{strict} \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} SeqSpec(Top_{cg})_{strict} \,.
Proof

Since the (acyclic) fibrations of SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} are by definition those morphisms that are degreewise (acylic) fibrations in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen}, the statement follows immediately from the fact that the right adjoint Ω\Omega is degreewise given by Maps(S 1,) *:(Top cg */) Quillen(Top cg */) QuillenMaps(S^1, -)_\ast \colon (Top^{\ast/}_{cg})_{Quillen} \to (Top^{\ast/}_{cg})_{Quillen}, which is a right Quillen functor by prop. .

In summary, prop. , prop. and prop. say that

Corollary

The commuting square of adjunctions in prop. is a square of Quillen adjunctions with respect to the classical model structure on pointed compactly generated topological spaces (thm., prop.) and the strict model structure on topological sequential spectra of theorem :

(Top cg */) Quillen ΩΣ (Top cg */) Quillen Σ Ω Σ Ω SeqSpec(Top cg) strict ΩΣ SeqSpec(Top cg) strict, \array{ (Top_{cg}^{\ast/})_{Quillen} & \underoverset{\underoverset{\Omega}{\bot}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{} & (Top^{\ast/}_{cg})_{Quillen} \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ SeqSpec(Top_{cg})_{strict} & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} & SeqSpec(Top_{cg})_{strict} } \,,

Further below we pass to the stable model structure in order to make the bottom adjunction in this diagram become a Quillen equivalence. This stable model structure will have more weak equivalences than the strict model structure, but will have the same cofibrations. Therefore we first consider now cofibrancy conditions already in the strict model structure.

CW-spectra

Definition

A sequential spectrum XX (def. ) is called a cell spectrum if

  1. all component spaces X nX_n are cell complexes (def.);

  2. all structure maps σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1} are relative cell complex inclusions.

A CW-spectrum is a cell spectrum such that all component spaces X nX_n are CW-complexes (def.).

Example

The suspension spectrum Σ X\Sigma^\infty X (example ) for XTop cg */X \in Top^{\ast/}_{cg} a CW-complex is a CW-spectrum (def. ).

Remark

Since, by definition , a pp-cell of a cell spectrum that appears at stage qq shows up as its kk-fold suspension at stage q+kq+k, its attachment to some spectrum XX is reflected by a pushout of spectra of the form

Σ S + 1[q] X * Σ (i p) +[q] (po) (po) Σ D + p[q] X^ Σ S p[q], \array{ \Sigma^\infty S^{-1}_+[-q] &\longrightarrow& X &\longrightarrow& \ast \\ {}^{\mathllap{\Sigma^\infty (i_{p})_+[-q]}} \downarrow &(po)& \downarrow &(po)& \downarrow \\ \Sigma^\infty D^{p}_+[-q] &\longrightarrow& \hat X &\longrightarrow& \Sigma^\infty S^p[-q] } \,,

where the left vertical morphism is the image under the q-qth shift spectrum functor (def. ) of the image under the suspension spectrum functor (example ) of the basic cell inclusion (i p) +(i_p)_+ of pointed topological spaces (def.). This is a cofibration by prop. , and so also the middle vertical morphism is a cofibration, by theorem . Using the pasting law for pushouts, we find that the cofiber of the middle vertical morphisms (hence its homotopy cofiber (def.) in the strict model structure) is Σ S p[q]\Sigma^\infty S^p[-q] (not Σ S + p[q]\Sigma^\infty S^p_+[-q]\;(!)). This is a shift of a trunction of the sphere spectrum.

After having set up the stable model category structure in theorem below, we find that this means that cell attachments to CW-spectra in the stable model structure are by cofibers of integer shifts of the sphere spectrum 𝕊\mathbb{S} (def. ), in that in the stable homotopy category (def. ) the above situation is reflected as a homotopy cofiber sequence of the form

Σ pq1𝕊XX^Σ pq𝕊. \Sigma^{p-q-1} \mathbb{S} \longrightarrow X \longrightarrow \hat X \longrightarrow \Sigma^{p-q} \mathbb{S} \,.
Lemma

Let κ\kappa be an regular cardinal and let XX be a κ\kappa-cell spectrum, hence a cell spectrum (def. ) obtained from at most κ\kappa stable cell attachments as in remark . Then XX is κ\kappa-small (def.) with respect to morphisms of spectra that are degreewise relative cell complex inclusions.

Proof

By remark the attachment of stable cells is by free spectra (def. ) on compact topological spaces. By prop. maps out of them are equivalently maps of component spaces in the lowest nontrivial degree. Since compact topological spaces are small with respect to relative cell complex inclusions (lemma), all these cells are small.

Now notice that κ\kappa-filtered colimits of sets commute with κ\kappa-small limtis of sets (prop.). By assumption XX is a κ\kappa-small transfinite composition of pushouts of κ\kappa-small coproducts, all three of which are κ\kappa-small colimits; and let YY be the codomain of a κ\kappa-small relative cell complex inclusion, hence itself a κ\kappa-small colimit.

Now if A=lim nσ nA = \underset{\longrightarrow}{\lim}_n \sigma_n is a κ\kappa-small colimit of κ\kappa-small objects σ n\sigma_n, and Y=lim iY iY = \underset{\longrightarrow}{\lim}_i Y_i is a κ\kappa-small colimit, then

Hom(A,lim iY i) Hom(lim σc σ,lim iY i) lim σHom(c σ,lim iY i) lim σlim iHom(c σ,Y i) lim ilim σHom(c σ,Y i) lim iHom(lim σc σ,Y i) lim iHom(A,Y i). \begin{aligned} Hom( A, \underset{\longrightarrow}{\lim}_i Y_i ) & \simeq Hom( \underset{\longrightarrow}{\lim}_\sigma c_\sigma, \underset{\longrightarrow}{\lim}_i Y_i ) \\ & \simeq \underset{\longleftarrow}{\lim}_\sigma Hom(c_\sigma, \underset{\longrightarrow}{\lim}_i Y_i ) \\ & \simeq \underset{\longleftarrow}{\lim}_\sigma \, \underset{\longrightarrow}{\lim}_i Hom(c_\sigma, Y_i ) \\ & \simeq \underset{\longrightarrow}{\lim}_i \, \underset{\longleftarrow}{\lim}_\sigma Hom(c_\sigma, Y_i ) \\ & \simeq \underset{\longrightarrow}{\lim}_i Hom(\underset{\longrightarrow}{\lim}_\sigma c_\sigma, Y_i ) \\ & \simeq \underset{\longrightarrow}{\lim}_i Hom(A, Y_i ) \end{aligned} \,.

Hence the claim follows.

Proposition

The class of CW-spectra is closed under various operations, including

Proposition

A sequential spectrum XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}) is cofibrant in the strict model structure SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} of theorem precisely if

  1. X 0X_0 is cofibrant;

  2. each structure map σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \to X_{n+1} is a cofibration

in the classical model structure (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} on pointed compactly generated topological spaces (thm., prop.).

In particular cell spectra and specifically CW-spectra (def. ) are cofibrant.

Proof

The initial object in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} is the spectrum *\ast that is constant on the point (example ). A morphism *X\ast \to X is a cofibration according to def. if

  1. the morphism *X 0\ast \to X_0 is a classical cofibration, hence if the object X 0X_0 is a classical cofibrant object, hence a retract of a cell complex;

  2. the morphisms

    * n+1S 1* nS 1X nX n+1 \ast_{n+1}\underset{S^1 \wedge \ast_n}{\sqcup} S^1 \wedge X_n \longrightarrow X_{n+1}

    are classical cofibrations. But since S 1***S^1 \wedge \ast \simeq \ast \overset{\simeq}{\to} \ast is an isomorphism in this case the pushout reduces to just its second summand, and so this is now equivalent to

    S 1X nX n+1 S^1 \wedge X_n \longrightarrow X_{n+1}

    being classical cofibrations; hence retracts of relative cell complexes.

Proposition

For XSeqSpec(Top) stableX\in SeqSpec(Top)_{stable} a CW-spectrum, def. , then its standard cylinder spectrum X(I +)X \wedge (I_+) of def. satisfies the conditions on an abstract cylinder object (def.) in that the inclusion

XXX(I +) X \vee X \longrightarrow X \wedge (I_+)

(of the wedge sum of XX with itself, example ) is a cofibration in SeqSpec(Top) stableSeqSpec(Top)_{stable}.

Proof

According to def. we need to check that for all nn the morphism

(XX) n+1S 1(XX) nS 1(X(I +)) n(X(I +)) n+1 (X \vee X)_{n+1} \underset{S^1 \wedge (X\vee X)_n}{\sqcup} S^1 \wedge (X \wedge (I_+))_n \longrightarrow (X \wedge (I_+))_{n+1}

is a retract of a relative cell complex. After distributing indices and smash products over wedge sums, this is equivalently

(X n+1X n+1)(S 1X n)(S 1X n))S 1X n(I +)X n+1(I +). (X_{n+1} \vee X_{n+1}) \underset{(S^1 \wedge X_n )\vee (S^1 \wedge X_n))}{\sqcup} S^1 \wedge X_n \wedge (I_+) \longrightarrow X_{n+1} \wedge (I_+) \,.

Now by the assumption that XX is a CW-spectrum, each X nX_{n} is a CW-complex, and this implies that X n(I +)X_n \wedge (I_+) is a relative cell complex in Top */Top^{\ast/}. With this, inspection shows that also the above morphism is a relative cell complex.

We now turn to discussion of CW-approximation of sequential spectra. First recall the relative version of CW-approximation for topological spaces.

For the following, recall that a continuous function f:XYf \colon X \to Y between topological spaces is called an n-connected map if the induced morphism on homotopy groups π (f):π (X,x)π (Y,f(x))\pi_\bullet(f)\colon \pi_\bullet(X,x) \to \pi_\bullet(Y,f(x)) is

  1. an isomorphism in degree <n\lt n;

  2. an epimorphism in degree nn.

(Hence an weak homotopy equivalence is an “\infty-connected map”.)

Lemma

Let f:AXf \;\colon\; A \longrightarrow X be a continuous function between topological spaces. Then there exists for each nn \in \mathbb{N} a relative CW-complex f^:AY^\hat f \colon A \hookrightarrow \hat Y together with an extension ϕ:YX\phi \colon Y \to X, i.e.

A f X f^ ϕ X^ \array{ A &\overset{f}{\longrightarrow}& X \\ {}^{\mathllap{\hat f}}\downarrow & \nearrow_{\mathrlap{\phi}} \\ \hat X }

such that ϕ\phi is n-connected.

Moreover:

  • if ff itself is k-connected, then the relative CW-complex f^\hat f may be chosen to have cells only of dimension k+1dimnk + 1 \leq dim \leq n.

  • if AA is already a CW-complex, then f^:AX\hat f \colon A \to X may be chosen to be a subcomplex inclusion.

(tomDieck 08, theorem 8.6.1)

Proposition

For every continuous function f:AXf \colon A \longrightarrow X out of a CW-complex AA, there exists a relative CW-complex f^:AX^\hat f \colon A \longrightarrow \hat X that factors ff followed by a weak homotopy equivalence

A f X f^ ϕWHE X^. \array{ A && \overset{f}{\longrightarrow} && X \\ & {}_{\mathllap{\hat f}}\searrow && \nearrow_{\mathrlap{{\phi} \atop {\in WHE}}} \\ && \hat X } \,.
Proof

Apply lemma iteratively for nn \in \mathbb{N} to produce a sequence with cocone of the form

A f 0 X 0 f 2 X 1 f ϕ 0 ϕ 1 X, \array{ A &\overset{f_0}{\longrightarrow}& X_0 &\overset{f_2}{\longrightarrow}& X_1 &\longrightarrow & \cdots \\ &{}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\phi_0}} & \swarrow_{\mathrlap{\phi_1}} & \cdots \\ && X } \,,

where each f nf_n is a relative CW-complex adding cells exactly of dimension nn, and where ϕ n\phi_n in n-connected.

Let then X^\hat X be the colimit over the sequence (its transfinite composition) and f^:AX\hat f \colon A \to X the induced component map. By definition of relative CW-complexes, this f^\hat f is itself a relative CW-complex.

By the universal property of the colimit this factors ff as

A f 0 X 0 f 1 X 1 X^ ϕ X. \array{ A &\overset{f_0}{\longrightarrow}& X_0 &\overset{f_1}{\longrightarrow}& X_1 &\longrightarrow & \cdots \\ &{}_{\mathllap{}}\searrow & \downarrow^{\mathrlap{}} & \swarrow_{\mathrlap{}} & \cdots \\ && \hat X \\ && \downarrow^{\mathrlap{\phi}} \\ && X } \,.

Finally to see that ϕ\phi is a weak homotopy equivalence: since n-spheres are compact topological spaces, then every map α:S nX^\alpha \colon S^n \to \hat X factors through a finite stage ii \in \mathbb{N} as S nX iX^S^n \to X_i \to \hat X (by this lemma). By possibly including further into higher stages, we may choose i>ni \gt n. But then the above says that further mapping along X^X\hat X \to X is the same as mapping along ϕ i\phi_i, which is (i>n)(i \gt n)-connected and hence an isomorphism on the homotopy class of α\alpha.

Proposition

For XX any topological sequential spectrum (def.), then there exists a CW-spectrum X^\hat X (def. ) and a homomorphism

ϕ:X^W strictX. \phi \;\colon\; \hat X \overset{\in W_{strict}}{\longrightarrow} X \,.

which is degreewise a weak homotopy equivalence, hence a weak equivalence in the strict model structure of theorem .

Proof

First let X^ 0X 0\hat X_0 \longrightarrow X_0 be a CW-approximation of the component space in degree 0, via prop. . Then proceed by induction: suppose that for nn \in \mathbb{N} a CW-approximation ϕ kn:X^ knX kn\phi_{k \leq n} \colon \hat X_{k \leq n} \to X_{k \leq n} has been found such that all the structure maps in degrees <n\lt n are respected. Consider then the composite continuous function

S 1X^ nS 1ϕ nS 1X nσ nX n+1. S^1 \wedge \hat X_n \overset{S^1 \wedge \phi_n}{\longrightarrow} S^1 \wedge X_n \overset{\sigma_n}{\longrightarrow} X_{n+1} \,.

Applying prop. to this function factors it as

S 1X^ nX^ n+1ϕ n+1X n+1. S^1 \wedge \hat X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1} \,.

Hence we have obtained the next stage X^ n+1\hat X_{n+1} of the CW-approximation. The respect for the structure maps is just this factorization property:

S 1X^ n S 1ϕ n S 1X n incl σ n X^ n+1 ϕ n+1 X n+1. \array{ S^1 \wedge \hat X_n &\overset{S^1 \wedge \phi_n}{\longrightarrow}& S^1 \wedge X_n \\ {}^{incl}\downarrow && \downarrow^{\mathrlap{\sigma_n}} \\ \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1} } \,.

Topological enrichment

We discuss here how the hom-set of homomorphisms between any two sequential spectra is naturally equipped with a topology, and how these hom-spaces interact well with the strict model structure on sequential spectra from theorem . This is in direct analogy to the compatibility of compactly generated mapping spaces (def.) with the classical model structure on compactly generated topological spaces discussed at Classical homotopy theory – Topological enrichment. It gives an improved handle on the analysis of morphisms of spectra below in the proof of the stable model structure and it paves the way to the discussion of fully fledge mapping spectra below in part 1.2. There we will give a fully general account of the principles underlying the following. Here we just consider a pragmatic minimum that allows us to proceed.

Definition

For X,YSeqSpec(Top cg)X, Y \in SeqSpec(Top_{cg}) two sequential spectra (def. ) let

SeqSpec(X,Y)Top cg */ SeqSpec(X,Y) \in Top_{cg}^{\ast/}

be the pointed topological space whose underlying set is the hom-set Hom SeqSpec(Top cg)(X,Y)Hom_{SeqSpec(Top_{cg})}(X,Y) of homomorphisms from XX to YY, and which is equipped with the final topology (def.) generated by those functions

ϕ:KHom SeqSpec(Top cg)(X,Y) \phi \;\colon\; K \longrightarrow Hom_{SeqSpec(Top_{cg})}(X,Y)

out of compact Hausdorff spaces KK, for which there exists a homomorphism of spectra

ϕ˜:XKY \tilde \phi \;\colon\; X\wedge K \longrightarrow Y

out of the smash tensoring of XX with KK (def. ) such that for all yKy \in K, nn \in \mathbb{N}, xX nx \in X_n

ϕ(y) n(x)=ϕ˜ n(x,y). \phi(y)_n(x) = \tilde \phi_n(x,y) \,.

By construction this makes SeqSpec(X,Y)SeqSpec(X,Y) indeed into a compactly generated topological space, and it gives a natural bijection

Hom Top cg */(K,SeqSpec(X,Y))Hom SeqSpec(Top cg */)(XK,Y). Hom_{Top^{\ast/}_{cg}}(K,\, SeqSpec(X,Y)) \;\simeq\; Hom_{SeqSpec(Top^{\ast/}_{cg})}( X \wedge K ,\, Y ) \,.

In Prelude – Classical homotopy theory we discussed, in the section Topological enrichment, that the classical model structure on topological spaces (when restricted to compactly generated topological spaces) interacts well with forming smash products and pointed mapping spaces. Concretely, the smash pushout product of two classical cofibrations is a classical cofibration, and is acyclic if either of the factors is:

Cof clCof clCof cl,(Cof clW cl)Cof clCof clW cl. Cof_{cl} \Box Cof_{cl} \subset Cof_{cl} \;\,, \;\;\;\;\;\;\; (Cof_{cl} \cap W_{cl}) \Box Cof_{cl} \subset Cof_{cl} \cap W_{cl} \,.

We also saw that, by Joyal-Tierney calculus (prop.), this is equivalent to the pullback powering satisfying the dual relations

Fib cl Cof clFib cl,Fib cl (Cof clW cl)Fib clW cl,(Fib clW cl) Cof clFib clW cl. Fib_{cl}^{\Box Cof_{cl}} \subset Fib_{cl} \;\,, \;\:\;\; Fib_{cl}^{\Box (Cof_{cl} \cap W_{cl})} \subset Fib_{cl} \cap W_{cl} \;\,, \;\;\; (Fib_{cl} \cap W_{cl})^{\Box Cof_{cl}} \subset Fib_{cl} \cap W_{cl} \,.

Now that we passed from spaces to spectra, def. generalizes the smash product of spaces to the smash tensoring of sequential spectra by spaces, and generalizes the pointed mapping space construction for spaces to the powering of a space into a sequential spectrum. Accordingly there is now the analogous concept of pushout product with respect to smash tensoring, and of pullback powering with respect to smash powering.

From the way things are presented, it is immediate that these operations on spectra satisfy the analogous compatibility condition with the strict model structure on spectra from theorem , in fact this follows generally for topologically enriched functor categories and is inherited via prop. . But since this will be important for some of the discussion to follow, we here make it explicit:

Definition

Let f:XYf \;\colon \; X \to Y be a morphism in SeqSpec(Top cg)SeqSpec(Top_{cg}) (def. ) and let i:ABi \;\colon\; A \to B a morphism in Top cg */Top_{cg}^{\ast/}.

Their pushout product with respect to smash tensoring is the universal morphism

fi((id,i),(f,id)) f \Box i \coloneqq \left((id,i), (f,id)\right)

in

XA (f,id) (id,i) YA (po) XB (YA)XA(XB) ((id,i),(f,id)) YB, \array{ && X \wedge A \\ & {}^{\mathllap{(f,id)}}\swarrow && \searrow^{\mathrlap{(id,i)}} \\ Y \wedge A && (po) && X \wedge B \\ & {}_{\mathllap{}}\searrow && \swarrow \\ && (Y \wedge A) \underset{X \wedge A}{\sqcup} (X \wedge B) \\ && \downarrow^{\mathrlap{((id, i), (f,id))}} \\ && Y \wedge B } \,,

where ()()(-)\wedge(-) denotes the smash tensoring from def. .

Dually, their pullback powering is the universal morphism

f i(Maps(B,f) *,Maps(i,X) *) f^{\Box i} \coloneqq (Maps(B,f)_\ast, Maps(i,X)_\ast)

in

Maps(B,X) * (Maps(B,f) *,Maps(i,X) *) Maps(B,Y) *×Maps(A,Y) *Maps(A,X) * Maps(B,Y) * (pb) Maps(A,X) * Maps(i,Y) * Maps(A,p) * Maps(A,Y) *, \array{ && Maps(B,X)_\ast \\ && \downarrow^{\mathrlap{(Maps(B,f)_\ast, Maps(i,X)_\ast)}} \\ && Maps(B,Y)_\ast \underset{Maps(A,Y)_\ast}{\times} Maps(A,X)_\ast \\ & \swarrow && \searrow \\ Maps(B,Y)_\ast && (pb) && Maps(A,X)_\ast \\ & {}_{\mathllap{Maps(i,Y)_\ast}}\searrow && \swarrow_{\mathrlap{Maps(A,p)_\ast}} \\ && Maps(A,Y)_\ast } \,,

where Maps(,) *Maps(-,-)_\ast denotes the smash powering from def. .

Similarly, for f:XYf \colon X \to Y and i:ABi \colon A \to B both morphisms of sequential spectra, then their pullback powering is the universal morphism

f i(SeqSpec(B,f),SeqSpec(i,X)) f^{\Box i} \coloneqq (SeqSpec(B,f), SeqSpec(i,X))

in

SeqSpec(B,X) * (SeqSpec(B,f) *,SeqSpec(i,X) *) SeqSpec(B,Y) *×SeqSpec(A,Y) *SeqSpec(A,X) * SeqSpec(B,Y) * (pb) SeqSpec(A,X) * SeqSpec(i,Y) * SeqSpec(A,p) * SeqSpec(A,Y) *, \array{ && SeqSpec(B,X)_\ast \\ && \downarrow^{\mathrlap{(SeqSpec(B,f)_\ast, SeqSpec(i,X)_\ast)}} \\ && SeqSpec(B,Y)_\ast \underset{SeqSpec(A,Y)_\ast}{\times} SeqSpec(A,X)_\ast \\ & \swarrow && \searrow \\ SeqSpec(B,Y)_\ast && (pb) && SeqSpec(A,X)_\ast \\ & {}_{\mathllap{SeqSpec(i,Y)_\ast}}\searrow && \swarrow_{\mathrlap{SeqSpec(A,p)_\ast}} \\ && SeqSpec(A,Y)_\ast } \,,

where now SeqSpec(,)SeqSpec(-,-) is the hom-space functor from def. .

Proposition

The operation of forming pushout products with respect to smash tensoring in def. is compatible with the strict model structure on sequential spectra from theorem and with the classical model structure on compactly generated pointed topological spaces (thm., prop.) in that it takes two cofibrations to a cofibration, and to an acyclic cofibration if at least one of the inputs is acyclic:

Cof strictCof cl Cof strict Cof strict(Cof clW cl) Cof strictW strict (Cof strictW strict)Cof cl Cof strictW strict. \begin{aligned} Cof_{strict} \Box Cof_{cl} & \subset\; Cof_{strict} \\ Cof_{strict} \Box (Cof_{cl} \Box W_{cl}) & \subset\; Cof_{strict} \cap W_{strict} \\ (Cof_{strict} \cap W_{strict}) \Box Cof_{cl} & \subset\; Cof_{strict} \cap W_{strict} \end{aligned} \,.

Dually, the pullback powering satisfies

Fib strict Cof cl Fib strict Fib strict (Cof clW cl) Fib strictW strict (Fib strictW strict) Cof cl Fib strictW strict. \begin{aligned} Fib_{strict}^{\Box Cof_{cl}} & \subset\; Fib_{strict} \\ Fib_{strict}^{\Box ( Cof_{cl} \cap W_{cl})} & \subset\; Fib_{strict}\cap W_{strict} \\ (Fib_{strict} \cap W_{strict})^{\Box Cof_{cl}} & \subset\; Fib_{strict} \cap W_{strict} \end{aligned} \,.
Proof

The statement concering the pullback powering follows directly form the analogous statement for topological spaces (prop.) by the fact that via theorem the fibrations and weak equivalences in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} are degree-wise those in (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen}. From this the statement about the pushout product follows dually by Joyal-Tierney calculus (prop.).

Remark

In the language of model category-theory, prop. says that SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} is an enriched model category, the enrichment being over (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen}. This is often referred to simply as a “topological model category”.

Proposition

For XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}) a sequential spectrum, fMor(SeqSpec(Top cg))f \in Mor(SeqSpec(Top_{cg})) any morphism of sequential spectra, and for gMor(Top cpt */)g \in Mor(Top_{cpt}^{\ast/}) a morphism of compact Hausdorff spaces, then the hom-spaces of def. interact with the pushout-product and pullback-powering from def. in that there is a natural isomorphism

SeqSpec(fg,X)SeqSpec(f,X) g. SeqSpec(f \Box g, X) \simeq SeqSpec(f,X)^{\Box g} \,.
Proposition

For X,YSeqSpec(Top cg)X,Y \in SeqSpec(Top_{cg}) two sequential spectra with XX a CW-spectrum (def. ), then there is a natural bijection

π 0SeqSpec(X,Y)[X,Y] strict \pi_0 SeqSpec(X,Y) \simeq [X,Y]_{strict}

between the connected components of the hom-space from def. and the hom-set in the homotopy category (def.) of the strict model structure from theorem .

Proof

By def. the path components of the hom-space are the left homotopy classes of morphisms of spectra with respect to the standard cylinder spectrum of def. :

I +SeqSpec(X,Y)X(I +)Y. \frac{ I_+ \longrightarrow SeqSpec(X,Y) }{ X \wedge (I_+) \longrightarrow Y } \,.

By prop. , for XX a CW-spectrum then the standard cylinder spectrum X(I +)X \wedge (I_+) is a good cyclinder object (def.) on a cofibrant object.

Since moreover every object in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} is fibrant, the statement follows (with this lemma).

The stable model structure on sequential spectra

The actual spectrum objects of interest in stable homotopy theory are not the pre-spectra of def. , but the Omega-spectra of def. among them. Hence we need to equip the category of sequential pre-spectra of def. with a model structure (def.) whose fibrant-cofibrant objects are, in particular Omega-spectra. More in detail, it is plausible to require that every pre-spectrum is weakly equivalent to a fibrant-cofibrant one which is both an Omega-spectrum and a CW-spectrum as in def. . By prop. this suggests to construct a model category structure on SeqSpec(Top cg)SeqSpec(Top_{cg}) that has the same cofibrations as the strict model structure of theorem , but more weak equivalences (and hence less fibrations), such as to make every sequential pre-spectrum weakly equivalent to an Omega cell spectrum.

Such a situation is called a Bousfield localization of a model category.

Bousfield localization

In plain category theory, a reflective localization of a category 𝒞\mathcal{C} is equivalently a full subcategory

i:𝒞 loc𝒞 i \;\colon\; \mathcal{C}_{loc} \hookrightarrow \mathcal{C}

such that the inclusion functor has a left adjoint LL

𝒞 lociL𝒞. \mathcal{C}_{loc} \underoverset {\underset{i}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{C} \,.

The adjunction unit η X:XL(X)\eta_X \colon X \to L(X) “reflects” every object XX of 𝒞\mathcal{C} into one in the 𝒞 loc\mathcal{C}_{loc}, and therefore this is also called a reflective subcategory inclusion.

It is a classical fact (Gabriel-Zisman 67, prop.) that in this situation

𝒞 loc𝒞[W L 1] \mathcal{C}_{loc} \simeq \mathcal{C}[W^{-1}_L]

is equivalently the localization (def.) of 𝒞\mathcal{C} at the “LL-equivalences”, namely at those morphisms ff such that L(f)L(f) is an isomorphism. Hence one also speaks of reflective localizations.

The following concept of Bousfield localization of model categories is the evident lift of this concept of reflective localization from the realm of categories to the realm of model categories (def.), where isomorphism is generealized to weak equivalence and where adjoint functors are taken to exhibit Quillen adjunctions.

Definition

A left Bousfield localization 𝒞 loc\mathcal{C}_{loc} of a model category 𝒞\mathcal{C} (def.) is another model category structure on the same underlying category with the same cofibrations,

Cof loc=Cof Cof_{loc} = Cof

but more weak equivalences

W locW. W_{loc} \supset W \,.

Notice that:

Proposition

Given a left Bousfield localization 𝒞 loc\mathcal{C}_{loc} of 𝒞\mathcal{C} as in def. , then

  1. Fib locFibFib_{loc} \subset Fib;

  2. W locFib loc=WFibW_{loc} \cap Fib_{loc} = W \cap Fib;

  3. the identity functors constitute a Quillen adjunction

    𝒞 locidid𝒞. \mathcal{C}_{loc} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot} \mathcal{C} \,.
  4. the induced adjunction of derived functors (prop.) exhibits a reflective subcategory inclusion of homotopy categories (def.)

    Ho(𝒞 loc)id𝕃idHo(𝒞). Ho(\mathcal{C}_{loc}) \underoverset {\underset{\mathbb{R} id}{\longrightarrow}} {\overset{\mathbb{L}id}{\longleftarrow}} {\bot} Ho(\mathcal{C}) \,.
Proof

Regarding the first two items:

Using the properties of the weak factorization systems (def.) of (acyclic cofibrations, fibrations) and (cofibrations, acyclic fibrations) for both model structures we get

Fib loc =(Cof locW loc)Inj (Cof locW)Inj =Fib \begin{aligned} Fib_{loc} &= (Cof_{loc} \cap W_{loc})Inj \\ &\subset (Cof_{loc} \cap W)Inj \\ & = Fib \end{aligned}

and

Fib locW loc =Cof locInj =CofInj =FibW. \begin{aligned} Fib_{loc} \cap W_{loc} & = Cof_{loc} Inj \\ & = Cof \, Inj \\ & = Fib \cap W \end{aligned} \,.

Regarding the third point:

By construction, id:𝒞𝒞 locid \colon \mathcal{C} \to \mathcal{C}_{loc} preserves cofibrations and acyclic cofibrations, hence is a left Quillen functor.

Regarding the fourth point:

Since Cof loc=CofCof_{loc} = Cof the notion of left homotopy in 𝒞 loc\mathcal{C}_{loc} is the same as that in 𝒞\mathcal{C}, and hence the inclusion of the subcategory of local cofibrant-fibrant objects into the homotopy category of the original cofibrant-fibrant objects is clearly a full inclusion. Since Fib locFibFib_{loc} \subset Fib by the first statement, on these cofibrant-fibrant objects the right derived functor of the identity is just the identity and hence does exhibit this inclusion. The left adjoint to this inclusion is given by 𝕃id\mathbb{L}id, by the general properties of Quillen adjunctions (prop).

In plain category theory, given a reflective subcategory

𝒞 lociL𝒞 \mathcal{C}_{loc} \underoverset {\underset{i}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{C}

then the composite

QiL:𝒞𝒞 Q \coloneqq i \circ L \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}

is an idempotent monad on 𝒞\mathcal{C}, hence, in particular, an endofunctor equipped with a natural transformation η X:XLX\eta_X \;\colon\; X \to L X (the adjunction unit) – which “reflects” every object into one in the image of LL – such that this reflection is a projection in that each L(η X)L(\eta_X) is an isomorphism. This characterizes the reflective subcategory 𝒞 loc𝒞\mathcal{C}_{loc} \hookrightarrow \mathcal{C} as the subcategory of those objects XX for which η X\eta_X is an isomorphism.

The following is the lift of this alternative perspective of reflective localization via idempotent monads from category theory to model category theory.

Definition

Let 𝒞\mathcal{C} be a model category (def.) which is right proper (def.), in that pullback along fibrations preserves weak equivalences.

Say that a Quillen idempotent monad on 𝒞\mathcal{C} is

  1. an endofunctor

    Q:𝒞𝒞Q \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}

  2. a natural transformation

    η:id 𝒞Q\eta \;\colon\; id_{\mathcal{C}} \longrightarrow Q

such that

  1. (homotopical functor) QQ preserves weak equivalences;

  2. (idempotency) for all X𝒞X \in \mathcal{C} the morphisms

    Q(η X):Q(X)WQ(Q(X)) Q(\eta_X) \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))

    and

    η Q(X):Q(X)WQ(Q(X)) \eta_{Q(X)} \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))

    are weak equivalences;

  3. (right-properness of the localization) if in a pullback square in 𝒞\mathcal{C}

    f *Z f *h X (pb) f Z h Y \array{ f^\ast Z &\overset{f^\ast h}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Z &\underset{h}{\longrightarrow}& Y }

    we have that

    1. ff is a fibration;

    2. η X\eta_X, η Y\eta_Y, and Q(h)Q(h) are weak equivalences

    then Q(f *h)Q(f^\ast h) is a weak equivalence.

Definition

For Q:𝒞𝒞Q \colon \mathcal{C} \longrightarrow \mathcal{C} a Quillen idempotent monad according to def. , say that a morphism ff in 𝒞\mathcal{C} is

  1. a QQ-weak equivalence if Q(f)Q(f) is a weak equivalence;

  2. a QQ-cofibation if it is a cofibration.

  3. a QQ-fibration if it has the right lifting property against the morphisms that are both (QQ-)cofibrations as well as QQ-weak equivalences.

Write

𝒞 Q \mathcal{C}_Q

for 𝒞\mathcal{C} equipped with these classes of morphisms.

Since QQ preserves weak equivalences (by def. ) then if the classes of morphisms in def. do constitute a model category structure, then this is a left Bousfield localization of 𝒞\mathcal{C}, according to def. .

We establish a couple of lemmas that will prove that the model structure indeed exists (prop. below).

Lemma

In the situation of def. , a morphism is an acyclic fibration in 𝒞 Q\mathcal{C}_Q precisely if it is an acyclic fibration in 𝒞\mathcal{C}.

Proof

Let ff be a fibration and a weak equivalence. Since QQ preserves weak equivalences by condition 1 in def. , ff is also a QQ-weak equivalence. Since QQ-cofibrations are cofibrations, the acyclic fibration ff has right lifting against QQ-cofibrations, hence in particular against against QQ-acyclic QQ-cofibrations, hence is a QQ-fibration.

In the other direction, let f:XYf \;\colon\; X \longrightarrow Y be a QQ-acyclic QQ-fibration. Consider its factorization into a cofibration followed by an acyclic fibration

f:XCofiZWFibpY. f \;\colon\; X \underoverset{\in Cof}{i}{\longrightarrow} Z \underoverset{\in W \cap Fib}{p}{\longrightarrow} Y \,.

Observe that QQ-equivalences satisfy two-out-of-three (def.), by functoriality and since the plain equivalences do. Now the assumption that QQ preserves weak equivalences together with two-out-of-three implies that ii is a QQ-weak equivalence, hence a QQ-acyclic QQ-cofibration. This implies that ff has the right lifting property against ii (since ff is assumed to be a QQ-fibration, which is defined by this lifting property). Hence the retract argument (prop.) implies that ff is a retract of the acyclic fibration pp, and so is itself an acyclic fibration.

Lemma

In the situation of def. , if a morphism f:XYf \colon X \longrightarrow Y is a fibration, and if η X,η Y\eta_X, \eta_Y are weak equivalences, then ff is a QQ-fibration.

(e.g. Goerss-Jardine 96, chapter X, lemma 4.4)

Proof

We need to show under the given assumptions that for every commuting square of the form

A α X W QCof Q i f B β Y \array{ A &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{i}}_{\mathllap{\in W_Q \cap Cof_Q}}\downarrow && \downarrow^{\mathrlap{f}} \\ B &\underset{\beta}{\longrightarrow}& Y }

there exists a lifting.

To that end, first consider a factorization of the image under QQ of this square as follows:

Q(A) Q(α) Q(X) Q(i) Q(f) Q(B) Q(β) Q(Y)Q(A) WCofj α Z Fibp α Q(X) Q(i) π Q(f) Q(B) j βWCof W p βFib Q(Y) \array{ Q(A) &\overset{Q(\alpha)}{\longrightarrow}& Q(X) \\ {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\mathrlap{Q(f)}} \\ Q(B) &\underset{Q(\beta)}{\longrightarrow}& Q(Y) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ Q(A) &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}& Z &\underoverset{\in Fib}{p_\alpha}{\longrightarrow}& Q(X) \\ {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\pi} && \downarrow^{\mathrlap{Q(f)}} \\ Q(B) &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}& W &\underoverset{p_\beta}{\in Fib}{\longrightarrow}& Q(Y) }

(This exists even without assuming functorial factorization: factor the bottom morphism, form the pullback of the resulting p βp_\beta, observe that this is still a fibration, and then factor (through j αj_\alpha) the universal morpism from the outer square into this pullback.)

Now consider the pullback of the right square above along the naturality square of η:idQ\eta \colon id \to Q, take this to be the right square in the following diagram

α: A (j αη A,α) Z×Q(X)X X i (π,f) f β: B (j βη B,β) W×Q(Y)Y Y, \array{ \alpha \colon & A &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}& Z \underset{Q(X)}{\times} X &\overset{}{\longrightarrow}& X \\ & {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{(\pi,f)}} && \downarrow^{\mathrlap{f}}& \\ \beta \colon & B &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}& W \underset{Q(Y)}{\times} Y &\underset{}{\longrightarrow}& Y } \,,

where the left square is the universal morphism into the pullback which is induced from the naturality squares of η\eta on α\alpha and β\beta.

We claim that (π,f)(\pi,f) here is a weak equivalence. This implies that we find the desired lift by factoring (π,f)(\pi,f) into an acyclic cofibration followed by an acyclic fibration and then lifting consecutively as follows

α: A (j αη A,α) Z×Q(X)X X id WCof Fib f A AAAAAAA Y Cof i WFib id β: B (j βη B,β) W×Q(Y)Y Y. \array{ \alpha \colon & A &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}& Z \underset{Q(X)}{\times} X &\overset{}{\longrightarrow}& X \\ & {}^{\mathllap{id}}\downarrow && {}^{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}}& \\ & A &\longrightarrow& &\overset{\phantom{AAAAAAA}}{\longrightarrow}& Y \\ & {}^{\mathllap{i}}_{\mathllap{\in Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{id}}& \\ \beta \colon & B &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}& W \underset{Q(Y)}{\times} Y &\longrightarrow& Y } \,.

To see that (ϕ,f)(\phi,f) indeed is a weak equivalence:

Consider the diagram

Q(A) WCofj α Z Wpr 1 Z×Q(X)X W Q(i) π (π,f) Q(B) j βWCof Z pr 2W W×Q(X)X. \array{ Q(A) &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}& Z &\underoverset{\in W}{pr_1}{\longleftarrow}& Z \underset{Q(X)}{\times} X \\ {}^{\mathllap{Q(i)}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\pi}} && \downarrow^{\mathrlap{(\pi,f)}} \\ Q(B) &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}& Z &\underoverset{pr_2}{\in W}{\longleftarrow}& W \underset{Q(X)}{\times} X } \,.

Here the projections are weak equivalences as shown, because by assumption in def. the ambient model category is right proper and these projections are the pullbacks along the fibrations p αp_\alpha and p βp_\beta of the morphisms η X\eta_X and η Y\eta_Y, respectively, where the latter are weak equivalences by assumption. Moreover Q(i)Q(i) is a weak equivalence, since ii is a QQ-weak equivalence.

Hence now it follows by two-out-of-three (def.) that π\pi and then (π,f)(\pi,f) are weak equivalences.

Proposition

(Bousfield-Friedlander theorem)

Let 𝒞\mathcal{C} be a right proper model category. Let Q:𝒞𝒞Q \colon \mathcal{C} \longrightarrow \mathcal{C} be a Quillen idempotent monad on 𝒞\mathcal{C}, according to def. .

Then the Bousfield localization model category 𝒞 Q\mathcal{C}_Q (def. ) at the QQ-weak equivalences (def. ) exists, in that the model structure on 𝒞\mathcal{C} with the classes of morphisms in def. exists.

(Bousfield-Friedlander 78, theorem 8.7, Bousfield 01, theorem 9.3, Goerss-Jardine 96, chapter X, lemma 4.5, lemma 4.6, theorem 4.1)

Proof

The existence of limits and colimits is guaranteed since 𝒞\mathcal{C} is already assumed to be a model category. The two-out-of-three poperty for QQ-weak equivalences is an immediate consequence of two-out-of-three for the original weak equivalences of 𝒞\mathcal{C}. Moreover, according to lemma the pair of classes (Cof Q,W QFib Q)(Cof_{Q}, W_Q \cap Fib_Q) equals the pair (Cof,WFib)(Cof, W \cap Fib), and this is a weak factorization system by the model structure 𝒞\mathcal{C}.

Hence it remains to show that (W QCof Q,Fib Q)(W_Q \cap Cof_Q, \; Fib_Q) is a weak factorization system. The condition Fib Q=RLP(W QCof Q)Fib_Q = RLP(W_Q \cap Cof_Q) holds by definition of Fib QFib_Q. Once we show that every morphism factors as W QCof QW_Q \cap Cof_Q followed by Fib QFib_Q, then the condition W QCof Q=LLP(Fib Q)W_Q \cap Cof_Q = LLP(Fib_Q) follows from the retract argument (lemma) and the fact that the classes W QW_Q and Cof QCof_Q are closed under retracts, because WW and Cof=Cof QCof = Cof_Q are (by this prop. and this prop., respectively).

So we may conclude by showing the existence of (W QCof Q,Fib Q)(W_Q \cap Cof_Q, \; Fib_Q) factorizations:

First we consider the case of morphisms of the form f:Q(Y)Q(Y)f \colon Q(Y) \to Q(Y). These may be factored with respect to 𝒞\mathcal{C} as

f:Q(X)WCofiZFibpQ(Y). f \;\colon\; Q(X) \underoverset{\in W \cap Cof}{\in i}{\longrightarrow} Z \underoverset{\in Fib}{p}{\longrightarrow} Q(Y) \,.

Here ii is already a QQ-acyclic QQ-cofibration, since QQ preserves weak equivalences by the first clause in def. . Now apply idηQid \overset{\eta}{\to} Q to obtain

f: Q(X) WCofi Z Fibp Q(Y) W η Q(X) η Z W η Q(Y) Q(Q(X)) Q(i)W Q(Z) Q(Q(Y)), \array{ f \colon & Q(X) &\underoverset{\in W \cap Cof}{i}{\longrightarrow}& Z &\underoverset{\in Fib}{p}{\longrightarrow}& Q(Y) \\ & \downarrow^{\mathrlap{\eta_{Q(X)}}}_{\mathrlap{\in W}} && \downarrow^{\mathrlap{\eta_Z}} && \downarrow^{\mathrlap{\eta_{Q(Y)}}}_{\mathrlap{\in W}} \\ & Q(Q(X)) &\underoverset{Q(i)}{\in W}{\longrightarrow}& Q(Z) &\underset{}{\longrightarrow}& Q(Q(Y)) } \,,

where η Q(X)\eta_{Q(X)} and η Q(Y)\eta_{Q(Y)} are weak equivalences by idempotency (the second clause in def. ), and Q(i)Q(i) is a weak equivalence since QQ preserves weak equivalences. Hence by two-out-of-three also η Z\eta_Z is a weak equivalence. Therefore lemma gives that pp is a QQ-fibration, and hence the above factorization is already as desired

f:Q(X)W QCof QiZFib QpQ(Y). f \;\colon\; Q(X) \underoverset{\in W_Q \cap Cof_Q}{\in i}{\longrightarrow} Z \underoverset{\in Fib_Q}{p}{\longrightarrow} Q(Y) \,.

Now for an arbitrary morphism g:XYg \colon X \to Y, form a factorization of Q(g)Q(g) as above and then decompose the naturality square for η\eta on gg into the pullback of the resulting QQ-fibration along η Y\eta_Y:

g: X i˜ Z×Q(Y)Y p˜Fib Q Y W Q η X η (pb) W Q η Y Q(g): Q(Y) iW Q Z pFib Q Q(Y). \array{ g \colon & X &\overset{\tilde i}{\longrightarrow}& Z \underset{Q(Y)}{\times} Y &\overset{\tilde p \in Fib_{Q}}{\longrightarrow}& Y \\ & {}^{\mathllap{\eta_X}}_{\mathllap{\in W_Q}}\downarrow && \downarrow^{\mathrlap{\eta'}}_{\mathrlap{}} &(pb)& \downarrow^{\mathrlap{\eta_Y}}_{\mathrlap{\in W_Q}} \\ Q(g) \colon & Q(Y) &\underoverset{i}{\in W_Q}{\longrightarrow}& Z &\underoverset{p}{\in Fib_Q}{\longrightarrow}& Q(Y) } \,.

This exhibits η\eta' as the pullback of a QQ-weak equivalence along a fibration between objects on which η\eta is a weak equivalence. Then the third clause in def. says that η\eta' is itself as a QQ-weak equivalence. This way, two-out-of-three implies that i˜\tilde i is a QQ-weak equivalence.

Observe that p˜\tilde p is a QQ-fibration, because it is the pullback of a QQ-fibration and because QQ-fibrations are defined by a right lifting property (def. ) and hence closed under pullback (prop.) Finally, apply factorization in (Cof,WFib)(Cof,\; W\cap Fib) to i˜\tilde i to obtain the desired factorization

f:W QCofi˜ LWFib=W QFib Qi˜ RFib Qp˜. f \;\colon\; \underoverset{W_Q \cap Cof}{\tilde i_L}{\longrightarrow} \underoverset{W \cap Fib = W_Q \cap Fib_Q}{\tilde i_R}{\longrightarrow} \underoverset{Fib_Q}{\tilde p}{\longrightarrow} \,.

While this establishes the QQ-model structure, so far this leaves open a more explicit description of the QQ-fibrations. This is provided by the next statement.

Proposition

For Q:𝒞𝒞Q \colon \mathcal{C} \longrightarrow \mathcal{C} a Quillen idempotent monad according to def. , then a morphism f:XYf \colon X \to Y in 𝒞\mathcal{C} is a QQ-fibration (def. ) precisely if

  1. ff is a fibration;

  2. the η\eta-naturality square on ff

    X η X Q(X) f (pb) h Q(f) Y η Y Q(Y) \array{ X &\stackrel{\eta_X}{\longrightarrow}& Q(X) \\ {}^{\mathllap{f}}\downarrow &{}^{(pb)^h}& \downarrow^{\mathrlap{Q(f)}} \\ Y &\underset{\eta_Y}{\longrightarrow}& Q(Y) }

    exhibits a homotopy pullback in 𝒞\mathcal{C} (def.), in that for any factorization of Q(f)Q(f) through a weak equivalence followed by a fibration pp, then the universally induced morphism

    Xp *Y X \longrightarrow p^\ast Y

    is weak equivalence (in 𝒞\mathcal{C}).

(e.g. Goerss-Jardine 96, chapter X, theorem 4.8)

Proof

First consider the case that ff is a fibration and that the square is a homotopy pullback. We need to show that then ff is a QQ-fibration.

Factor Q(f)Q(f) as

Q(f):Q(X)WCofiZFibpQ(Y). Q(f) \;\colon\; Q(X) \underoverset{\in W \cap Cof}{i}{\longrightarrow} Z \underoverset{\in Fib}{p}{\longrightarrow} Q(Y) \,.

By the proof of prop. , the morphism pp is also a QQ-fibration. Hence by the existence of the QQ-local model structure, also due to prop. , its pullback p˜\tilde p is also a QQ-fibration

X η X Q(X) W i˜ W i Y×Q(Y)Z p *η Y Z Fib Q p˜ (pb) Fib Q p Y η Y Q(Y). \array{ X &\overset{\eta_X}{\longrightarrow}& Q(X) \\ {}^{\mathllap{\tilde i}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{i}}_{\mathrlap{\in W}} \\ Y \underset{Q(Y)}{\times} Z &\overset{p^\ast \eta_Y}{\longrightarrow}& Z \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow &(pb)& \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q}} \\ Y &\underset{\eta_Y}{\longrightarrow}& Q(Y) } \,.

Here i˜\tilde i is a weak equivalence by assumption that the diagram exhibits a homotopy pullback. Hence it factors as

i˜:XWCofjX^WFib=W QFib QπY×Q(Y)Z. \tilde i \;\colon\; X \underoverset{\in W \cap Cof}{j}{\longrightarrow} \hat X \underoverset{\in W \cap Fib = W_Q \cap Fib_Q}{\pi}{\longrightarrow} Y \underset{Q(Y)}{\times} Z \,.

This yields the situation

X = X WCof j Fib f X^ p˜πFib Q YX j X^ X f p˜π f Y = Y = Y. \array{ X &\overset{=}{\longrightarrow}& X \\ {}^{\mathllap{j}}_{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ \hat X &\underoverset{\tilde p \circ \pi}{\in Fib_Q}{\longrightarrow}& Y } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ X &\overset{j}{\longrightarrow}& \hat X &\overset{\exists}{\longrightarrow}& X \\ {}^{\mathllap{}f}\downarrow && \downarrow^{\mathrlap{\tilde p \circ \pi}} && \downarrow^{\mathrlap{f}} \\ Y &=& Y &=& Y } \,.

As in the retract argument (prop.) this diagram exhibits ff as a retract (in the arrow category, rmk.) of the QQ-fibration p˜π\tilde p \circ \pi. Hence by the existence of the QQ-model structure (prop. ) and by the closure properties for fibrations (prop.), also ff is a QQ-fibration.

Now for the converse. Assume that ff is a QQ-fibration. Since 𝒞 Q\mathcal{C}_Q is a left Bousfield localization of 𝒞\mathcal{C} (prop. ), ff is also a fibration (prop. ). We need to show that the η\eta-naturality square on ff exhibits a homotopy pullback.

So factor Q(f)Q(f) as before, and consider the pasting composite of the factorization of the given square with the naturality squares of η\eta:

X W Qη X Q(X) WW Qη Q(X) Q(Q(X)) W Q i˜ WW Q i W Q(i) Y×Q(Y)Z W Qp *η Y Z Wη Z Q(Z) Fib Q p˜ (pb) Fib QFib p Q(p) Y η YW Q Q(Y) η Q(Y)WW Q Q(Q(Y)). \array{ X &\underoverset{\in W_Q}{\eta_X}{\longrightarrow}& Q(X) &\underoverset{\in W \subset W_Q}{\eta_{Q(X)}}{\longrightarrow}& Q(Q(X)) \\ {}^{\mathllap{\tilde i}}_{\mathllap{\in W_Q}}\downarrow && {}^{\mathllap{i}}_{\mathllap{\in W\subset W_Q}}\downarrow && \downarrow^{\mathrlap{Q(i)}}_{\mathrlap{\in W}} \\ Y \underset{Q(Y)}{\times} Z &\underoverset{\in W_Q}{p^\ast \eta_Y}{\longrightarrow}& Z &\underoverset{\in W}{\eta_Z}{\longrightarrow}& Q(Z) \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow &(pb)& \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q \subset Fib}} && \downarrow^{\mathrlap{Q(p)}} \\ Y &\underoverset{\eta_Y}{\in W_Q}{\longrightarrow}& Q(Y) &\underoverset{\eta_{Q(Y)}}{\in W \subset W_Q}{\longrightarrow}& Q(Q(Y)) } \,.

Here the top and bottom horizontal morphisms are weak (QQ-)equivalences by the idempotency of QQ, and Q(i)Q(i) is a weak equivalence since QQ preserves weak equivalences (first and second clause in def. ). Hence by two-out-of-three also η Z\eta_Z is a weak equivalence. From this, lemma gives that pp is a QQ-fibration. Then p *η Yp^\ast \eta_Y is a QQ-weak equivalence since it is the pullback of a QQ-weak equivalence along a fibration between objects whose η\eta is a weak equivalence, via the third clause in def. . Finally two-out-of-three implies that i˜\tilde i is a QQ-weak equivalence.

In particular, the bottom right square is a homotopy pullback (since two opposite edges are weak equivalences, by this prop.), and since the left square is a genuine pullback of a fibration, hence a homotopy pullback, the total bottom rectangle here exhibits a homotopy pullback by the pasting law for homotopy pullbacks (prop.).

Now by naturality of η\eta, that total bottom rectangle is the same as the following rectangle

Y×Q(Y)Z η (Y×Q(Y)Z) Q(Y×Q(Y)Z) WQ(p *η Y) Q(Z) Fib Q p˜ Q(p˜) Q(p) Y η Y Q(Y) Q(η Y)W Q(Q(Y)), \array{ Y \underset{Q(Y)}{\times} Z &\overset{\eta_{\left(Y \underset{Q(Y)}{\times} Z\right)}}{\longrightarrow}& Q(Y \underset{Q(Y)}{\times} Z) &\underoverset{\in W}{Q(p^\ast \eta_Y)}{\longrightarrow}& Q(Z) \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow && \downarrow^{\mathrlap{Q(\tilde p)}}_{\mathrlap{}} && \downarrow^{\mathrlap{Q(p)}} \\ Y &\underset{\eta_Y}{\longrightarrow}& Q(Y) &\underoverset{Q(\eta_Y)}{\in W}{\longrightarrow}& Q(Q(Y)) } \,,

where now Q(p *η Y)WQ(p^\ast \eta_Y) \in W since p *η YW Qp^\ast \eta_Y \in W_Q, as we had just established. This means again that the right square is a homotopy pullback (prop.), and since the total rectangle still is a homotopy pullback itself, by the previous remark, so is now also the left square, by the other direction of the pasting law for homotopy pullbacks (prop.).

So far this establishes that the η\eta-naturality square of p˜\tilde p is a homotopy pullback. We still need to show that also the η\eta-naturality square of ff is a homotopy pullback.

Factor i˜\tilde i as a cofibration followed by an acyclic fibration. Since i˜\tilde i is also a QQ-weak equivalence, by the above, two-out-of-three for QQ-fibrations gives that this factorization is of the form

X W QCof=W QCof Qj X^ WFib=W QFib Qπ Y×Q(Y)Z. \array{ X &\underoverset{\in W_Q \cap Cof = W_Q \cap Cof_Q}{j}{\longrightarrow}& \hat X &\underoverset{\in W \cap Fib = W_Q \cap Fib_Q }{\pi}{\longrightarrow}& Y\underset{Q(Y)}{\times} Z } \,.

As in the first part of the proof, but now with (WCof,Fib)(W \cap Cof, Fib) replaced by (W QCof Q,Fib Q)(W_Q \cap Cof_Q, Fib_Q) and using lifting in the QQ-model structure, this yields the situation

X = X W QCof Q j Fib Q f X^ p˜π YX j X^ X f p˜π f Y = Y = Y. \array{ X &\overset{=}{\longrightarrow}& X \\ {}^{\mathllap{j}}_{\mathllap{\in W_Q \cap Cof_Q}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib_Q}} \\ \hat X &\underoverset{\tilde p \circ \pi}{}{\longrightarrow}& Y } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ X &\overset{j}{\longrightarrow}& \hat X &\overset{\exists}{\longrightarrow}& X \\ {}^{\mathllap{}f}\downarrow && \downarrow^{\mathrlap{\tilde p \circ \pi}} && \downarrow^{\mathrlap{f}} \\ Y &=& Y &=& Y } \,.

As in the retract argument (prop.) this diagram exhibits ff as a retract (in the arrow category, rmk.) of p˜π\tilde p \circ \pi.

Observe that the η\eta-naturality square of the weak equivalence π\pi is a homotopy pullback, since QQ preserves weak equivalences (first clause of def. ) and since a square with two weak equivalences on opposite sides is a homotopy pullback (prop.). It follows that also the η\eta-naturality square of p˜π\tilde p \circ \pi is a homotopy pullback, by the pasting law for homotopy pullbacks (prop.).

In conclusion, we have exhibited ff as a retract (in the arrow category, rmk.) of a morphism p˜π\tilde p \circ \pi whose η\eta-naturality square is a homotopy pullback. By naturality of η\eta, this means that the whole η\eta-naturality square of ff is a retract (in the category of commuting squares in 𝒞\mathcal{C}) of a homotopy pullback square. This means that it is itself a homotopy pullback square (prop.).

Proof of the stable model structure

We show now that the operation of Omega-spectrification of topological sequental spectra, from def. , is a Quillen idempotent monad in the sense of def. . Via the Bousfield-Friedlander theorem (prop. ) this establishes the stable model structure on topological sequential spectra in theorem below.

Lemma

The Omega-spectrification (Q,η)(Q,\eta) from def. preserves homotopy pullbacks (def.) in the strict model structure SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} from theorem .

(Schwede 97, lemma 2.1.3 (e))

Proof

Since, by prop. , QQ preserves weak equivalences, it is sufficient to show that every pullback square in SeqSpec(Top cg)SeqSpec(Top_{cg}) of a fibration

B×YX X (pb) Fib B Y \array{ B \underset{Y}{\times} X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ B &\longrightarrow& Y }

is taken by QQ to a homotopy pullback square. By prop. we need to check that this is the case for the kkth component space of the sequential spectra in the diagram, for all kk \in \mathbb{N}.

Let Z i,k XZ^X_{i,k}, Z i,k YZ^Y_{i,k} etc. denote the objects appearing in the definition of (QX) klim iZ i,k X(Q X)_k \coloneqq \underset{\longrightarrow}{\lim}_i Z^X_{i,k}, (QY) klim iZ i,k Y(Q Y)_k \coloneqq \underset{\longrightarrow}{\lim}_i Z^Y_{i,k} , etc. (def. ).

Use the small object argument (prop.) for the set J (Top */)J_{(Top^{\ast/})} of acyclic generating cofibrations in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} (def.) to construct a functorial factorization (def.) through acyclic relative cell complex inclusions (def.) followed by Serre fibrations (def.) in each degree:

Z i,k XJ TopCellW iFib clZ i,k Y. Z^X_{i,k} \overset{\in J_{Top} Cell}{\longrightarrow} W_i \overset{\in Fib_{cl}}{\longrightarrow} Z^Y_{i,k} \,.

Notice that by construction Z ,k KZ^K_{\bullet,k} and Z ,k YZ^Y_{\bullet,k} are sequences of relative cell complexes. This implies, by the way the small object argument works and by the commutativity of each

Z i,k X J (Top */)Cell W i I (Top */)Cell Z i+1,k X J (Top */)Cell W i+1, \array{ Z^X_{i,k} &\overset{\in J_{(Top^{\ast/})} Cell}{\longrightarrow}& W_i \\ {}_{\mathllap{\in I_{(Top^{\ast/})}Cell }}\downarrow && \downarrow \\ Z^X_{i+1,k} &\overset{\in J_{(Top^{\ast/})} Cell}{\longrightarrow}& W_{i+1} } \,,

that also W W_\bullet is a sequence of relative cell complex inclusions: a cell in W iW_i is given by the top square in the following diagram, and the total rectangle is the image of that cell as a cell in W i+1W_{i+1}:

S n1 i n D n1 Z i,k X J (Top */)Cell W i I (Top */)Cell Z i+1,k X J (Top */)Cell W i+1, \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^{n-1} \\ \downarrow && \downarrow \\ Z^X_{i,k} &\overset{\in J_{(Top^{\ast/})} Cell}{\longrightarrow}& W_i \\ {}_{\mathllap{\in I_{(Top^{\ast/})}Cell }}\downarrow && \downarrow \\ Z^X_{i+1,k} &\overset{\in J_{(Top^{\ast/})} Cell}{\longrightarrow}& W_{i+1} } \,,

Therefore, forming the colimit over iIi \in I of these sequences sends the degreewise Serre fibration to a Serre fibration (prop.): because we test for a Serre fibration by lifting against the morphism in J Top */J_{Top^{\ast/}}, which have compact domain and codomain, and these may be taken inside the colimit over relative cell complex inclusions (by this lemma)). So we have a Serre fibration

lim iW iW cl(QY) k \underset{\longrightarrow}{\lim}_i W_i \overset{\in W_{cl}}{\longrightarrow} (Q Y)_k

for each kk \in \mathbb{N}.

Consider then the commuting diagrams

Z i,k B Z i,k Y Fib cl W i W clCof cl Z i,k X W cl ϕ B W cl ϕ Y W cl W cl ϕ X Ω iB k+i Ω iY k+i Fib cl Ω iX k+i, \array{ Z^B_{i,k} &\longrightarrow& Z^Y_{i,k} &\overset{\in Fib_{cl}}{\longleftarrow}& W_i &\overset{\in W_{cl}\in Cof_{cl}}{\longleftarrow}& Z^X_{i,k} \\ \downarrow^{\mathrlap{\phi^B}}_{\mathrlap{\in W_{cl}}} && \downarrow^{\mathrlap{\phi^Y}}_{\mathrlap{\in W_{cl}}} && &_{\mathllap{\exists \in W_{cl}}}\searrow& \downarrow^{\mathrlap{\phi^X}}_{\mathrlap{\in W_{cl}}} \\ \Omega^i B_{k+i} &\longrightarrow& \Omega^i Y_{k+i} &\longleftarrow& &\underset{\in Fib_{cl}}{\longleftarrow}& \Omega^i X_{k+i} } \,,

where the vertical morphisms are composites of the weak equivalences ϕ i,k:Z i+1,kϕ i,kΩZ i,k+1 \phi_{i,k} \colon Z_{i+1,k} \overset{\phi_{i,k}}{\longrightarrow} \Omega Z_{i,k+1} from def. .

The diagonal is a chosen lift (where we use that Ω=Maps(S 1,) *\Omega = Maps(S^1,-)_{\ast} preserves Serre fibrations by prop. ). This lift is a weak equivalence by two-out-of-three. On the left of the diagram this exhibits now a weak equivalence of cospan-diagrams with right leg a fibration. Therefore, since forming the limit over these cospan diagrams is a homotopy pullback (def., all objects here being fibrant), this induces a weak equivalence on these limits (prop.)

κ:Z i,k B×Z i,k YW iW clΩ iB k+i×Ω iY k+iΩ iX k+iΩ i(B k+i×Y k+iX k+i). \kappa \;\colon\; Z^B_{i,k} \underset{Z^Y_{i,k}}{\times} W_i \overset{\in W_{cl}}{\longrightarrow} \Omega^i B_{k+i} \underset{\Omega^i Y_{k+i}}{\times} \Omega^i X_{k+i} \simeq \Omega^i ( B_{k+i} \underset{Y_{k+i}}{\times} X_{k+i} ) \,.

By universality of the pullback there is a commuting triangle

Z i,k B× YX ρ i Z i,k B×Z i,k YW i ϕW cl κW cl Ω i(B i+k×Y i+kX i+k) \array{ Z^{B\times_Y X}_{i,k} && \overset{\rho_i}{\longrightarrow} && Z^B_{i,k} \underset{Z^Y_{i,k}}{\times} W_i \\ & {}_{\mathllap{\phi \in W_{cl}}}\searrow && \swarrow_{\mathrlap{\kappa \in W_{cl}}} \\ && \Omega^i( B_{i+k}\underset{Y_{i+k}}{\times} X_{i+k} ) }

and hence by two-out-of-three also the top morphism is a weak equivalence.

Now observe that colimits over sequences of relative cell inclusions preserve finite limits up to weak equivalence (prop.). This follows again by using that nn-spheres may be taken inside the colimits from the classical fact that filtered colimits preserve finite limits. In conclusion then, we have a weak equivalence of the form

(Q(B×YX)) k=lim iZ i,k B× YXW cllim iρ ilim i(Z i,k B×Z i,k YW i)W cl(lim iZ i,k B)×lim iZ i,k Y(lim iW i)=(QB) k×(QY) k(lim iW i). (Q(B \underset{Y}{\times} X))_k = \underset{\longrightarrow}{\lim}_i Z^{B \times_Y X}_{i,k} \underoverset{\in W_{cl}}{\underset{\longrightarrow}{\lim}_i \rho_i}{\longrightarrow} \underset{\longrightarrow}{\lim}_i \left( Z^B_{i,k} \underset{Z^Y_{i,k}}{\times} W_i \right) \overset{\in W_{cl}}{\longrightarrow} \left(\underset{\longrightarrow}{\lim}_i Z^B_{i,k}\right) \underset{ \underset{\longrightarrow}{\lim}_i Z^Y_{i,k} }{\times} \left( \underset{\longrightarrow}{\lim}_i W_i \right) = (Q B)_k \underset{(Q Y)_k}{\times} \left( \underset{\longrightarrow}{\lim}_i W_i \right) \,.

This exhibits (degreewise and hence globally) the homotopy pullback property to be show.

Proposition

The Omega-spectrification (Q,η)(Q,\eta) from def. is a Quillen idempotent monad in the sense of def. on the strict model structre theorem :

Q:SeqSpec(Top cg) strictSeqSpec(Top cg) strict. Q \;\colon\; SeqSpec(Top_{cg})_{strict} \longrightarrow SeqSpec(Top_{cg})_{strict} \,.

(Schwede 97, prop. 2.1.5)

Proof

First notice that the strict model structure is indeed right proper, as demanded in def. : Since every object in SeqSpec(Top cg)SeqSpec(Top_{cg}) is fibrant (this being so degreewise in (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen}) this follows from this lemma.

The first two conditions required on a Quillen idempotent monad in def. are explicit in prop. .

The third condition follows from lemma : A pullback of a QQ-equivalence along a fibration is a homotopy pullback and is hence sent by QQ to another homotopy pullback square.

f *Z f *h X (pb) fFib Z hW Q YQ(f *Z) Q(f *h)W Q(X) (pb) h Q(f) Q(Z) Q(h)W Q(Y). \array{ f^\ast Z &\stackrel{f^\ast h}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f \in Fib}} \\ Z &\underset{h \in W_{Q}}{\longrightarrow}& Y } \;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\; \array{ Q(f^\ast Z) &\overset{Q(f^\ast h) \in W}{\longrightarrow}& Q(X) \\ \downarrow &(pb)^h& \downarrow^{\mathrlap{Q(f)}} \\ Q(Z) &\underset{Q(h) \in W}{\longrightarrow}& Q(Y) } \,.

By definition of QQ-equivalence that resulting homotopy pullback square has the bottom edge a weak equivalence, and hence also the top edge is a weak equivalence (prop.).

Theorem

The left Bousfield localization of the strict model structure on sequential spectra (theorem ) at the class of stable weak homotopy equivalences (def. ) exists, called the stable model structure on topological sequential spectra

SeqSpec(Top cg) stableididSeqSpec(Top cg) strict. SeqSpec(Top_{cg})_{stable} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot} SeqSpec(Top_{cg})_{strict} \,.

Moreover, its fibrant objects are precisely the Omega-spectra (def.).

Proof

Let (Q,η)(Q,\eta) be the Omega-spectrification operation from def. . According to prop. this is a Quillen-idempotent monad (def. ) on SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict}. Hence the Bousfield-Friedlander theorem (prop. ) asserts that the Bousfield localization of the strict model structure at the QQ-equivalences exists. By prop. these are precisely the stable weak homotopy equivalences.

Finally, by prop. an object XSeqSpec(Top cg) stableX \in SeqSpec(Top_{cg})_{stable} is fibrant in SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} precisely if

X η X Q(X) * * \array{ X &\overset{\eta_X}{\longrightarrow}& Q(X) \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \ast }

exhibits a homotopy pullback in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict}. Since every object in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} is fibrant, the vertical morphisms here are fibrations. The pullback of Q(X)Q(X) along id *id_\ast is just Q(X)Q(X) itself, and the universally induced morphism into this pullback is just η X\eta_X itself. Hence the square is a homotopy pullback precisely if η X\eta_X is a weak equivalence in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict}, hence degreewise a weak homotopy equivalence. Since Q(X)Q(X) is an Omega-spectrum by prop. , this means precisely that XX is an Omega-spectrum.

Stability of the homotopy theory

We discuss that the stable model structure SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} of theorem is indeed a stable model category, in that the canonical reduced suspension operation is an equivalence of categories from the stable homotopy category (def. ) to itself. This is theorem below.

Definition

A pointed model category 𝒞\mathcal{C} (exmpl.) is called a stable model category if the canonically induced reduced suspension and loop space object-functors (prop.) on its homotopy category (defn.) constitute an equivalence of categories

(ΣΩ):Ho(𝒞)ΩΣHo(𝒞). (\Sigma \dashv \Omega) \;\colon\; Ho(\mathcal{C}) \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq} Ho(\mathcal{C}) \,.

Literature (Jardine 15, sections 10.3 and 10.4)

\,

First we observe that the alternative suspension induces an equivalence of homotopy categories:

Lemma

With Σ\Sigma and Ω\Omega the alternative suspension and alternative looping functors from def. :

  1. Ω\Omega preserves Omega-spectra (def. );

  2. Σ\Sigma preserves stable weak homotopy equivalences (def. ).

Proof

Regarding the first statement:

By prop. , Ω\Omega acts on component spaces and adjunct structure maps as the right Quillen functor

Maps(S 1,) *:(Top cg */) Quillen(Top cg */) Quillen Maps(S^1,-)_\ast \;\colon\; (Top_{cg}^{\ast/})_{Quillen} \longrightarrow (Top_{cg}^{\ast/})_{Quillen}

on the classical model structure on pointed compactly generated topological spaces (thm., prop.). Since in this model structure all objects are fibrant, Ken Brown's lemma (prop.) implies that with σ˜ n X\tilde \sigma^X_n a weak homotopy equivalence, so is σ˜ n ΩX=Maps(S 1,σ˜ n X)\tilde \sigma^{\Omega X}_n = Maps(S^1,\tilde \sigma^X_n).

Regarding the second point:

Let f:XYf \colon X \to Y be a stable weak homotopy equivalence. By the existence of the model structure SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} from theorem , Σf\Sigma f is a stable weak homotopy equivalence precisely if its image in the homotopy category Ho(SeqSpec(Top cg) stable)Ho(SeqSpec(Top_{cg})_{stable}) is an isomorphism (prop.). By the Yoneda lemma (fully faithfulness of the Yoneda embedding), this is the case if for all ZHo(SeqSpec(Top cg) stable)Z \in Ho(SeqSpec(Top_{cg})_{stable}) the function

[Σf,Z] stable:[ΣY,Z] stable[ΣX,Z] stable [\Sigma f, Z]_{stable} \;\colon\; [\Sigma Y,Z]_{stable} \longrightarrow [\Sigma X,Z]_{stable}

is a bijection. By the fact that the stable model structure is a left Bousfield localization of the strict model structure with fibrant objects the Omega-spectra, this is the case equivalently (using this lemma) if

[Σf,Z] strict:[ΣY,Z] strict[ΣX,Z] strict [\Sigma f, Z]_{strict} \;\colon\; [\Sigma Y,Z]_{strict} \longrightarrow [\Sigma X,Z]_{strict}

is a bijection for all Omega-spectra ZZ. Now by the Quillen adjunction ΣΩ\Sigma \dashv \Omega on the strict model category (prop. ) this is equivalent to

[f,ΩZ] strict:[Y,ΩZ] strict[X,ΩZ] strict [f, \Omega Z]_{strict} \;\colon\; [Y,\Omega Z]_{strict} \longrightarrow [X,\Omega Z]_{strict}

being a bijection for all Omega-spectra ZZ. But since Ω\Omega preserves Omega-spectra by the first point above, this is still maps into a fibrant objects, hence is again equivalent (using again the property of the left Bousfield localization) to the hom in the strict model structure

[f,ΩZ] stable:[Y,ΩZ] stable[X,ΩZ] stable [f, \Omega Z]_{stable} \;\colon\; [Y,\Omega Z]_{stable} \longrightarrow [X,\Omega Z]_{stable}

being a bijection for all ΩZ\Omega Z. But this is indeed a bijection, since ff is a stable weak homotopy equivalence, hence an isomorphism in the homotopy category.

Lemma

For XX a sequential spectrum, then (using remark to suppress parenthesis)

  1. the structure maps constitute a homomorphism

    ΣX[1]X \Sigma X[-1] \longrightarrow X

    (from the shift, def. , of the alternative suspension, def. ) and this is a stable weak homotopy equivalence,

  2. the adjunct structure maps constitute a homomorphism

    XΩX[1] X \longrightarrow \Omega X[1]

    (to the shift, def. , of the alternative looping, def. )

    If XX is an Omega-spectrum (def. ) then this is a weak equivalence in the strict model structure (def. ), hence in particular a stable weak homotopy equivalence.

Proof

The diagrams that need to commute for the structure maps to give a homomorphism as claimed are in degree 0 this one

S 1S 1* 0 X 0 S 10 σ 0 S 1X 0 σ 0 X 1 \array{ S^1 \wedge S^1 \wedge \ast &\overset{0}{\longrightarrow}& X_0 \\ {}^{\mathllap{S^1 \wedge 0}}\downarrow && \downarrow^{\mathrlap{\sigma_0}} \\ S^1 \wedge X_0 &\underset{\sigma_0}{\longrightarrow}& X_1 }

and in degree n1n \geq 1 these:

S 1S 1X n1 S 1σ n1 X n S 1σ n1 σ n S 1X n σ n X n+1. \array{ S^1 \wedge S^1 \wedge X_{n-1} &\overset{S^1 \wedge \sigma_{n-1}}{\longrightarrow}& X_n \\ {}^{\mathllap{S^1 \wedge \sigma_{n-1}}}\downarrow && \downarrow^{\mathrlap{\sigma_n}} \\ S^1 \wedge X_{n} &\underset{\sigma_n}{\longrightarrow}& X_{n+1} } \,.

But in all these cases commutativity it trivially satisfied.

That the adjunct structure maps constitute a morphism XΩX[1]X \to \Omega X[1] follows dually.

If XX is an Omega-spectrum, then by definition this last morphism is already a weak equivalence in the strict model structure, hence in particular a weak equivalence in the stable model structure.

From this it follows that also ΣX[1]X\Sigma X[-1]\to X is a stable weak homotopy equivalence, because for every Omega-spectrum YY then by the adjunctions in prop. we have a commuting diagram of the form

[X,Y] strict [ΣX[1],Y] strict id [X,Y] strict [X,ΩY[1]] strict. \array{ [X, Y]_{strict} &\overset{}{\longrightarrow}& [\Sigma X[-1],Y]_{strict} \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ [X,Y]_{strict} &\underset{\simeq}{\longrightarrow}& [X, \Omega Y[1]]_{strict} } \,.

(To see the commutativity of this diagram in detail, consider for any [f][X,Y] strict[f] \in [X,Y]_{strict} chasing the element σ n Y\sigma_n^Y in the two possible ways through the natural adjunction isomorphism:

[S 1Y n1,Y n] [Y n1,ΩY n] [S 1f n1,Y n] [f n1,ΩY n] [S 1X n1,Y n] [X n1,ΩY n]. \array{ [S^1 \wedge Y_{n-1}, Y_n] &\simeq& [Y_{n-1}, \Omega Y_n] \\ {}^{\mathllap{[S^1 \wedge f_{n-1}, Y_n]}}\downarrow && \downarrow^{\mathrlap{[f_{n-1}, \Omega Y_n]}} \\ [S^1 \wedge X_{n-1}, Y_n] &\simeq& [X_{n-1}, \Omega Y_n] } \,.

Sending σ n Y\sigma_n^Y down gives σ n YS 1f n1\sigma_n^Y \circ S^1 \wedge f_{n-1} which equals (by the homomorphism property) f nσ n Xf_n \circ \sigma_n^X. Instead sending σ n Y\sigma_n^Y to the right yields σ˜ n Y\tilde \sigma_n^Y and then down yields σ˜ n Yf n1\tilde \sigma_n^Y \circ f_{n-1}. By commutativity this is adjunct to f nσ n Xf_n \circ \sigma_n^X.)

Hence

[X,Y] strict[ΣX[1],Y] strict [X, Y]_{strict} \overset{}{\longrightarrow} [\Sigma X[-1],Y]_{strict}

is a bijection for all Omega-spectra YY, and so the conclusion that ΣX[1]X\Sigma X[-1]\to X is a stable weak homotopy equivalence follows as in the proof of lemma .

Lemma

The total derived functor of the alternative suspension operation Σ\Sigma of def. exists and constitutes an equivalence of categories from the stable homotopy category to itself:

Σ:Ho(SeqSpec(Top) stable)Ho(SeqSpec(Top) stable). \Sigma \;\colon\; Ho(SeqSpec(Top)_{stable}) \overset{\simeq}{\longrightarrow} Ho(SeqSpec(Top)_{stable}) \,.
Proof

The total derived functor of Σ\Sigma exists, because by lemma Σ\Sigma preserves stable weak homotopy equivalences. Also the shift functor [1][-1] from def. clearly preserves stable equivalences, hence both descend to the homotopy category. There, by prop. and remark , they are inverses of each other, up to isomorphism.

Lemma

The canonical suspension functor on the homotopy category of any model category (from this prop.) in the case of the stable homotopy category (def. ) Ho(Spectra)=Ho(SeqSpec(Top cg) stable)Ho(Spectra) = Ho(SeqSpec(Top_{cg})_{stable}) is represented by the “standard suspension” operation of def. .

Proof

By CW-approximation (prop. ), every object in the stable homotopy category is represented by a CW-spectrum. By prop. , on CW-spectra the canonical suspension functor on the homotopy category (from this prop.) is represented by the “standard suspension” operation of def. .

The combination of lemma with lemma gives that in order to show that SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} is indeed a stable model category according to def. , we are reduced to showing that in the homotopy category the alternative suspension operation (which we know gives an equivalence) is naturally isomorphic to the standard suspension operation (which we know is the correct suspension operation). This we turn to now.

According to remark , both should be directly comparable and isomorphic in the homotopy category “in even degrees”, but non-comparable in odd degree. In order to make this precise, we now introduce the concept of sequential spectra with components only in even degree and then use an adjunction back to ordinary sequential spectra.

Observe that the definition of the category SeqSpec(Top cg)SeqSpec(Top_{cg}) of sequential spectra in def. does not require anything specific of the circle S 1S^1: the same kind of definition may be considered for any other pointed topological space TT in place of S 1S^1. The construction of the stable model structure SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} in theorem does depend on the nature of S 1S^1, but only in that it uses that the n-spheres S n=(S 1) nS^n = (S^1)^{\wedge n}

  1. co-represent homotopy groups in the classical pointed homotopy category: [S n,] *π n()[S^n, -]_{\ast}\simeq \pi_n(-);

  2. are compact, so that maps out of them factor through finite stages of transfinite compositions of relative cell complex inclusions.

Both points still hold with S 1S^1 replaced by S 1K +S^1 \wedge K_+, for KK any contractible compact topological space. Moreover, since only the stable homotopy groups matter for the construction of the stable model category, one could replace S 1S^1 by any S kS^k: While the smash powers (S k) n(S^k)^{\wedge n} co-represent only every kkth homotopy group, this is still sufficient for co-represent all the stable homotopy groups.

The following is an immediate variant of the definition of sequential spectra:

Definition

Let T=K +Top cg */T = K_+ \in Top^{\ast/}_{cg} be a compact contractible topological space with a basepoint freely adjoined, and let kk \in \mathbb{N}, k1k \geq 1.

A sequential TS kT \wedge S^k-spectrum is a sequence of component spaces X knTop cgX_{k n} \in Top_{cg} for nn \in \mathbb{N}, and a sequence of structure maps of the form

σ k,n:TS kX knX k(n+1). \sigma_{k,n} \;\colon\; T \wedge S^k \wedge X_{k n} \longrightarrow X_{k(n+1)} \,.

A homomorphism of sequential TS kT \wedge S^k-spectra f:XYf \colon X \to Y is a sequence of component maps f kn:X knY knf_{k n} \;\colon\; X_{k n} \to Y_{k n} such that all these diagrams commute:

TS kX kn TS kf kn TS kY kn σ k,n X σ k,n Y X k(n+1) f k(n+1) Y k(n+1). \array{ T \wedge S^k \wedge X_{k n } &\overset{T \wedge S^k \wedge f_{k n}}{\longrightarrow}& T \wedge S^k \wedge Y_{k n} \\ {}^{\mathllap{\sigma_{k,n}^{X}}}\downarrow && \downarrow^{\mathrlap{\sigma_{k,n}^Y}} \\ X_{k(n+1)} &\underset{f_{k(n+1)}}{\longrightarrow}& Y_{k(n+1)} } \,.

Write

Seq TS kSpec(Top cg) Seq_{T\wedge S^k}Spec(Top_{cg})

for the resulting category of sequential TS kT \wedge S^k-spectra.

Proposition

For any TS kT \wedge S^k as in def. , there exists a model category structure

Seq TS kSpec(Top cg) stable Seq_{T\wedge S^k}Spec(Top_{cg})_{stable}

on the category of sequential TS kT\wedge S^k-spectra, where

  • the weak equivalences are the morphisms that induce isomorphisms under lim knkπ kn()\underset{\longrightarrow}{\lim}_{k n \in k \mathbb{N}}\pi_{k n}(-);

  • the fibrations are the morphisms whose η k\eta_k-naturality square is a homotopy pullback, where η K:idQ k\eta_K \colon id \to Q_k is the KS kK \wedge S^k-spectrification functor defined as in def. but with S 1S^1 replaced by TS kT \wedge S^k throughout.

Proof

The proof is verbatim that of theorem , with S 1S^1 replaced by TS kT \wedge S^k throughout.

Lemma

For kk \in \mathbb{N}, k1k \geq 1, there is a pair of adjoint functors

SeqSpec(Top cg)R kL kSeq S kSpec(Top cg) SeqSpec(Top_{cg}) \underoverset {\underset{R_k}{\longrightarrow}} {\overset{L_k}{\longleftarrow}} {\bot} Seq_{S^k}Spec(Top_{cg})

between sequential spectra (def. ) and sequential S kS^k-spectra (def. )

  • where (R kX) knX kn(R_k X)_{k n} \coloneqq X_{k n} and

    σ n R kX:S kX knS k1S 1X knS 1σ kn XS k1X kn+1S 1X kn+(k1)σ kn+(k1) XX k(n+1) \sigma_n^{R_k X} \;\colon\; S^k X_{k n} \simeq S^{k-1}\wedge S^1 \wedge X_{k n} \overset{S^1 \wedge \sigma_{k n}^X}{\longrightarrow} S^{k-1} \wedge X_{k n +1} \longrightarrow \cdots \longrightarrow S^1 \wedge X_{k n+ (k-1)} \overset{\sigma_{k n + (k-1)}^X}{\longrightarrow} X_{k(n+1)}
  • and where

    (L k𝒳) n{𝒳 n ifnk S q𝒳 nq ifq<kandnqk (L_k \mathcal{X})_n \coloneqq \left\{ \array{ \mathcal{X}_n & if \; n \in k \mathbb{N} \\ S^q \wedge \mathcal{X}_{n-q} & if \; q \lt k \; and \; n-q \in k \mathbb{N} } \right.

    and

    σ n L k𝒳={σ n(k1) 𝒳 ifn+1k id S 1𝒳 n otherwise . \sigma^{L_k \mathcal{X}}_{n} = \left\{ \array{ \sigma^{\mathcal{X}}_{n - (k-1)} & if \; n+1 \in k \mathbb{N} \\ id_{S^1 \wedge \mathcal{X}_n} & otherwise \\ } \right. \,.

Moreover, for each XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}), the adjunction unit

L kR kXX L_k R_k X \longrightarrow X

is a stable weak homotopy equivalence (def. ).

Proof

For ease of notation we discuss this for k=2k = 2. The general case is directly analogous. To see that we have an adjunction, consider a homomorphism

f:L 2𝒳Y. f \;\colon\; L_2 \mathcal{X} \longrightarrow Y \,.

Given its even-graded component maps, then its odd-graded component maps f 2n+1f_{2n+1} need to fit into commuting squares of the form

S 1𝒳 2n S 1f 2n S 1Y 2n id σ 2n Y S 1𝒳 2n f 2n+1 Y 2n+1. \array{ S^1 \wedge \mathcal{X}_{2n} &\overset{S^1 \wedge f_{2n}}{\longrightarrow}& S^1 \wedge Y_{2n} \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\sigma_{2n}^Y}} \\ S^1 \wedge \mathcal{X}_{2n} &\underset{f_{2n+1}}{\longrightarrow}& Y_{2n+1} } \,.

Since here the left map is an identity, this uniquely fixes the odd-graded components f 2n+1f_{2n+1} in terms of the even-graded components. Moreover, these components then make the following pasting rectangles comute

S 2𝒳 2n S 2f 2n S 2Y 2n S 1σ 2n Y S 2𝒳 2n S 1f 2n+1 S 1Y 2n+1 σ 2n 𝒳 σ 2n+1 Y 𝒳 2n+2 f 2n+2 Y 2n+2. \array{ S^2 \wedge \mathcal{X}_{2n} &\overset{S^2 \wedge f_{2n}}{\longrightarrow}& S^2 \wedge Y_{2 n} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{S^1 \wedge \sigma_{2n}^Y}} \\ S^2 \wedge \mathcal{X}_{2n} &\overset{S^1 \wedge f_{2n+1}}{\longrightarrow}& S^1 \wedge Y_{2n+1} \\ {}^{\mathllap{\sigma_{2n}^{\mathcal{X}}}}\downarrow && \downarrow^{\mathrlap{\sigma_{2n+1}^Y}} \\ \mathcal{X}_{2n+2} &\overset{f_{2n+2}}{\longrightarrow}& Y_{2n+2} } \,.

This equivalently exhibits ff as a homomorphism of the form

f˜:𝒳R 2Y \tilde f \;\colon\; \mathcal{X} \longrightarrow R_2 Y

and hence establishes the adjunction isomorphism.

Finally to see that the adjunction unit is a stable weak homotopy equivalence: for XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}) then the morphism of stable homotopy groups induced from

L 2R 2XX L_2 R_2 X \longrightarrow X

is in degree qq given by

lim( π q+2k(X 2k) π q+2k+2(X q+2k+2) ) = π q(L 2R 2X) lim( π q+2k(X 2k) π 2+2k+1(X 2k+1) π q+2k+2(X q+2k+2) ) = π q(X). \array{ \underset{\longrightarrow}{\lim} (\cdots &\to& \pi_{q+2k}(X_{2k}) && \longrightarrow && \pi_{q+2k+2}(X_{q+2k+2}) &\to& \cdots ) &=& \pi_q(L_2 R_2 X) \\ && {}^{\mathllap{\simeq}}\downarrow && && {}^{\mathllap{\simeq}}\downarrow && && \downarrow \\ \underset{\longrightarrow}{\lim} (\cdots &\to& \pi_{q+2k}(X_{2k}) &\longrightarrow& \pi_{2+2k+1}(X_{2k+1}) &\longrightarrow& \pi_{q+2k+2}(X_{q+2k+2}) &\to& \cdots ) &=& \pi_q(X) } \,.

From this it is clear by inspection that the induced vertical map on the right is an isomorphism. Stated more abstractly: the inclusion of partially ordered sets even \mathbb{N}_{even}^{\leq} \hookrightarrow \mathbb{N}^{\leq} is a cofinal functor and hence restriction along it preserves colimits.

Definition

For

α:T 1S kT 2S k \alpha \;\colon\; T_1 \wedge S^{k} \longrightarrow T_2 \wedge S^{k}

any morphism, write

α *:Seq T 2S kSpect(Top cg)Seq T 1S kSpect(Top cg) \alpha^\ast \;\colon\; Seq_{T_2 \wedge S^{k}}Spect(Top_{cg}) \longrightarrow Seq_{T_1 \wedge S^{k}}Spect(Top_{cg})

for the functor from the category of sequential T 2S kT_2 \wedge S^{k}-spectra (def. ) to that of T 1S kT_1 \wedge S^{k}-spectra which sends any XX to α *X\alpha^\ast X with

(α *X) knX kn (\alpha^\ast X)_{k n} \coloneqq X_{k n}

and

σ k,n α *X:T 1S kX knαidT 2S kX knσ k,n XX k(n+1). \sigma_{k,n}^{\alpha^\ast X} \;\colon\; T_1 \wedge S^{k} \wedge X_{k n} \overset{\alpha \wedge id}{\longrightarrow} T_2 \wedge S^{k} \wedge X_{k n} \overset{\sigma_{k,n}^X}{\longrightarrow} X_{k (n+1)} \,.
Lemma

For TK +T \coloneqq K_+ a compact contractible topological space with base point adjoined, and for kk \in \mathbb{N}, write i:S kTS ki \colon S^k \longrightarrow T \wedge S^k for the canonical inclusion. Then the induced functor i *i^\ast from def. is the right adjoint in a Quillen equivalence (def.)

Seq TS 1Spec(Top cg) stable Qui *LSeqSpec(Top cg) stable Seq_{T\wedge S^1}Spec(Top_{cg})_{stable} \underoverset {\underset{i^\ast}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\simeq_{Qu}} SeqSpec(Top_{cg})_{stable}

between the stable model structures of sequential S kS^k-spectra and of sequential TS kT \wedge S^k-spectra (prop. ), respectively.

(Jardine 15, theorem 10.40)

Proof

Write p:TS 1S 1p \colon T \wedge S^1 \to S^1 for the canonical projection.

A morphism

f:Xi *Y f \;\colon\; X \longrightarrow i^\ast Y

is given by components fitting into commuting squares of the form

S 1X n S 1f n S 1Y n id iid S 1X n TS 1Y n σ n X σ n Y X n+1 f n+1 Y n+1. \array{ S^1 \wedge X_n &\overset{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{i \wedge id}} \\ S^1 \wedge X_n && T \wedge S^1 \wedge Y_n \\ {}^{\mathllap{\sigma_n^X}}\downarrow && \downarrow^{\mathrlap{\sigma_n^Y}} \\ X_{n+1} &\underset{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.

Since pi=idp \circ i = id, every such diagram factors as

S 1X n S 1f n S 1Y n iid iid TS 1X n TS 1f n TS 1Y n pid S 1X n σ n X σ n Y X n+1 f n+1 Y n+1. \array{ S^1 \wedge X_n &\overset{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n \\ {}^{\mathllap{i \wedge id}}\downarrow && \downarrow^{\mathrlap{i \wedge id}} \\ T \wedge S^1 \wedge X_n &\overset{T \wedge S^1 \wedge f_n}{\longrightarrow}& T \wedge S^1 \wedge Y_n \\ {}^{\mathllap{p \wedge id}}\downarrow && \downarrow \\ S^1 \wedge X_n && \\ {}^{\mathllap{\sigma_n^X}}\downarrow && \downarrow^{\mathrlap{\sigma_n^Y}} \\ X_{n+1} &\underset{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.

Here the bottom square exhibits the components of a morphism

f˜:p *XY \tilde f \;\colon\; p^\ast X \longrightarrow Y

and this correspondence is clearly naturally bijective

This establishes the adjunction p *i *p^\ast \dashv i^\ast. This is a Quillen equivalence because for every ZTop cg */Z \in Top^{\ast/}_{cg} then by the contractibility of KK there is an equivalence

[TS q,Z] *[S q,Z] * [T \wedge S^q,Z]_\ast \simeq [S^q, Z]_\ast

and hence the concept of stable weak homotopy equivalences in both categories agrees. Hence any f˜:p *XY\tilde f \colon p^\ast X \to Y is a stable weak homotopy equivalence precisely if f:Xi *yf \colon X \to i^\ast y is.

With this in hand, we now finally state the comparison between standard and alternative suspension:

Lemma

There is a natural isomorphism in the homotopy category Ho(SeqSpec(Top cg) stable)Ho(SeqSpec(Top_{cg})_{stable}) of the stable model structure, between the total derived functors (prop.) of the standard suspension (def. ) and of the alternative suspension (def. ):

Σ()()S 1Ho(SeqSpec(Top cg) stable) \Sigma (-) \; \simeq \; (-) \wedge S^1 \;\;\;\;\; \in Ho(SeqSpec(Top_{cg})_{stable})

Notice that we agreed in Part P to suppress the notation 𝕃\mathbb{L} for left derived functors of the suspension functor, not to clutter the notation. If we re-instantiate this then the above says that there is a natural isomorphism

𝕃Σ𝕃(()S 1). \mathbb{L} \Sigma \; \simeq \; \mathbb{L}((-) \wedge S^1) \,.

(Jardine 15, corollary 10.42, prop. 10.53)

Proof

Consider the adjunction (L 2R 2):SeqSpec(Top)Seq 2Spec(Top)(L_2 \dashv R_2) \colon SeqSpec(Top) \leftrightarrow Seq_2Spec(Top) from lemma . We claim that there is a natural isomorphism

τ:R 2(Σ())R 2(()S 1), \tau \;\colon\; R_2 (\Sigma(-)) \simeq R_2((-)\wedge S^1) \,,

in Ho(Seq S 2Spec(Top cg) stable)Ho(Seq_{S^2}Spec(Top_{cg})_{stable}).

This implies the statement, since by lemma the adjunction unit is a stable weak equivalence, so that we get natural isomorphisms

ΣXL 2R 2(ΣX)L 2τL 2R 2(XS 1)XS 1 \Sigma X \simeq L_2 R_2 (\Sigma X) \underoverset{\simeq}{L_2 \tau}{\longrightarrow} L_2 R_2 (X \wedge S^1) \simeq X \wedge S^1

in Ho(SeqSpec(Top cg) stable)Ho(SeqSpec(Top_{cg})_{stable}) (where we are using that R 2R_2 evidently preserves cofibrant spectra, so that L 2L_2 applied to τ\tau represents the correct derived functor of L 2L_2 and hence preserves this isomorphism).

Now to see that the isomorphism τ\tau exists. Write

τ S 2,S 1:S 2S 1S 1S 2 \tau_{S^2,S^1} \;\colon\; S^2 \wedge S^1 \overset{\simeq}{\longrightarrow} S^1 \wedge S^2

for the braiding isomorphism, which swaps the first two canonical coordinates with the third. Since the homotopy class of this map is trivial in that

[τ S 2,S 1]=1π 3(S 3) [\tau_{S^2, S^1}] = 1 \in \mathbb{Z} \simeq \pi_3(S^3)

is the trivial element in the homotopy groups of spheres (and that is the point of passing to S 2S^2-spectra here, because for S 1S^1-spectra the analogous map τ S 1,S 1\tau_{S^1, S^1} has non-trivial class, remark ) it follows that there is a left homotopy (def.) of the form

S 3 i 0 (I +)S 3 i 1 S 3 id τ S 2,S 1 S 3. \array{ S^3 &\overset{i_0}{\longrightarrow}& (I_+) \wedge S^3 &\overset{i_1}{\longleftarrow}& S^3 \\ & {}_{\mathllap{id}}\searrow & \downarrow & \swarrow_{\mathrlap{\tau_{S^2, S^1}}} \\ && S^3 } \,.

By forming the smash product of the entire diagram with X 2nX_{2n} and pasting on the right the naturality square for the braiding with S 1S^1

S 1S 2X 2n τ S 2X 2n,S 1 S 2X 2nS 1 S 1(σ 2n+1(S 1σ 2n)) (σ 2n+1(S 1σ 2n))S 1 S 1X 2(n+1) τ X 2n,S 1 X 2nS 1 \array{ S^1 \wedge S^2 \wedge X_{2n} &\overset{\tau_{S^2 \wedge X_{2n}, S^1} }{\longleftarrow}& S^2 \wedge X_{2n} \wedge S^1 \\ {}^{\mathllap{S^1 \wedge (\sigma_{2n+1} \circ (S^1 \wedge \sigma_{2n}))}}\downarrow && \downarrow^{\mathrlap{(\sigma_{2n+1} \circ (S^1 \wedge \sigma_{2n})) \wedge S^1 }} \\ S^1 \wedge X_{2(n+1)} &\underset{\tau_{X_{2n}, S^1}}{\longleftarrow}& X_{2n} \wedge S^1 }

this yields the diagram

S 3X 2n i 0 (I +)S 3X 2n i 1 S 3X 2n S 2τ X 2n,S 1 S 2X 2nS 1 id τ S 2,S 1X n S 3X 2n (σ 2n+1(S 1σ 2n))S 1 S 1(σ 2n+1(S 1σ 2n)) S 1X 2n τ X 2n,S 1 X 2nS 1. \array{ S^3 \wedge X_{2n} &\overset{i_0}{\longrightarrow}& (I_+) \wedge S^3 \wedge X_{2n} &\overset{i_1}{\longleftarrow}& S^3 \wedge X_{2n} &\underoverset{\simeq}{S^2\wedge \tau_{ X_{2n}, S^1}}{\longleftarrow}& S^2 \wedge X_{2n} \wedge S^1 \\ & {}_{\mathllap{id}}\searrow & \downarrow & \swarrow_{\mathrlap{\tau_{S^2, S^1} \wedge X_n}} && & \downarrow \\ && S^3 \wedge X_{2n} && && \downarrow^{\mathrlap{(\sigma_{2n+1}\circ (S^1 \wedge \sigma_{2n})) \wedge S^1}} \\ && & {}_{\mathllap{S^1 \wedge (\sigma_{2n+1}\circ (S^1 \wedge \sigma_{2n}))}}\searrow & && \downarrow \\ && && S^1 \wedge X_{2 n} &\underoverset{\tau_{X_{2n}, S^1}}{\simeq}{\longleftarrow}& X_{2n} \wedge S^1 } \,.

Here the left diagonal composite is the structure map of R 2(ΣX)R_2 (\Sigma X) in degree nn, while the right vertical morphism is the structure map of R 2(XS 1)R_2 ( X \wedge S^1 ) in degree nn. In the middle we have the structure map of an auxiliary (I +)S 2(I_+) \wedge S^2-spectrum (def. )

ZSeq I +S 2Spec(Top cg), Z \in Seq_{I_+ \wedge S^2} Spec(Top_{cg}) \,,

and the horizontal morphisms exhibit the functors of def. from (I +)S 2(I_+)\wedge S^2-spectra to S 2S^2-spectra with

i 0 *Z=R 2(ΣX),i 1 *Z=R 2(XS 1). i_0^\ast Z = R_2 (\Sigma X) \;\;\,, \;\;\;\;\; i_1^\ast Z = R_2 (X \wedge S^1) \,.

By lemma and since II is contractible, these functors are equivalences of categories on the Ho(Seq S 2Spec(Top cg))Ho(Seq_{S^2}Spec(Top_{cg})), and moreover they have the same inverse, namely p *p^\ast for p:I +S 2S 2p \colon I_+ \wedge S^2 \to S^2 the canonical projection. This implies the isomorphism.

Explicitly, due to the equivalence there exists VV with Zp *VZ\simeq p^\ast V and with this we may form the composite isomorphism

R 2(ΣX)i 0 *Zi 0 *p *VVi 1 *p *Vi 1 *ZR 2(XS 1). R_2 (\Sigma X) \simeq i_0^\ast Z \simeq i_0^\ast p^\ast V \simeq V \simeq i_1^\ast p^\ast V \simeq i_1^\ast Z \simeq R_2 (X \wedge S^1) \,.

We conclude:

Theorem

The stable model structure SeqSpec(Top) stableSeqSpec(Top)_{stable} from theorem indeed gives a stable model category in the sense of def. , in that the canonically induced reduced suspension functor (prop.) on its homotopy category is an equivalence of categories

Σ:Ho(SeqSpec(Top) stable)Ho(SeqSpec(Top) stable). \Sigma \;\colon\; Ho(SeqSpec(Top)_{stable}) \overset{\simeq}{\longrightarrow} Ho(SeqSpec(Top)_{stable}) \,.
Proof

By lemma , the canonical suspension functor is represented, on fibrant-cofibrant objects, by the standard suspension functor of def. . By prop. this is naturally isomorphic – on the level of the homotopy category – to the alternative suspension operation of def. . Therefore the claim follows with prop. .

In fact this lifts to a Quillen equivalence:

Proposition

The (ΣΩ)(\Sigma \dashv \Omega)-adjunction from prop. is a Quillen equivalence (def.) with respect to the stable model structure of theorem :

SeqSpec(Top cg) stable QΩΣSeqSpec(Top cg) stable. SeqSpec(Top_{cg})_{stable} \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq_{\mathrlap{Q}}} SeqSpec(Top_{cg})_{stable} \,.

Its derived functors (prop.) exhibit the canonical reduced suspension and looping operation as an adjoint equivalence on the stable homotopy category

Ho(Spectra)ΩΣHo(Spectra). Ho(Spectra) \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq} Ho(Spectra) \,.
Proof

By prop. and the fact that the stable model structure has the same cofibrations as the strict model structure, Σ\Sigma preserves stable cofibrations. Moreover, by lemma Σ\Sigma preserves in fact all stable weak equivalences. Hence Σ\Sigma is a left Quillen functor and so (ΣΩ)(\Sigma \dashv \Omega) is a Quillen adjunction. Finally lemma gives that this Quillen adjunction is a Quillen equivalence.

In summary, this concludes the characterization of the stable homotopy category as the result of stabilizing the canonical (ΣΩ)(\Sigma \dashv \Omega)-adjunction on the classical homotopy category:

Theorem

The classical model structure (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} on pointed compactly generated topological spaces (thm., prop.) and the stable model structure on topological sequential spectra SeqSpec(Top cg)SeqSpec(Top_{cg}) (theorem ) sit in a commuting diagram of Quillen adjunctions of the form

(Top cg */) Quillen ΩΣ (Top cg */) Quillen Σ Ω Σ Ω SeqSpec(Top cg) strict ΩΣ SeqSpec(Top cg) strict id id id id SeqSpec(Top cg) stable QΩΣ SeqSpec(Top cg) stable, \array{ (Top_{cg}^{\ast/})_{Quillen} & \underoverset{\underoverset{\Omega}{\bot}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{} & (Top^{\ast/}_{cg})_{Quillen} \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ SeqSpec(Top_{cg})_{strict} & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} & SeqSpec(Top_{cg})_{strict} \\ {}^{\mathllap{id}}\downarrow \dashv \uparrow^{\mathrlap{id}} && {}^{\mathllap{id}}\downarrow \dashv \uparrow^{\mathrlap{id}} \\ SeqSpec(Top_{cg})_{stable} & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq_{\mathrlap{Q}}} & SeqSpec(Top_{cg})_{stable} } \,,

where the top parts is from corollary , the bottom vertical Quillen adjunction is the Bousfield localization of theorem and the bottom horizontal adjunction is the Quillen equivalence of prop. .

Hence (by this prop.) the derived functors of the functors in this diagram yield a commuting square of adjoint functors between the classical homotopy category (def.) and the stable homotopy category (def. ) of the form

Ho(Top */) ΩΣ Ho(Top */) Σ Ω Σ Ω Ho(Spectra) ΩΣ Ho(Spectra), \array{ Ho(Top^{\ast/}) & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} & Ho(Top^{\ast/}) \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ Ho(Spectra) & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq} & Ho(Spectra) } \,,

where the horizontal adjunctions are the canonically induced (via this prop.)suspension/looping functors by prop. and by lemma and theorem .

Cofibrant generation

We show that the stable model structure SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} from theorem is a cofibrantly generated model category (def.).

We will not use the result of this section in the remainder of part 1.1, but the following argument is the blueprint for the proof of the model structure on orthogonal spectra that we consider in part 1.2, in the section The stable model structure on structured spectra, and it will be used in the proof of the Quillen equivalence of SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} to the stable model structure on orthogonal spectra (thm.).

Moreover, that SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} is cofibrantly generated means that for 𝒞\mathcal{C} any topologically enriched category (def.) then there exists a projective model structure on functors [𝒞,SeqSpec(Top cg) stable] proj[\mathcal{C}, SeqSpec(Top_{cg})_{stable}]_{proj} on the category of topologically enriched functors 𝒞SeqSpec(Top cg)\mathcal{C} \to SeqSpec(Top_{cg}) (def.), in direct analogy to the projective model structure [𝒞,(Top cg */) Quillen] proj[\mathcal{C},(Top^{\ast/}_{cg})_{Quillen}]_{proj} (thm.). This is the model structure for parameterized stable homotopy theory. Just as the stable homotopy theory discussed here is the natural home of generalized (Eilenberg-Steenrod) cohomology theories (example ) so parameterized stable homotopy theory is the natural home of twisted cohomology theories.

In order to express the generating (acyclic) cofibrations, we need the following simple but important concept.

Definition

For KTop cg */K \in Top_{cg}^{\ast/}, and nn \in \mathbb{N}, write F nKSeqSpec(Top cg)F_n K \in SeqSpec(Top_{cg}) for the free spectrum on KK at nn, with components

(F nK) q{* forq<n S qnK forqn (F_n K)_q \coloneqq \left\{ \array{ \ast & for \; q \lt n \\ S^{q-n} \wedge K & for \; q \geq n } \right.

and with structure maps σ q\sigma_q the canonical identifications for qnq \geq n

σ q:S 1(F nK) q=S 1S qnKS q+1nK=(F nK) q+1. \sigma_q \;\colon\; S^1 \wedge (F_n K)_q = S^1 \wedge S^{q-n} \wedge K \overset{\simeq}{\longrightarrow} S^{q+1-n} \wedge K = (F_n K)_{q+1} \,.

For nn \in \mathbb{N}, write

k n:F n+1S 1F nS 0 k_n \;\colon\; F_{n+1}S^1 \longrightarrow F_n S^0

for the canonical morphisms of free sequential spectra with the following components

(k n) n+3 S 3 id S 3 (k n) n+2 S 2 id S 2 (k n) n+1 S 1 id S 1 (k n) n: * 0 S 0 * * * * k n: F n+1S 1 F nS 0 \array{ & \vdots && \vdots \\ (k_n)_{n+3} & S^3 &\stackrel{id}{\longrightarrow}& S^3 \\ (k_n)_{n+2} & S^2 &\stackrel{id}{\longrightarrow}& S^2 \\ (k_n)_{n+1} & S^1 &\stackrel{id}{\longrightarrow}& S^1 \\ (k_n)_n \colon & \ast &\stackrel{0}{\longrightarrow}& S^0 \\ & \ast &\longrightarrow& \ast \\ & \vdots && \vdots \\ & \ast &\longrightarrow& \ast \\ & \underbrace{\,\,\,} && \underbrace{\,\,\,} \\ k_n \colon & F_{n+1} S^1 &\stackrel{}{\longrightarrow}& F_n S^0 }
Example

The free spectrum F 0S 0F_0 S^0 (def. ) is the standard sequential sphere spectrum from def.

F 0S 0𝕊 std. F_0 S^0 \simeq \mathbb{S}_{std} \,.

Generally the free spectrum F 0KF_0 K is the suspension spectrum (def. ) on KK:

F 0KΣ K. F_0 K \simeq \Sigma^\infty K \,.

Just as forming suspension spectra is left adjoint to extracting the 0th component space of a sequential spectrum (prop. ), so forming the nnth free spectrum is left adjoint to extracting the nnth component space:

Proposition

For nn \in \mathbb{N}, let

Ev n:SeqSpec(Top cg)Top cg */ Ev_n \;\colon\; SeqSpec(Top_{cg}) \longrightarrow Top_{cg}^{\ast/}

be the functor from sequential spectra (def. ) to pointed topological spaces given by extracting the nnth component space

Ev n(X)X n. Ev_n(X) \coloneqq X_n \,.

Then this functor is right adjoint to forming nnth free spectra (def. ):

(F nEv n):SeqSpec(Top cg)Ev nF nTop cg */. (F_n \dashv Ev_n) \;\colon\; SeqSpec(Top_{cg}) \underoverset {\underset{Ev_n}{\longrightarrow}} {\overset{F_n}{\longleftarrow}} {\bot} Top_{cg}^{\ast/} \,.
Proof

The proof is verbatim as that of prop. , just with nn zeros inserted at the bottom of the sequences of components maps.

Definition

Write

I seq stableI seq strictSeqSpec(Top) I_{seq}^{stable} \coloneqq I_{seq}^{strict} \;\; \in SeqSpec(Top)

for the set of morphisms appearing already in def. , and write

J seq stableJ seq strict{k ni +} n,i +(I Top */) J_{seq}^{stable} \coloneqq J_{seq}^{strict} \;\sqcup\: \{ k_n \Box i_+ \}_{n \in \mathbb{N},i_+ \in \left(I_{Top^{\ast/}}\right)}

for the disjoint union of the other set of morphisms appearing in def. with the set {k ni +} n,i + \{k_n \Box i_+\}_{n,i_+} of pushout-products under smash tensoring (according to def. ) of the morphisms k nk_n from def. with the generating cofibrations of the classical model structure on pointed topological spaces (def.).

Theorem

The stable model structure SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} from theorem is cofibrantly generated (def.) with generating (acyclic) cofibrations the sets I seq stableI_{seq}^{stable} (and J seq stableJ_{seq}^{stable}) from def. .

This is one of the cofibrantly model categories considered in (Mandell-May-Schwede-Shipley 01) .

Proof

It is clear (as in theorem ) that the two classes have small domains (def.). Moreover, since I seq stable=I seq strictI_{seq}^{stable} = I_{seq}^{strict} and Cof stable=Cof strictCof_{stable} = Cof_{strict} by definition, the fact that the ccofibrations are the retracts of relative I seq stableI_{seq}^{stable}-cell complexes is part of theorem . It only remains to show that the stable acyclic cofibrations are precisely the retracts of relative J seq stableJ_{seq}^{stable}-cell complexes. This we is the statement of lemma below.

Lemma

The morphisms of free spectra {k n} n\{k_n\}_{n \in \mathbb{N}} from def. co-represent the adjunct structure maps of sequential spectra from def. , in that for XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}), then

SeqSpec(F nS 0,X) X n SeqSpec(k n,X) σ˜ n X SeqSpec(F n+1S 1,X) ΩX n+1, \array{ SeqSpec(F_n S^0, X) &\simeq& X_n \\ {}^{\mathllap{SeqSpec(k_n,X)}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma_n^X}} \\ SeqSpec(F_{n+1}S^1, X) &\simeq& \Omega X_{n+1} } \,,

where on the left we have the hom-spaces of def. , and where the horizontal equivalences are via prop. .

Proof

Recall that we are precomposing with

(k n) n+3 S 3 id S 3 (k n) n+2 S 2 id S 2 (k n) n+1 S 1 id S 1 (k n) n: * 0 S 0 * * * * k n: F n+1S 1 F nS 0 \array{ & \vdots && \vdots \\ (k_n)_{n+3} & S^3 &\stackrel{id}{\longrightarrow}& S^3 \\ (k_n)_{n+2} & S^2 &\stackrel{id}{\longrightarrow}& S^2 \\ (k_n)_{n+1} & S^1 &\stackrel{id}{\longrightarrow}& S^1 \\ (k_n)_n \colon & \ast &\stackrel{0}{\longrightarrow}& S^0 \\ & \ast &\longrightarrow& \ast \\ & \vdots && \vdots \\ & \ast &\longrightarrow& \ast \\ & \underbrace{\,\,\,} && \underbrace{\,\,\,} \\ k_n \colon & F_{n+1} S^1 &\stackrel{}{\longrightarrow}& F_n S^0 }

Now for XX any sequential spectrum, then a morphism f:F nS 0Xf \colon F_n S^0 \to X is uniquely determined by its nnth component f n:S 0X nf_n \colon S^0 \to X_n: the compatibility with the structure maps forces the next component, in particular, to be σ n XΣf\sigma_n^X\circ \Sigma f:

ΣS 0 Σf ΣX n σ n X S 1 σ n XΣf X n. \array{ \Sigma S^0 &\stackrel{\Sigma f}{\longrightarrow}& \Sigma X_n \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\sigma_n^X}} \\ S^1 &\stackrel{\sigma_n^X \circ \Sigma f}{\longrightarrow}& X_n } \,.

But that (n+1)(n+1)st component is just the component that similarly determines the precompositon of ff with k nk_n, hence fk nf\circ k_n is uniquely determined by the map σ n XΣf\sigma_n^X \circ \Sigma f. Therefore SeqSpec(k n,)SeqSpec(k_n,-) is the function

SeqSpec(k n,):X n=SeqSpec(S 0,X n)fσ n XΣfMaps(S 1,X n+1) *=ΩX n+1. SeqSpec(k_n,-) \;\colon\; X_n = SeqSpec(S^0, X_n) \stackrel{f \mapsto \sigma_n^X \circ \Sigma f}{\longrightarrow} \Maps(S^1, X_{n+1})_\ast = \Omega X_{n+1} \,.

It remains to see that this is indeed the (ΣΩ)(\Sigma \dashv \Omega)-adjunct of σ n X\sigma_n^X. By the general formula for adjuncts, this is

σ˜ n X:X nηΩΣX nΩσ n XΩX n+1. \tilde \sigma_n^X \;\colon\; X_n \stackrel{\eta}{\longrightarrow} \Omega \Sigma X_n \stackrel{\Omega \sigma_n^X}{\longrightarrow} \Omega X_{n+1} \,.

To compare to the above, we check what this does on points: S 0fX nS^0 \stackrel{f}{\longrightarrow} X_n is sent to the composite

S 0fX nηΩΣX nΩσ 0 XΩX n+1. S^0 \stackrel{f}{\longrightarrow} X_n \stackrel{\eta}{\longrightarrow} \Omega \Sigma X_n \stackrel{\Omega \sigma_0^X}{\longrightarrow} \Omega X_{n+1} \,.

To identify this as a map S 1X n+1S^1 \to X_{n+1}, we use the adjunction isomorphism once more to throw all the Ω\Omega-s on the right back to Σ\Sigma-s the left, to finally find that this is indeed

σ n XΣf:S 1=ΣS 0ΣfΣX nσ n XX n+1. \sigma_n^X \circ \Sigma f \;\colon\; S^1 = \Sigma S^0 \stackrel{\Sigma f}{\longrightarrow} \Sigma X_n \stackrel{\sigma_n^X}{\longrightarrow} X_{n+1} \,.
Lemma

Every element in J seq stableJ_{seq}^{stable} (def. ) is an acyclic cofibration in the model structure SeqSpec(Top cg) stableSeqSpec(Top_{cg})_{stable} from theorem .

Proof

For the elements in J seq strictJ_{seq}^{strict} this is part of theorem . It only remains to see that the morphisms k ni +k_n \Box i_+ are stable acyclic cofibrations.

To see that they are stable cofibrations, hence strict cofibrations:

By Joyal-Tierney calculus (prop.) k ni +k_n \Box i_+ has left lifting against any strict acyclic fibration ff precisely if k nk_n has left lifting against the pullback powering f i +f^{\Box i_+} (def. ). By prop. the latter is still a strict acyclic fibration. Since k nk_n is evidently a strict cofibration, the lifting follows and hence also k ni +k_n \Box i_+ is a strict cofibration, hence a stable cofibration.

To see that they are stable weak equivalences: For each qq the morphisms k nS q1k_n \wedge S^{q-1} are stable acyclic cofibrations, and since stable acyclic cofibrations are preserved under pushout, it follows by two-out-of-three that also k ni +k_n \Box i_+ is a stable weak equivalence.

The reason for considering the set {k ni +}\{k_n \Box i_+\} is to make the following true:

Lemma

A morphism f:XYf \colon X \to Y in SeqSpec(Top)SeqSpec(Top) is a J seq stableJ_{seq}^{stable}-injective morphism (def.) precisely if

  1. it is fibration in the strict model structure (hence degreewise a fibration);

  2. for all nn \in \mathbb{N} the commuting squares of structure map compatibilities on the underlying sequential spectra

    X n σ˜ n X ΩX n+1 f n Ωf n+1 Y n σ˜ n Y ΩY n+1 \array{ X_n &\overset{\tilde\sigma^X_n}{\longrightarrow}& \Omega X_{n+1} \\ {}^{\mathllap{f_n}}\downarrow && \downarrow^{\mathrlap{\Omega f_{n+1}}} \\ Y_n &\underset{\tilde \sigma^Y_n}{\longrightarrow}& \Omega Y_{n+1} }

    exhibit homotopy pullbacks (def.) in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict}, in that the comparison map

    X nY n×ΩY n+1ΩX n1 X_n \longrightarrow Y_n \underset{\Omega Y_{n+1}}{\times} \Omega X_{n-1}

    is a weak homotopy equivalence (notice that Ωf n+1\Omega f_{n+1} is a fibration by the previous item and since Ω=Maps(S 1,) *\Omega = Maps(S^1,-)_\ast is a right Quillen functor by prop. ).

In particular, the J seq stableJ_{seq}^{stable}-injective objects are precisely the Omega-spectra, def. .

Proof

By theorem , lifting against J seq stricJ_{seq}^{stric} alone characterizes strict fibrations, hence degreewise fibrations. Lifting against the remaining pushout product morphism k ni +k_n \Box i_+ is, by Joyal-Tierney calculus (prop.), equivalent to left lifting i +i_+ against the pullback powering f k nf^{\Box k_n} from def. . Since the {i +}\{i_+\} are the generating cofibrations in Top cg */Top_{cg}^{\ast/} such lifting means that f k nf^{\Box k_n} is a weak equivalence in the strict model sructure. But by lemma , f k nf^{\Box k_n} is precisely the comparison morphism in question.

Lemma

A morphism in SeqSpec(Top)SeqSpec(Top) which is both

  1. a stable weak homotopy equivalence (def. );

  2. a J seq stableJ_{seq}^{stable}-injective morphism (def. , def.)

is an acyclic fibration in the strict model structure, hence is degreewise a weak homotopy equivalence and Serre fibration of topological spaces;

Proof

Let f:XBf \colon X \to B be both a stable weak homotopy equivalence as well as a KK-injective morphism. Since KK contains the generating acyclic cofibrations for the strict model structure, ff is in particular a strict fibration, hence a degreewise fibration.

Consider the fiber FF of ff, hence the morphism F*F \to \ast which is the pullback of ff along *B\ast \to B. Notice that since ff is a strict fibration, this is the homotopy fiber (def.) of ff in the strict model structure.

We claim that

  1. FF is an Omega-spectrum;

  2. F*F\to \ast is a stable weak homotopy equivalence.

The first item follows since FF, being the pullback of a KK-injective morphisms, is a KK-injective object (prop.), so that, by lemma , FF it is an Omega-spectrum.

For the second item:

Since FXfBF \to X \overset{f}{\to} B is degreewise a homotopy fiber sequence, there are degreewise its long exact sequences of homotopy groups (exmpl.)

π +1(B n)π (F n)π (X n)(f n) *π (B n)π 1(B n)π 0(F n)π 0(X n)π 0(B) n \cdots \to \pi_{\bullet + 1}(B_n) \longrightarrow \pi_\bullet(F_n) \longrightarrow \pi_\bullet(X_n) \overset{(f_n)_\ast}{\longrightarrow} \pi_\bullet(B_n) \to \cdots \to \pi_1(B_n) \longrightarrow \pi_0(F_n) \longrightarrow \pi_0(X_n) \longrightarrow \pi_0(B)_n

Since in the category Ab of abelian group forming filtered colimits is an exact functor (prop.), it follows that after passing to stable homotopy groups the resulting sequence

π +1(X)f *π +1(B)π (F)π (X)(f *π (B) \cdots \pi_{\bullet + 1}(X) \overset{f_\ast}{\longrightarrow} \pi_{\bullet + 1}(B) \longrightarrow \pi_\bullet(F) \longrightarrow \pi_\bullet(X) \overset{(f_\ast}{\longrightarrow} \pi_\bullet(B) \to \cdots

is still a long exact sequence.

Since, by assumption, f *f_\ast is an isomorphism, this exactness implies that π (F)=0\pi_\bullet(F) = 0, and hence that F*F \to \ast is a stable weak homotopy equivalence. But since, by the first item above, FF is an Omega-spectrum, it follows (via example ) that F*F \to \ast is even a degreewise weak homotopy equivalence, hence that π (F n)0\pi_\bullet(F_n)\simeq 0 for all nn \in \mathbb{N}.

Feeding this back into the above degreewise long exact sequence of homotopy groups now implies that π 1(f n)\pi_{\bullet \geq 1}(f_n) is a weak homotopy equivalence for all nn and for each homotopy group in positive degree.

To deduce the remaining case that also π 0(f 0)\pi_0(f_0) is an isomorphism, observe that by assumption of KK-injectivity, lemma gives that f 0f_0 is the pullback (in topological spaces) of Ω(f 1)\Omega (f_{1}). But by the above Ωf 1\Omega f_{1} is a weak homotopy equivalence, and since Ω=Maps(S 1,) *\Omega = Maps(S^1,-)_\ast is a right Quillen functor (prop. ) it is also a Serre fibration. Therefore f 0f_0 is the pullback of an acyclic Serre fibration and hence itself a weak homotopy equivalence.

Lemma

The retracts (rmk.) of J seq stableJ_{seq}^{stable}-relative cell complexes are precisely the stable acyclic cofibrations.

Proof

Since all elements of J seq stableJ_{seq}^{stable} are stable weak equivalences and strict cofibrations by lemma , it follows that every retract of a relative J seq stableJ_{seq}^{stable}-cell complex has the same property.

In the other direction, let ff be a stable acyclic cofibration. Apply the small object argument (prop.) to factor it

f:J seq stableCelliJ seq stableInjp f \colon \underoverset{J_{seq}^{stable}Cell}{i}{\to} \underoverset{J_{seq}^{stable} Inj}{p}{\to}

as a J seq stableJ_{seq}^{stable}-relative cell complex ii followed by a J seq stableJ_{seq}^{stable}-injective morphism pp. By the previous statement ii is a stable weak homotopy equivalence, and hence by assumption and by two-out-of-three so is pp. Therefore lemma implies that pp is a strict acyclic fibration. But then the assumption that ff is a strict cofibration means that it has the left lifting property against pp, and so the retract argument (prop.) implies that ff is a retract of the relative J seq stableJ_{seq}^{stable}-cell complex ii.

This completes the proof of theorem .

The stable homotopy category

Definition

Write

Ho(Spectra)Ho(SeqSpec(Top cg) stable) Ho(Spectra) \coloneqq Ho(SeqSpec(Top_{cg})_{stable})

for the homotopy category (defn.) of the stable model structure on topological sequential spectra from theorem .

This is called the stable homotopy category.

The stable homotopy category of def. inherits particularly nice properties that are usefully axiomatized for themselves. This axiomatics is called triangulated category structure (def. below) where the “triangles” are referring to the structure of the long fiber sequences and long cofiber sequences (prop.) which happen to coincide in stable homotopy theory.

Additivity

The stable homotopy category Ho(Spectra)Ho(Spectra) is the analog in homotopy theory of the category Ab of abelian groups in homological algebra. While the stable homotopy category is not an abelian category, as Ab is, but a homotopy-theoretic version of that to which we turn below, it is an additive category.

\,

Lemma

The stable homotopy category (def. ) has finite coproducts. They are represented by wedge sums (example ) of CW-spectra (def. ).

Proof

Having finite coproducts means

  1. having empty coproducts, hence initial objects,

  2. and having binary coproducts.

Regarding the initial object:

The spectrum Σ *\Sigma^\infty \ast (suspension spectrum (example ) on the point) is both an initial object and a terminal object in SeqSpec(Top cg)SeqSpec(Top_{cg}). This implies in particular that it is both fibrant and cofibrant. Finally its standard cylinder spectrum (example ) is trivial (Σ *)(I +)Σ *(\Sigma^\infty \ast) \wedge (I_+)\simeq \Sigma^\infty \ast. All together with means that for XX any fibrant-cofibrant spectrum, then

Hom Ho(Spectra)(Σ *,Z)Hom SeqSpec(Σ *,Z)/ * Hom_{Ho(Spectra)}(\Sigma^\infty \ast, Z) \simeq Hom_{SeqSpec}(\Sigma^\infty \ast, Z)/_\sim \simeq \ast

and so Σ *\Sigma^\infty \ast also represents the initial object in the stable homotopy category.

Now regarding binary coproducts:

By prop. and prop. , every spectrum has a cofibrant replacement by a CW-spectrum. By prop. the wedge sum XYX \vee Y of two CW-spectra is still a CW-spectrum, hence still cofibrant.

Let PP and QQ be fibrant and cofibrant replacement functors, respectively, as in the section_Classical homotopy theory – The homotopy category.

We claim now that P(XY)Ho(Spectra)P(X \vee Y) \in Ho(Spectra) is the coproduct of PXP X with PYP Y in Ho(Spectra)Ho(Spectra). By definition of the homotopy category (def.) this is equivalent to claiming that for ZZ any stable fibrant spectrum (hence an Omega-spectrum by theorem ) then there is a natural isomorphism

Hom SeqSpec(P(XY),QZ)/ Hom SeqSpec(PX,QZ)/ ×Hom SeqSpec(PY,QZ)/ Hom_{SeqSpec}(P(X \vee Y), Q Z)/_\sim \simeq Hom_{SeqSpec}(P X, Q Z)/_\sim \times Hom_{SeqSpec}(P Y, Q Z)/_\sim

between left homotopy-classes of morphisms of sequential spectra.

But since XYX \vee Y is cofibrant and ZZ is fibrant, there is a natural isomorphism (prop.)

Hom SeqSpec(P(XY),QZ)/ Hom SeqSpec(XY,Z)/ . Hom_{SeqSpec}(P(X \vee Y), Q Z)/_\sim \overset{\simeq}{\longrightarrow} Hom_{SeqSpec}(X \vee Y, Z)/_\sim \,.

Now the wedge sum XYX \vee Y is the coproduct in SeqSpec(Top cg)SeqSpec(Top_{cg}), and hence morphisms out of it are indeed in natural bijection with pairs of morphisms out of the two summands. But we need this property to hold still after dividing out left homotopy. The key is that smash tensoring (def. ) distributes over wedge sum

(XY)(I +)(X(I +))(Y(I +)) (X \vee Y) \wedge (I_+) \simeq (X \wedge (I_+)) \vee (Y\wedge (I_+))

(due to the fact that the smash product of compactly generated pointed topological spaces distributes this way over wedge sum of pointed spaces). This means that also left homotopies out of XYX \vee Y are in natural bijection with pairs of left homotopies out of the summands separately, and hence that there is a natural isomorphism

Hom SeqSpec(XY,Z)/ Hom SeqSpec(X,Z)/ ×Hom SeqSpec(Y,Z)/ . Hom_{SeqSpec}(X \vee Y, Z)/_\sim \overset{\simeq}{\longrightarrow} Hom_{SeqSpec}(X,Z)/_\sim \times Hom_{SeqSpec}(Y,Z)/_\sim \,.

Finally we may apply the inverse of the natural isomorphism used before (prop.) to obtain in total

Hom SeqSpec(X,Z)/ ×Hom SeqSpec(Y,Z)/ Hom SeqSpec(PX,QZ)/ ×Hom SeqSpec(PY,QZ)/ . Hom_{SeqSpec}(X,Z)/_\sim \times Hom_{SeqSpec}(Y,Z)/_\sim \overset{\simeq}{\longrightarrow} Hom_{SeqSpec}(P X,Q Z)/_\sim \times Hom_{SeqSpec}(P Y,Q Z)/_\sim \,.

The composite of all these isomorphisms proves the claim.

Definition

Define group structure on the pointed hom-sets of the stable homotopy category (def. )

[X,Y]Grp [X,Y] \in Grp

induced from the fact (prop.) that the hom-sets of any homotopy category into an object in the image of the canonical loop space functor Ω\Omega inherit group structure, together with the fact (theorem ) that on the stable homotopy category Ω\Omega and Σ\Sigma are inverse to each other, so that

[X,Y][X,ΩΣY], [X,Y] \simeq [X, \Omega \Sigma Y] \,,
Lemma

The group structure on [X,Y][X,Y] in def. is abelian and composition in Ho(Spectra)Ho(Spectra) is bilinear with respect to this group structure. (Hence this makes Ho(Spectra)Ho(Spectra) an Ab-enriched category.)

Proof

Recall (prop, rmk.) that the group structure is given by concatenation of loops

XΔ XX×X(f,g)ΩΣX×ΩΣXΩΣX. X \overset{\Delta_X}{\to} X \times X \overset{(f,g)}{\longrightarrow} \Omega \Sigma X \times \Omega \Sigma X \overset{}{\longrightarrow} \Omega \Sigma X \,.

That the group structure is abelian follows via the Eckmann-Hilton argument from the fact that there is always a compatible second (and indeed arbitrarily many compatible) further group structures, since, by stability

[X,Y][X,ΩΣY][X,Ω(ΩΣ)ΣY]=[X,Ω 2Σ 2Y]. [X,Y] \simeq [X, \Omega \Sigma Y] \simeq [X, \Omega \circ (\Omega \Sigma) \circ \Sigma Y] = [X, \Omega^2 \Sigma^2 Y] \,.

That composition of morphisms distributes over the operation in this group is evident for precomposition. Let f:WXf \colon W \to X then clearly

f *:[X,ΩΣY][W,ΩΣY] f^\ast \;\colon\; [X,\Omega \Sigma Y] \longrightarrow [W,\Omega \Sigma Y]

preserves the group structure induced by the group structure on ΩΣY\Omega\Sigma Y. That the same holds for postcomposition may be immediately deduced from noticing that this group structure is also the same as that induced by the cogroup structure on ΣΩX\Sigma \Omega X, so that with g:YZg \colon Y \to Z then

g *:[ΣΩX,Y][ΣΩX,Z] g_\ast \;\colon\; [\Sigma \Omega X,Y] \longrightarrow [\Sigma\Omega X,Z]

preserves group structure.

More explicitly, we may see the respect for groupstructure structure of the postcomposition opeation from the naturality of the loop composition map which is manifest when representing loop spectra via the standard topological loop space object ΩX=fib(Maps(I +,X)X×X)\Omega X = fib( Maps(I_+,X)\to X \times X ) (rmk.) under smash powering (def. ).

To make this fully explicit, consider the following diagram in Ho(Spectra)Ho(Spectra):

Y×Y ΩΣY×ΩΣY Q(Maps(S 1,ΣY) *×Maps(S 1,ΣY) *) Q(Maps(S [0,2] 1,ΣY)) * ΩΣY Y g×g ΩΣg×ΩΣg Q(Maps(S 1,Σg) *×Maps(S 1,ΩΣg) *) Q(Maps(S [0,2] 1,Σg) *) ΩΣg g Z×Z ΩΣZ×ΩΣZ Q(Maps(S 1,ΣZ) *×Maps(S 1,ΣZ) *) Q(Maps(S [0,2] 1,ΣZ) *) ΩΣZ Z, \array{ Y \times Y &\overset{\simeq}{\longrightarrow}& \Omega \Sigma Y \times \Omega \Sigma Y &\overset{\simeq}{\longrightarrow}& Q(Maps(S^1,\Sigma Y)_\ast \times Maps(S^1,\Sigma Y)_\ast) &\longrightarrow& Q(Maps(S_{[0,2]}^1, \Sigma Y))_\ast &\simeq& \Omega \Sigma Y &\simeq& Y \\ {}^{\mathllap{g \times g}}\downarrow && \downarrow^{\mathrlap{\Omega \Sigma g \times \Omega \Sigma g}} && \downarrow^{\mathrlap{Q(Maps(S^1, \Sigma g)_\ast \times Maps(S^1,\Omega \Sigma g)_\ast)}} && \downarrow^{\mathrlap{Q(Maps(S_{[0,2]}^1,\Sigma g)_\ast)}} && \downarrow^{\mathrlap{\Omega \Sigma g}} && \downarrow^{g} \\ Z \times Z &\overset{\simeq}{\longrightarrow}& \Omega \Sigma Z \times \Omega \Sigma Z &\overset{\simeq}{\longrightarrow}& Q(Maps(S^1,\Sigma Z)_\ast \times Maps(S^1,\Sigma Z)_\ast) &\longrightarrow& Q(Maps(S_{[0,2]}^1, \Sigma Z)_\ast) &\simeq& \Omega \Sigma Z &\simeq& Z } \,,

where S [0,2] 1S^1_{[0,2]} denotes the sphere of length 2.

Here the leftmost square and the rightmost square are the naturality squares of the equivalence of categories (ΣΩ)(\Sigma\dashv \Omega) (theorem ).

The second square from the left and the second square from the right exhibit the equivalent expression of Ω\Omega as the right derived functor of (either the standard or the alternative, by lemma ) degreewise loop space functor. Here we let ΣX\Sigma X denote any fibrant representative, for notational brevity, and use that the derived functor of a right Quillen functor is given on fibrant objects by the original functor followed by cofibrant replacement (prop.).

The middle square is the image under QQ of the evident naturality square for concatenation of loops. This is where we use that we have the standard model for forming loop spaces and concatenation of loops (rmk.): the diagram commutes because the loops are always poinwise pushed forward along the map ff.

It is conventional (Adams 74, p. 138) to furthermore make the following definition:

Definition

For X,YHo(Spectra)X, Y \in Ho(Spectra) two spectra, define the \mathbb{Z}-graded abelian group

[X,Y] Ab [X,Y]_\bullet \; \in Ab^{\mathbb{Z}}

to be in degree nn the abelian hom group of lemma out of XX into the nn-fold suspension of YY (lemma ):

[X,Y] n[X,Σ nY]. [X,Y]_n \;\coloneqq\; [X, \Sigma^{-n} Y] \,.

Defining the composition of f 1[X,Y] n 1f_1 \in [X,Y]_{n_1} with f 2[Y,Z] n 2f_2 \in [Y,Z]_{n_2} to be the composite

Xf 1Σ n 1YΣ n 2(f 2)Σ n 2(Σ n 1Z)Σ n 1n 2Z X \overset{f_1}{\longrightarrow} \Sigma^{-n_1} Y \overset{\Sigma^{-n_2}(f_2)}{\longrightarrow} \Sigma^{-n_2}(\Sigma^{-n_1} Z) \simeq \Sigma^{-n_1 - n_2} Z

gives the stable homotopy category the structure of an Ab Ab^{\mathbb{Z}}-enriched category.

Example

(generalized cohomology groups)

Let ESeqSpec(Top cg)E \in SeqSpec(Top_{cg}) be an Omega-spectrum (def. ) and let XTop cg */X\in Top^{\ast/}_{cg} be a pointed topological space with Σ X\Sigma^\infty X its suspension spectrum (example ). Then the graded abelian group (by prop. , def. )

E˜ (X) [Σ X,E] =[Σ X,Σ E] [X,Ω Σ E] * [X,E ] * \begin{aligned} \tilde E^\bullet(X) & \coloneqq [\Sigma^\infty X, E]_{-\bullet} \\ & = [\Sigma^\infty X, \Sigma^\bullet E] \\ & \simeq [X, \Omega^\infty \Sigma^\bullet E]_\ast \\ & \simeq [X, E_\bullet]_\ast \end{aligned}

is also called the reduced cohomology of XX in the generalized (Eilenberg-Steenrod) cohomology theory that is represented by EE.

Here the equivalences used are

  1. the adjunction isomorphism of (Σ Ω )(\Sigma^\infty \dashv \Omega^\infty) from theorem ;

  2. the isomorphism Σ[1]\Sigma \simeq [1] of suspension with the shift spectrum (def. ) on Ho(Spectra)Ho(Spectra) of lemma , together with the nature of Ω \Omega^\infty from prop. .

The latter expression

E˜ n(X)[X,E n] * \tilde E^n(X) \simeq [X, E_n]_\ast

(on the right the hom in in the classical homotopy category Ho(Top */)Ho(Top^{\ast/}) of pointed topological spaces) is manifestly the definition of reduced generalized (Eilenberg-Steenrod) cohomology as discussed in part S in the section on the Brown representability theorem.

Suppose EE here is not necessarily given as an Omega-spectrum. In general the hom-groups [X,E]=[X,E] stable[X,E] = [X,E]_{stable} in the stable homotopy category are given by the naive homotopy classes of maps out of a cofibrant resolution of XX into a fibrant resolution of EE (by this lemma). By theorem a fibrant replacement of EE is given by Omega-spectrification QEQ E (def. ). Since the stable model structure of theorem is a left Bousfield localization of the strict model structure from theorem , and since for the latter all objects are fibrant, it follows that

[X,E] stable[X,QE] strict, [X,E]_{stable} \simeq [X,Q E]_{strict} \,,

and hence

E 0(X) [Σ X,E] stable [Σ X,QE] strict [X,Ω QE] * =[X,(QE) 0] *, \begin{aligned} E^0(X) &\coloneqq [\Sigma^\infty X, E]_{stable} \\ & \simeq [\Sigma^\infty X, Q E]_{strict} \\ & \simeq [X, \Omega^\infty Q E]_\ast \\ & = [X, (Q E)_0]_\ast \end{aligned} \,,

where the last two hom-sets are again those of the classical homotopy category. Now if EE happens to be a CW-spectrum, then by remark its Omega-spectrification is given simply by (QE) nlim kΩ kE n+k)(Q E)_n \simeq \underset{\longrightarrow}{\lim}_k \Omega^k E_{n+k}) and hence in this case

E 0(X)[X,lim kΩ kE k] *. E^0(X) \simeq [X, \underset{\longrightarrow}{\lim}_k \Omega^k E_k]_\ast \,.

If XX here is moreover a compact topological space, then it may be taken inside the colimit (e.g. Weibel 94, topology exercise 10.9.2), and using the (ΣΩ)(\Sigma \dashv \Omega)-adjunction this is rewritten as

E 0(X) lim k[X,Ω kE k] * lim k[Σ kX,E k] *. \begin{aligned} E^0(X) & \simeq \underset{\longrightarrow}{\lim}_k [X, \Omega^k E_k]_\ast \\ & \simeq \underset{\longrightarrow}{\lim}_k [\Sigma^k X, E_k]_\ast \end{aligned} \,.

(e.g. Adams 74, prop. 2.8).

This last expression is sometimes used to define cohomology with coefficients in an arbitrary spectrum. For examples see in the part S the section Orientation in generalized cohomology.

More generally, it is immediate now that there is a concept of EE-cohomology not only for spaces and their suspension spectra, but also for general spectra: for XHo(Spectra)X \in Ho(Spectra) be any spectrum, then

E˜ (X)[X,Σ E] \tilde E^\bullet(X) \coloneqq [X,\Sigma^\bullet E]

is called the reduced EE-cohomology of the spectrum XX.

Beware that here one usually drops the tilde sign.

In summary, lemma and lemma state that the stable homotopy category is an Ab-enriched category with finite coproducts. This is called an additive category:

Definition

An additive category is a category which is

  1. an Ab-enriched category;

    (sometimes called a pre-additive category–this means that each hom-set carries the structure of an abelian group and composition is bilinear)

  2. which admits finite coproducts

    (and hence, by prop. below, finite products which coincide with the coproducts, hence finite biproducts).

Proposition

In an Ab-enriched category, a finite product is also a coproduct, and dually.

This statement includes the zero-ary case: any terminal object is also an initial object, hence a zero object (and dually), hence every additive category (def. ) has a zero object.

More precisely, for {X i} iI\{X_i\}_{i \in I} a finite set of objects in an Ab-enriched category, then the unique morphism

iIX ijIX j, \underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j \,,

whose components are identities for i=ji = j and are zero otherwise, is an isomorphism.

Proof

Consider first the zero-ary case. Given an initial object \emptyset and a terminal object *\ast, observe that since the hom-sets Hom(,)Hom(\emptyset,\emptyset) and Hom(*,*)Hom(\ast,\ast) by definition contain a single element, this element has to be the zero element in the abelian group structure. But it also has to be the identity morphism, and hence id =0id_\emptyset = 0 and id *=0id_{\ast} = 0. It follows that the 0-element in Hom(*,)Hom(\ast, \emptyset) is a left and right inverse to the unique element in Hom(,*)Hom(\emptyset,\ast), and so this is an isomorphism

0:*. 0 \;\colon\; \emptyset \overset{\simeq}{\longrightarrow} \ast \,.

Consider now the case of binary (co-)products. Using the existence of the zero object, hence of zero morphisms, then in addition to its canonical projection maps p i:X 1×X 2X ip_i \colon X_1 \times X_2 \to X_i, any binary product also receives “injection” maps X iX 1×X 2X_i \to X_1 \times X_2, and dually for the coproduct:

X 1 X 2 (id,0) (0,id) id X 1 X 1×X 2 id X 2 p X 1 p X 2 X 1 X 2,X 1 X 2 i X 1 i X 2 id X 1 X 1X 2 id X 2 (id,0) (0,id) X 1 X 2. \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow \\ X_1 && && X_2 } \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\; \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow \\ X_1 && && X_2 } \,.

Observe some basic compatibility of the AbAb-enrichment with the product:

First, for (α 1,β 1),(α 2,β 2):RX 1×X 2(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2 then

()(α 1,β 1)+(α 2,β 2)=(α 1+α 2,β 1+β 2) (\star) \;\;\;\;\;\; (\alpha_1,\beta_1) + (\alpha_2, \beta_2) = (\alpha_1+ \alpha_2 , \; \beta_1 + \beta_2)

(using that the projections p 1p_1 and p 2p_2 are linear and by the universal property of the porduct).

Second, (id,0)p 1(id,0) \circ p_1 and (0,id)p 2(0,id) \circ p_2 are two projections on X 1×X 2X_1\times X_2 whose sum is the identity:

()(id,0)p 1+(0,id)p 2=id X 1×X 2. (\star\star) \;\;\;\;\;\; (id, 0) \circ p_1 + (0, id) \circ p_2 = id_{X_1 \times X_2} \,.

(We may check this, via the Yoneda lemma on generalized elements: for (α,β):RX 1×X 2(\alpha, \beta) \colon R \to X_1\times X_2 any morphism, then (id,0)p 1(α,β)=(α,0)(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0) and (0,id)p 2(α,β)=(0,β)(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta), so the statement follows with equation ()(\star).)

Now observe that for f i:X iQf_i \;\colon\; X_i \to Q any two morphisms, the sum

ϕf 1p 1+f 2p 2:X 1×X 2Q \phi \;\coloneqq\; f_1 \circ p_1 + f_2 \circ p_2 \;\colon\; X_1 \times X_2 \longrightarrow Q

gives a morphism of cocones

X 1 X 2 (id,0) (0,id) id X 1 X 1×X 2 id X 2 X 1 ϕ X 2 f 1 f 2 Q. \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & && \\ X_1 && \downarrow^{\mathrlap{\phi}} && X_2 \\ & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}} \\ && Q } \,.

Moreover, this is unique: suppose ϕ\phi' is another morphism filling this diagram, then, by using equation ()(\star \star), we get

(ϕϕ) =(ϕϕ)id X 1×X 2 =(ϕϕ)((id X 1,0)p 1+(0,id X 2)p 2) =(ϕϕ)(id X 1,0)=0p 1+(ϕϕ)(0,id X 2)=0p 2 =0 \begin{aligned} (\phi-\phi') & = (\phi - \phi') \circ id_{X_1 \times X_2} \\ &= (\phi - \phi') \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ & = \underset{ = 0}{\underbrace{(\phi - \phi') \circ (id_{X_1}, 0)}} \circ p_1 + \underset{ = 0}{\underbrace{(\phi - \phi') \circ (0, id_{X_2})}} \circ p_2 \\ & = 0 \end{aligned}

and hence ϕ=ϕ\phi = \phi'. This means that X 1×X 2X_1\times X_2 satisfies the universal property of a coproduct.

By a dual argument, the binary coproduct X 1X 2X_1 \sqcup X_2 is seen to also satisfy the universal property of the binary product. By induction, this implies the statement for all finite (co-)products.

Remark

Finite coproducts coinciding with products as in prop. are also called biproducts or direct sums, denoted

X 1X 2X 1X 2X 1×X 2. X_1 \oplus X_2 \coloneqq X_1 \sqcup X_2 \simeq X_1 \times X_2 \,.

The zero object is denoted “0”, of course.

Conversely:

Definition

A semiadditive category is a category that has all finite products which, moreover, are biproducts in that they coincide with finite coproducts as in def. .

Proposition

In a semiadditive category, def. , the hom-sets acquire the structure of commutative monoids by defining the sum of two morphisms f,g:XYf,g \;\colon\; X \longrightarrow Y to be

f+gXΔ XX×XXXfgYYYY XY. f + g \;\coloneqq\; X \overset{\Delta_X}{\to} X \times X \simeq X \oplus X \overset{f \oplus g}{\longrightarrow} Y \oplus Y \simeq Y \sqcup Y \overset{\nabla_X}{\to} Y \,.

With respect to this operation, composition is bilinear.

Proof

The associativity and commutativity of ++ follows directly from the corresponding properties of \oplus. Bilinearity of composition follows from naturality of the diagonal Δ X\Delta_X and codiagonal X\nabla_X:

W Δ W W×W WW e e×e ee X Δ X X×X XX fg YY YY X Y hh hh h ZZ ZZ Z Z \array{ W &\overset{\Delta_W}{\longrightarrow}& W \times W &\overset{\simeq}{\longrightarrow}& W \oplus W \\ \downarrow^{\mathrlap{e}} && \downarrow^{\mathrlap{e \times e}} && \downarrow^{\mathrlap{e \oplus e}} \\ X &\overset{\Delta_X}{\to}& X \times X &\simeq& X \oplus X &\overset{f \oplus g}{\longrightarrow}& Y \oplus Y &\simeq& Y \sqcup Y &\overset{\nabla_X}{\to}& Y \\ && && && \downarrow^{\mathrlap{h \oplus h}} && \downarrow^{\mathrlap{h \sqcup h}} && \downarrow^{\mathrlap{h}} \\ && && && Z \oplus Z &\simeq& Z \sqcup Z &\overset{\nabla_Z}{\to}& Z }
Proposition

Given an additive category according to def. , then the enrichement in commutative monoids which is induced on it via prop. and prop. from its underlying semiadditive category structure coincides with the original enrichment.

Proof

By the proof of prop. , the codiagonal on any object in an additive category is the sum of the two projections:

X:XXp 1+p 2X. \nabla_X \;\colon\; X \oplus X \overset{p_1 + p_2}{\longrightarrow} X \,.

Therefore (checking on generalized elements, as in the proof of prop. ) for all morphisms f,g:XYf,g \colon X \to Y we have commuting squares of the form

X f+g Y Δ X p 1+p 2 Y= XX fg YY. \array{ X &\overset{f+g}{\longrightarrow}& Y \\ {}^{\mathllap{\Delta_X}}\downarrow && \uparrow^{\mathrlap{\nabla_Y =}}_{\mathrlap{p_1 + p_2}} \\ X \oplus X &\underset{f \oplus g}{\longrightarrow}& Y\oplus Y } \,.
Remark

Prop. says that being an additive category is an extra property on a category, not extra structure. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.

In conclusion we have:

Proposition

The stable homotopy category (def. ) is an additive category (def. ).

Hence prop. implies that in the stable homotopy category finite coproducts (wedge sums) and finite products agree, in that they are finite biproducts (direct sums).

×Ho(Spectra). \vee \simeq \times \simeq \oplus \;\;\in \;\; Ho(Spectra) \,.
Proof

By lemma and lemma .

Triangulated structure

We have seen above that the stable homotopy category Ho(Spectra)Ho(Spectra) is an additive category. In the context of homological algebra, when faced with an additive category one next asks for the existence of kernels (fibers) and cokernels (cofibers) to yield a pre-abelian category, and then asks that these are suitably compatible, to yield an abelian category.

Now here in stable homotopy theory, the concept of kernels and cokernels is replaced by that of homotopy fibers and homotopy cofibers. That these certainly exist for homotopy theories presented by model categories was the topic of the general discussion in the section Homotopy theory – Homotopy fibers. Various of the properties they satisfy was the topic of the sections Homotopy theory – Long sequences and Homotopy theory – Homotopy pullbacks.. For the special case of stable homotopy theory we will find a crucial further property relating homotopy fibers to homotopy cofibers.

The axiomatic formulation of a subset of these properties of stable homotopy fibers and stable homotopy cofibers is called a triangulated category structure. This is the analog in stable homotopy theory of abelian category structure in homological algebra.

category of abelian groupsstable homotopy category
direct sums and hom-abelian groupsadditive categoryadditive category
(homotopy) fibers and cofibers existpre-additive categoryhomotopy category of a model category
(homotopy) fibers and cofibers are compatibleabelian categorytriangulated category

Literature (Hubery, Schwede 12, II.2)

\,

Definition

A triangulated category is

  1. an additive category HoHo (def. );

  2. a functor, called the suspension functor or shift functor

    Σ:HoHo \Sigma \;\colon\; Ho \overset{\simeq}{\longrightarrow} Ho

    which is required to be an equivalence of categories;

  3. a sub-class CofSeqMor(Ho Δ[3])CofSeq \subset Mor(Ho^{\Delta[3]}) of the class of triples of composable morphisms, called the class of distinguished triangles, where each element that starts at AA ends at ΣA\Sigma A; we write these as

    ABB/AΣA, A \overset{}{\longrightarrow} B \overset{}{\longrightarrow} B/A \overset{}{\longrightarrow} \Sigma A \,,

    or

    A f B [1] B/A \array{ A && \overset{f}{\longrightarrow} && B \\ & {}_{\mathllap{[1]}}\nwarrow && \swarrow_{\mathrlap{}} \\ && B/A }

    (whence the name triangle);

such that the following conditions hold:

  • T0 For every morphism f:ABf \colon A \to B, there does exist a distinguished triangle of the form

    AfBB/AΣA. A \overset{f}{\longrightarrow} B \longrightarrow B/A \longrightarrow \Sigma A \,.

    If (f,g,h)(f,g,h) is a distinguished triangle and there is a commuting diagram in HoHo of the form

    A f B g B/A h ΣA Iso Iso Iso Iso A f B g B/A h ΣA \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{\in Iso}} && \downarrow^{\mathrlap{\in Iso}} && \downarrow^{\mathrlap{\in Iso}} && \downarrow^{\mathrlap{\in Iso}} \\ A' &\overset{f'}{\longrightarrow}& B' &\overset{g'}{\longrightarrow}& B'/A' &\overset{h'}{\longrightarrow}& \Sigma A' }

    (with all vertical morphisms being isomorphisms) then (f,g,h)(f',g',h') is also a distinguished triangle.

  • T1 For every object XHoX \in Ho then (0,id X,0)(0,id_X,0) is a distinguished triangle

    0Xid XX0; 0 \longrightarrow X \overset{id_X}{\longrightarrow} X \overset{}{\longrightarrow} 0 \,;
  • T2 If (f,g,h)(f,g,h) is a distinguished triangle, then so is (g,h,Σf)(g,h, - \Sigma f); hence if

    AfBgB/AhΣA A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A

    is, then so is

    BgB/AhΣAΣfΣB. B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A \overset{-\Sigma f}{\longrightarrow} \Sigma B \,.
  • T3 Given a commuting diagram in HoHo of the form

    A B B/A ΣA ϕ A ϕ B A B B/A ΣA \array{ A &\overset{}{\longrightarrow}& B &\overset{}{\longrightarrow}& B/A &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{\phi_A}} && \downarrow^{\mathrlap{\phi_B}} && && \\ A' &\overset{}{\longrightarrow}& B' &\overset{}{\longrightarrow}& B'/A' &\overset{}{\longrightarrow}& \Sigma A' }

    where the top and bottom are distinguished triangles, then there exists a morphism B/AB/AB/A \to B'/A' such as to make the completed diagram commute

    A B B/A ΣA ϕ A ϕ B Σϕ A A B B/A ΣA \array{ A &\overset{}{\longrightarrow}& B &\overset{}{\longrightarrow}& B/A &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{\phi_A}} && \downarrow^{\mathrlap{\phi_B}} && \downarrow^{\mathrlap{\exists}} && \downarrow^{\mathrlap{\Sigma \phi_A}} \\ A' &\overset{}{\longrightarrow}& B' &\overset{}{\longrightarrow}& B'/A' &\overset{}{\longrightarrow}& \Sigma A' }
  • T4 (octahedral axiom) For every pair of composable morphisms f:ABf \colon A \to B and f:BDf' \colon B \to D then there is a commutative diagram of the form

    A f B g B/A h ΣA = f x = A ff D g D/A h ΣA g y D/B D/B h (Σg)h ΣB Σg ΣB/A \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ {}^{\mathllap{=}}\downarrow && {}^{\mathllap{f'}}\downarrow && \downarrow^{\mathrlap{x}} && \downarrow^{\mathrlap{=}} \\ A &\underset{f' \circ f}{\longrightarrow}& D &\underset{g''}{\longrightarrow}& D/A &\underset{h''}{\longrightarrow}& \Sigma A \\ && {}^{\mathllap{g'}}\downarrow && \downarrow^{\mathrlap{y}} \\ && D/B &\underset{\simeq}{\longrightarrow}& D/B \\ && {}^{\mathllap{h'}}\downarrow && \downarrow^{\mathrlap{(\Sigma g)\circ h'}} \\ && \Sigma B &\underset{\Sigma g}{\longrightarrow}& \Sigma B/A }

    such that the two top horizontal sequences and the two middle vertical sequences each are distinguished triangles.

Proposition

The stable homotopy category Ho(Spectra)Ho(Spectra) from def. , equipped with the canonical suspension functor Σ:Ho(Spectra)Ho(Spectra)\Sigma \colon Ho(Spectra) \overset{\simeq}{\longrightarrow} Ho(Spectra) (according to this prop.) is a triangulated category (def. ) for the distinguished triangles being the closure under isomorphism of triangles of the images (under localization SeqSpec(Top cg) stableHo(Spectra)SeqSpec(Top_{cg})_{stable} \to Ho(Spectra) (prop.) of the stable model category of theorem ) of the canonical long homotopy cofiber sequences (prop.)

AfBhocofib(f)ΣA. A \overset{f}{\longrightarrow} B \overset{}{\longrightarrow} hocofib(f) \longrightarrow \Sigma A \,.

(e.g. Schwede 12, chapter II, theorem 2.9)

Proof

By prop. the stable homotopy category is additive, by theorem the functor Σ\Sigma is an equivalence.

The axioms T0 and T1 are immediate from the definition of homotopy cofiber sequences.

The axiom T2 is the very characterization of long homotopy cofiber sequences (from this prop.).

Regarding axiom T3:

By the factorization axioms of the model category we may represent the morphisms AAA \to A' and BBB \to B' in the homotopy category by cofibrations in the model category. Then BB/AB \to B/A and BB/AB' \to B'/A' are represented by their ordinary cofibers (def., prop.).

We may assume without restriction (lemma) that the commuting square

A f B ϕ A ϕ B A f B \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{\phi_A}}\downarrow && \downarrow^{\mathrlap{\phi_B}} \\ A' &\underset{f'}{\longrightarrow}& B' }

in the homotopy category is the image of a commuting square (not just commuting up to homotopy) in SeqSpec(Top cg)SeqSpec(Top_{cg}). In this case then the morphism B/AB/AB/A \to B'/A' is induced by the universal property of ordinary cofibers. To see that this also completes the last vertical morphism, observe that by the small object argument (prop.) we have functorial factorization (def.).

With this, again the universal property of the ordinary cofiber gives the fourth vertical morphism needed for T3.

Axiom T4 follows in the same fashion: we may represent all spectra by CW-spectra and represent ff and ff', hence also fff'\circ f, by cofibrations. Then forming the functorial mapping cones as above produces the commuting diagram

A f B g B/A h ΣA = (1) f (2) x = A ff D g D/A h ΣA g (3) y D/B D/B h (Σg)h ΣB Σg ΣB/A \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ {}^{\mathllap{=}}\downarrow &(1)& {}^{\mathllap{f'}}\downarrow &(2)& \downarrow^{\mathrlap{x}} && \downarrow^{\mathrlap{=}} \\ A &\underset{f' \circ f}{\longrightarrow}& D &\underset{g''}{\longrightarrow}& D/A &\underset{h''}{\longrightarrow}& \Sigma A \\ && {}^{\mathllap{g'}}\downarrow &(3)& \downarrow^{\mathrlap{y}} && \\ && D/B &\underset{\simeq}{\longrightarrow}& D/B \\ && {}^{\mathllap{h'}}\downarrow && \downarrow^{\mathrlap{(\Sigma g)\circ h'}} \\ && \Sigma B &\underset{\Sigma g}{\longrightarrow}& \Sigma B/A }

The fact that the second horizontal morphism from below is indeed an isomorphism follows by applying the pasting law for homotopy pushouts twice (prop.):

Draw all homotopy cofibers as homotopy pushout squares (def.) with one edge going to the point. Then assemble the squares (1)-(3) in the pasting composite of two cubes on top of each other: (1) as the left face of the top cube, (2) as the middle face where the two cubes touch, and (3) as the front face of the bottom cube. All remaining edges are points. This way the rear and front face of the top cube and the left and right face of the bottom cube are homotopy pushouts by construction. Also the top face

A * A * \array{ A &\longrightarrow & \ast \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ A &\longrightarrow& \ast }

is a homotopy pushout, since two opposite edges of it are weak equivalences (prop.). From this the pasting law for homotopy pushouts (prop.) gives that also the middle square (2) is a homotopy pushout. Applying the pasting law once more this way, now for the bottom cube, gives that the bottom square

* * D/B (D/A)/(B/A) \array{ \ast &\longrightarrow& \ast \\ \downarrow && \downarrow \\ D/B &\longrightarrow& (D/A)/(B/A) }

is a homotopy pushout. Since here the left edge is a weak equivalence, necessarily, so is the right edge (prop.), which hence exhibits the claimed identification

D/B(D/A)/(B/A). D/B \simeq (D/A)/(B/A) \,.
Remark

All we used in the proof (of prop. ) of the octahedral axiom (T4) is the existence and nature of homotopy pushouts. In fact one may show that the octahedral axiom is equivalent to the existence of homotopy pushouts, in the sense of axiom B in (Hubery).

Long fiber-cofiber sequences

In homotopy theory there are generally long homotopy fiber sequences to the left and long homotopy cofiber sequences to the right, as discussed in the section Homotopy theory – Long sequences. We prove now, in the generality of the axiomatics of triangulated categories (since the stable homotopy category is triangulated by prop. ), that in stable homotopy theory both these sequences are long in both directions, and in fact coincide.

Literature (Schwede 12, II.2)

\,

Lemma

For (Ho,Σ,CofSeq)(Ho,\Sigma, CofSeq) a triangulated category, def. , and

AfBgB/AhΣA A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A

a distinguished triangle, then

gf=0 g\circ f = 0

is the zero morphism.

Proof

Consider the commuting diagram

A id A 0 ΣA id f A f B g B/A h ΣA. \array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} \\ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A } \,.

Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. . Hence by T3 there is an extension to a commuting diagram of the form

A id A 0 ΣA id f Σf A f B g B/A h ΣA. \array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{\Sigma f}} \\ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A } \,.

Now the commutativity of the middle square proves the claim.

Proposition

Let (Ho,Σ,CofSeq)(Ho,\Sigma, CofSeq) be a triangulated category, def. , with hom-functor denoted by [,] *:Ho op×HoAb[-,-]_\ast \colon Ho^{op}\times Ho \to Ab. For XHoX\in Ho any object, and for DCofSeqD\in CofSeq any distinguished triangle

D=(AfBgB/AhΣA) D = (A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A)

then the sequences of abelian groups

  1. (long cofiber sequence)

    [ΣA,X] *[h,X] *[B/A,X] *[g,X] *[B,X] *[f,X] *[A,X] * [\Sigma A, X]_\ast \overset{[h,X]_\ast}{\longrightarrow} [B/A,X]_\ast \overset{[g,X]_\ast}{\longrightarrow} [B,X]_\ast \overset{[f,X]_\ast}{\longrightarrow} [A,X]_\ast
  2. (long fiber sequence)

    [X,A] *[X,f] *[X,B] *[X,g] *[X,B/A] *[X,h] *[X,ΣA] * [X,A]_\ast \overset{[X,f]_\ast}{\longrightarrow} [X,B]_\ast \overset{[X,g]_\ast}{\longrightarrow} [X,B/A]_\ast \overset{[X,h]_\ast}{\longrightarrow} [X,\Sigma A]_\ast

are long exact sequences.

Proof

Regarding the first case:

Since gf=0g \circ f = 0 by lemma , we have an inclusion im([g,X] *)ker([f,X] *)im([g,X]_\ast) \subset ker([f,X]_\ast). Hence it is sufficient to show that if ψ:BX\psi \colon B \to X is in the kernel of [f,X] *[f,X]_\ast in that ψf=0\psi \circ f = 0, then there is ϕ:B/AX\phi \colon B/A \to X with ϕg=ψ\phi \circ g = \psi. To that end, consider the commuting diagram

A f B g B/A h ΣA ψ 0 X id X 0, \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow && {}^{\mathllap{\psi}}\downarrow \\ 0 &\overset{}{\longrightarrow}& X &\overset{id}{\longrightarrow}& X &\overset{}{\longrightarrow}& 0 } \,,

where the commutativity of the left square exhibits our assumption.

The top part of this diagram is a distinguished triangle by assumption, and the bottom part is by condition T1T1 in def. . Hence by condition T3 there exists ϕ\phi fitting into a commuting diagram of the form

A f B g B/A h ΣA ψ ϕ 0 X id X 0. \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow && {}^{\mathllap{\psi}}\downarrow && \downarrow^{\mathrlap{\phi}} && \downarrow \\ 0 &\overset{}{\longrightarrow}& X &\overset{id}{\longrightarrow}& X &\overset{}{\longrightarrow}& 0 } \,.

Here the commutativity of the middle square exhibits the desired conclusion.

This shows that the first sequence in question is exact at [B,X] *[B,X]_\ast. Applying the same reasoning to the distinguished triangle (g,h,Σf)(g,h,-\Sigma f) provided by T2 yields exactness at [B/A,X] *[B/A,X]_\ast.

Regarding the second case:

Again, from lemma it is immediate that

im([X,f] *)ker([X,g] *) im([X,f]_\ast) \subset ker([X,g]_\ast)

so that we need to show that for ψ:XB\psi \colon X \to B in the kernel of [X,g] *[X,g]_\ast, hence such that gψ=0g\circ \psi = 0, then there exists ϕ:XA\phi \colon X \to A with fϕ=ψf \circ \phi = \psi.

To that end, consider the commuting diagram

X 0 ΣX Σid ΣX ψ B g B/A h ΣA Σf ΣB, \array{ X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- \Sigma id}{\longrightarrow}& \Sigma X \\ \downarrow^{\mathrlap{\psi}} && \downarrow \\ B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B } \,,

where the commutativity of the left square exhibits our assumption.

Now the top part of this diagram is a distinguished triangle by conditions T1 and T2 in def. , while the bottom part is a distinguished triangle by applying T2 to the given distinguished triangle. Hence by T3 there exists ϕ˜:ΣXΣA\tilde \phi \colon \Sigma X \to \Sigma A such as to extend to a commuting diagram of the form

X 0 ΣX Σid ΣX ψ ϕ˜ Σψ B g B/A h ΣA Σf ΣB, \array{ X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- \Sigma id}{\longrightarrow}& \Sigma X \\ \downarrow^{\mathrlap{\psi}} && \downarrow && \downarrow^{\mathrlap{\tilde \phi}} && \downarrow^{\mathrlap{\Sigma \psi}} \\ B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B } \,,

At this point we appeal to the condition in def. that Σ:HoHo\Sigma \colon Ho \to Ho is an equivalence of categories, so that in particular it is a fully faithful functor. It being a full functor implies that there exists ϕ:XA\phi \colon X \to A with ϕ˜=Σϕ\tilde \phi = \Sigma \phi. It being faithful then implies that the whole commuting square on the right is the image under Σ\Sigma of a commuting square

X id X ϕ ψ A f B. \array{ X &\overset{-id}{\longrightarrow}& X \\ {}^{\mathllap{\phi}}\downarrow && \downarrow^{\mathrlap{\psi}} \\ A &\underset{-f}{\longrightarrow}& B } \,.

This concludes the exactness of the second sequence at [X,B] *[X,B]_\ast. As before, exactness at [X,B/A] *[X,B/A]_\ast follows with the same argument applied to the shifted triangle, via T2.

Lemma

Consider a morphism of distinguished triangles in a triangulated category (def. ):

A B g B/A h ΣA a b c Σa A B B/A ΣA. \array{ A &\overset{}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{a}} && \downarrow^{\mathrlap{b}} && \downarrow^{\mathrlap{c}} && \downarrow^{\mathrlap{\Sigma a}} \\ A' &\overset{}{\longrightarrow}& B' &\overset{}{\longrightarrow}& B'/A' &\overset{}{\longrightarrow}& \Sigma A' } \,.

If two out of {a,b,c}\{a,b,c\} are isomorphisms, then so is the third.

Proof

Consider the image of the situation under the hom-functor [X,] *[X,-]_\ast out of any object XX:

[X,A] * [X,B] * g [X,B/A] * h [X,ΣA] * [X,ΣB] * a * b * c * (Σa) * (Σb) * [X,A] * [X,B] * [X,B/A] * [X,ΣA] * [X,ΣB] *, \array{ [X,A]_\ast &\overset{}{\longrightarrow}& [X,B]_\ast &\overset{g}{\longrightarrow}& [X,B/A]_\ast &\overset{h}{\longrightarrow}& [X,\Sigma A]_\ast &\longrightarrow& [X,\Sigma B]_\ast \\ \downarrow^{\mathrlap{a_\ast}} && \downarrow^{\mathrlap{b_\ast}} && \downarrow^{\mathrlap{c_\ast}} && \downarrow^{\mathrlap{(\Sigma a)_\ast }} && \downarrow^{\mathrlap{(\Sigma b)_\ast }} \\ [X,A']_\ast &\overset{}{\longrightarrow}& [X,B']_\ast &\overset{}{\longrightarrow}& [X,B'/A']_\ast &\overset{}{\longrightarrow}& [X,\Sigma A']_\ast &\longrightarrow& [X,\Sigma B']_\ast } \,,

where we extended one step to the right using axiom T2 (def. ).

By prop. here the top and bottom are exact sequences.

So assume the case that aa and bb are isomorphisms, hence that a *a_\ast, b *b_\ast, (Σa) *(\Sigma a)_\ast and (Σb) *(\Sigma b)_\ast are isomorphisms. Then by exactness of the horizontal sequences, the five lemma implies that c *c_\ast is an isomorphism. Since this holds naturally for all XX, the Yoneda lemma (fully faithfulness of the Yoneda embedding) then implies that cc is an isomorphism.

If instead bb and cc are isomorphisms, apply this same argument to the triple (b,c,Σa)(b,c,\Sigma a) to conclude that Σa\Sigma a is an isomorphism. Since Σ\Sigma is an equivalence of categories, this implies then that aa is an isomorphism.

Analogously for the third case.

Lemma

If (g,h,Σf)(g,h,-\Sigma f) is a distinguished triangle in a triangulated category (def. ), then so is (f,g,h)(f,g,h).

Proof

By T0 there is some distinguished triangle of the form (f,g,h)(f,g',h'). By T2 this gives a distinguished triangle (Σf,Σg,Σh)(-\Sigma f, -\Sigma g', -\Sigma h'). By T3 there is a morphism cc' giving a commuting diagram

ΣA Σf ΣB Σg ΣC Σh Σ 2A = = c = ΣA Σf ΣB Σg ΣC Σh Σ 2A. \array{ \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B &\overset{-\Sigma g}{\longrightarrow}& \Sigma C &\overset{-\Sigma h}{\longrightarrow}& \Sigma^2 A \\ {}^{\mathllap{=}}\downarrow && {}^{\mathllap{=}}\downarrow && {}^{\mathllap{c'}}\downarrow && {}^{\mathllap{=}}\downarrow \\ \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B &\overset{-\Sigma g'}{\longrightarrow}& \Sigma C &\overset{-\Sigma h'}{\longrightarrow}& \Sigma^2 A } \,.

Now lemma gives that cc' is an isomorphism. Since Σ\Sigma is an equivalence of categories, there is an isomorphism cc such that c=Σcc' = \Sigma c. Since Σ\Sigma is in particular a faithful functor, this cc exhibits an isomorphism between (f,g,h)(f,g,h) and (f,g,h)(f,g',h'). Since the latter is distinguished, so is the former, by T0.

In conclusion:

Proposition

Let

XfYgZ X \overset{f}{\longrightarrow} Y \overset{g}{\longrightarrow} Z

be a homotopy cofiber sequence (def.) of spectra in the stable homotopy category (def. ) Ho(Spectra)Ho(Spectra). Let AHo(Spectra)A \in Ho(Spectra) be any other spectrum. Then the abelian hom-groups of the stable homotopy category (def. , lemma ) sit in long exact sequences of the form

[A,ΩY](Ωg) *[A,ΩZ][A,X]f *[A,Y]g *[A,Z][A,ΣX](Σf) *[A,ΣY]. \cdots \longrightarrow [A, \Omega Y] \overset{-(\Omega g)_\ast}{\longrightarrow} [A, \Omega Z] \overset{}{\longrightarrow} [A,X] \overset{f_\ast}{\longrightarrow} [A,Y] \overset{g_\ast}{\longrightarrow} [A,Z] \overset{}{\longrightarrow} [A,\Sigma X] \overset{-(\Sigma f)_\ast}{\longrightarrow} [A, \Sigma Y] \longrightarrow \cdots \,.
Proof

By prop. the above abstract reasoning in triangulated categories applies. By prop. we have long exact sequences to the right as shown. By lemma these also extend to the left as shown.

This suggests that homotopy cofiber sequences coincide with homotopy fiber sequence in the stable homotopy category. This is indeed the case:

Proposition

In the stable homotopy category, a sequence of morphisms is a homotopy cofiber sequence precisely if it is a