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sequential spectrum

Contents

Contents

Idea

In stable homotopy theory, a sequential (pre-)spectrum EE (also Boardman spectrum, after (Boardman 65)) is a sequence of pointed homotopy types (pointed topological spaces, pointed simplicial sets) E nE_n, for nn \in \mathbb{N}, together with maps ΣE nE n+1\Sigma E_n \to E_{n+1} from the reduced suspension of one into the next space in the sequence.

This is the original definition of spectrum (or pre-spectrum) and still the one predominently meant be default, as used in, say, the Brown representability theorem. But in view of many other definitions (all giving rise to equivalent stable homotopy theory) that involve systems of spaces indexed on more than just the integers (such as coordinate-free spectra, excisive functors, equivariant spectra) or that are of different flavor altogether (such as combinatorial spectra), one says sequential spectrum for emphasis (e.g. Schwede 12, def. 2.1).

Definition

In components

In what follows, sSet denotes the category of simplicial sets and sSet */sSet^{\ast/} the category */sSet\ast/sSet of pointed simplicial sets (the undercategory under the terminal object *\ast), which may be thought of as a possible base of enrichment.

Definition

A sequential pre-spectrum in simplicial sets, is an \mathbb{N}-graded pointed simplicial set X X_\bullet equipped with morphisms 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}, where S 1Δ[1]/Δ[1]S^1 \coloneqq \Delta[1]/\partial\Delta[1] is the minimal simplicial circle, and where \wedge is the smash product of pointed objects.

A homomorphism f:XYf \colon X \to Y of sequential prespectra is a collection f :X Y f_\bullet \colon X_\bullet \to Y_\bullet of homomorphisms of pointed simplicial sets, such 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 \array{ S^1 \wedge X_n &\stackrel{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma_n^X}} && \downarrow^{\mathrlap{\sigma_n^Y}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} }

commute.

This gives a category SeqSpec(sSet)SeqSpec(sSet) of sequential prespectra.

Example

For XSeqSpec(sSet)X \in SeqSpec(sSet) and KK \in sSet, hence K +sSet */K_+ \in sSet^{\ast/} then XK +X \wedge K_+ is the spectrum which is degreewise given by the smash product of pointed objects

(XK +) n(X nK +) (X \wedge K_+)_n \coloneqq (X_n \wedge K_+)

and whose structure maps are given by

S 1(X nK +)(S 1X n)K +σ nK +X n+1K +. S^1 \wedge (X_n \wedge K_+) \simeq (S^1 \wedge X_n) \wedge K_+ \stackrel{\sigma_n \wedge K_+}{\longrightarrow} X_{n+1}\wedge K_+ \,.
Proposition

The category SeqSpecSeqSpec of def. becomes a simplicially enriched category (in fact an sSet */sSet^{\ast/}-enriched category) with hom objects [X,Y]sSet[X,Y]\in sSet given by

[X,Y] nHom SeqSpec(sSet)(XΔ[n] +,Y). [X,Y]_n \coloneqq Hom_{SeqSpec(sSet)}(X\wedge \Delta[n]_+,Y) \,.
Definition

An Omega-spectrum in the following is a sequential prespectrum XX, def. , such that after geometric realization/Kan fibrant replacement ||{\vert -\vert} the smash\dashvpointed-hom adjuncts

|X n||X n+1| |S 1| {\vert X_n\vert} \stackrel{}{\longrightarrow} {\vert X^{n+1}\vert}^{{\vert S^1\vert}}

of the structure maps |σ n|{\vert \sigma_n\vert} are weak homotopy equivalences.

As diagram spectra

Definition

Write S std 1Δ[1]/Δ[1]sSet */S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/} for the standard minimal pointed simplicial 1-sphere.

Write

ι:StdSpheressSet fin */ \iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}

for the non-full sSet */sSet^{\ast/}-enriched subcategory of pointed simplicial finite sets, def. whose

  • objects are the smash product powers S std n(S std 1) nS^n_{std} \coloneqq (S^1_{std})^{\wedge^n} (the standard minimal simplicial n-spheres);

  • hom-objects are

    [S std n,S std n+k] StdSpheres{* for k<0 im(S std k[S std n,S std n+k] sSet fin */) otherwise [S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres} \coloneqq \left\{ \array{ \ast & for & k \lt 0 \\ im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}}) & otherwise } \right.

(Lydakis 98, def. 4.2), see also (MMSS 00) this example.

Proposition

There is an sSet */sSet^{\ast/}-enriched functor

() seq:[StdSpheres,sSet */]SeqSpec(sSet) (-)^seq \;\colon\; [StdSpheres,sSet^{\ast/}] \longrightarrow SeqSpec(sSet)

(from the category of sSet */sSet^{\ast/}-enriched copresheaves on the categories of standard simplicial spheres of def. to the category of sequential prespectra in sSet) given on objects by sending X[StdSpheres,sSet */]X \in [StdSpheres,sSet^{\ast/}] to the sequential prespectrum X seqX^{seq} with components

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

and with structure maps

S std 1X n seqσ nX n seqS std 1[X n seq,X n+1 seq] \frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}

given by

S std 1id˜[S std n,S std n+1]X S std n,S std n+1[X n seq,X n+1 seq]. S^1_{std} \stackrel{\widetilde{id}}{\longrightarrow} [S^n_{std}, S^{n+1}_{std}] \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow} [X^{seq}_n, X^{seq}_{n+1}] \,.

This is an sSet */sSet^{\ast/} enriched equivalence of categories.

(Lydakis 98, prop. 4.3), see also (MMSS 00)

Remark

Prop. is a special case of a more general statement expressing structured spectra equivalently as enriched functors. Analogous statements hold for symmetric spectra and orthogonal spectra. See at Model categories of diagram spectra this lemma and this example.

Properties

Limits and colimits

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 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} \,.

Tensoring and powering over pointed spaces

The following defines tensoring and powering of sequential spectra over pointed topological spaces/pointed simplicial sets.

Definition

Let XX be a sequential spectrum and KK a pointed topological space/pointed simplicial set. Then

  1. XKX \wedge K is the sequential spectrum with

    • (XK) nX nK(X \wedge K)_n \coloneqq X_n \wedge K (smash product)

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

  2. X KX^K is the sequential spectrum with

    • (X K) n(X n) K(X^K)_n \coloneqq (X_n)^K (pointed mapping space)

    • σ n (X k):S 1X n K(S 1X n) K(σ n) K(X n+1) K\sigma_n^{(X^k)} \colon S^1 \wedge X_n^K \to (S^1 \wedge X_n)^K \overset{(\sigma_n)^K}{\longrightarrow} (X_{n+1})^K.

Model category structures

There is a standard model structure on spectra for sequential spectra in Top the model structure on topological sequential spectra (Kan 63, MMSS 00) and in simplicial sets, the Bousfield-Friedlander model structure (Bousfield-Friedlander 78).

The strict Bousfield-Friedlander model structure (of which the actual stable version is the left Bousfield localization at the stable weak homotopy equivalences) is equivalently the projective model structure on enriched functors for the presentation of sequential spectra from prop. :

SeqSpec(sSet) stableBousf.locSeqSpec(sSet) strict=[StdSpheres,Top Quillen */] proj. SeqSpec(sSet)_{stable} \stackrel{\longleftarrow}{\overset{Bousf.\;loc}{\longrightarrow}} SeqSpec(sSet)_{strict} = [StdSpheres, Top^{\ast/}_{Quillen}]_{proj} \,.

Suspension and looping

There are three common constructions of looping and suspension of sequential spectra (with analogues for highly structured spectra). While they are not isomorphic, they are stably equivalent.

Definition

For XX a sequential spectrum 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, then

  1. the real suspension of XX is XS 1X \wedge S^1 according to def. ;

  2. the real looping of XX is X S 1X^{S^1} according to def. .

Definition

For XX a sequential spectrum, then

  1. the fake 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

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

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

    1. (ΩX) n(X n) S 1(\Omega X)_n \coloneqq (X_n)^{S^1};

    2. σ˜ n ΩX(σ n) S 1\tilde \sigma_n^{\Omega X} \coloneqq (\sigma_n)^{S^1}.

Here Σ˜ n\tilde \Sigma_n denotes the (ΣΩ)(\Sigma\dashv \Omega)-adjunct of σ n\sigma_n.

e.g. (Jardine 15, section 10.4).

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.

Definition

The canonical morphism

  1. ΣXX[1]\Sigma X \longrightarrow X[1] is given in degree nn by σ n X\sigma_n^X.

  2. X[1]ΩXX[-1] \longrightarrow \Omega X is given in degree nn by σ˜ n1 X\tilde \sigma^X_{n-1}.

Proposition

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

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

  2. ()S 1() S 1(-)\wedge S^1 \dashv (-)^{S^1};

  3. ΣΩ\Sigma \dashv \Omega.

Proof

The first is immediate from the definition.

The second is just degreewise the adjunction smash product\dashvpointed mapping space (discussed here), since by definition the smash product and mapping spaces here do not interact non-trivially with the structure maps.

The third follows by applying the smash product\dashvpointed mapping space-adjunction isomorphism twice, like so:

Morphisms f:ΣXYf\colon \Sigma X \to Y 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 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 (Y n+1) S 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{\tilde f_{n+1}}{\longrightarrow}& (Y_{n+1})^{S^1} } \,.

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

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

This finally equivalently exhibits morphisms of the form

XΩY. X \longrightarrow \Omega Y \,.
Example

For XX a sequential spectrum, then X[1][1]=XX[-1][1] = X while X[1][1]X[1][-1] is XX with its 0-th component space set to the point. The adjunction unit XX[1][1]X \to X[1][-1] has components

X 2 id X 2 X 1 id X 1 X 0 0 * X η X[1][1]. \array{ \vdots && \vdots \\ X_2 &\overset{id}{\longrightarrow}& X_2 \\ X_1 &\overset{id}{\longrightarrow}& X_1 \\ \underbrace{X_0} &\overset{0}{\longrightarrow}& \underbrace{\;\ast\;} \\ X &\overset{\eta}{\longrightarrow}& X[1][-1] } \,.
Proposition

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

    and this is a stable equivalence.

  2. the adjunct structure maps constitute a homomorphism

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

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

Now as in the proof of prop. , under applying the (S 1())() S 1(S^1\wedge (-)) \dashv (-)^{S^1}-adjunction isomorphism twice, these diagrams are in bijection to diagrams for n1n \geq 1 of the form

X n1 σ˜ n1 (X n) S 1 σ˜ n1 σ˜ n (X n) S 1 (σ˜ n) S 1 ((X n) S 1) S 1. \array{ X_{n-1} &\overset{\tilde \sigma_{n-1}}{\longrightarrow}& (X_n)^{S^1} \\ {}^{\mathllap{\tilde \sigma_{n-1}}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma_n}} \\ (X_n)^{S^1} &\underset{(\tilde \sigma_n)^{S^1}}{\longrightarrow}& \left((X_n)^{S^1}\right)^{S^1} } \,.

This gives the claimed morphism XΩX[1]X \to \Omega X[-1].

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 the first morphism is a stable equivalence, because for every Omega-spectrum YY then by the adjunctions in prop.

[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} } \,.
Proposition

For XX a sequential spectrum in simplicial sets. Then there are stable equivalences

XS 1ΣXX[1] X\wedge S^1 \longrightarrow \Sigma X \longrightarrow X[1]

between the real suspension (def. ), the fake suspension (def. ) and the shift by +1 (def. ) of XX.

If each X nX_n is a Kan complex, then there are stable equivalences

X S 1ΩXX[1] X^{S^1} \longrightarrow \Omega X \longrightarrow X[-1]

between the real looping (def. ), the fake looping (def. ) and the shift by -1 (def. ) of XX.

(Jardine 15, corollary 10.54)

Relation to excisive functors

We discuss aspects of the equivalence of sequential spectra carrying the Bousfield-Friedlander model structure with excisive (infinity,1)-functors, modeled as simplicial functors carrying a model structure for excisive functors.

Definition

Write

Write

sSet */u() +sSet sSet^{\ast/} \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}} sSet

for the free-forgetful adjunction, where the left adjoint functor () +(-)_+ freely adjoins a base point.

Write

:sSet */×sSet */sSet */ \wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}

for the smash product of pointed simplicial sets, similarly for its restriction to sSet fin *sSet_{fin}^{\ast}:

XYcofib(((u(X),*)(*,u(Y)))u(X)×u(Y)). X \wedge Y \coloneqq cofib\left( \; \left(\, (u(X),\ast) \sqcup (\ast, u(Y)) \,\right) \longrightarrow u(X) \times u(Y) \; \right) \,.

This gives sSet */sSet^{\ast/} and sSet fin */sSet^{\ast/}_{fin} the structure of a closed monoidal category and we write

[,] *:(sSet */) op×sSet */sSet */ [-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}

for the corresponding internal hom, the pointed function complex functor.

We regard all the categories in def. canonically as simplicially enriched categories, and in fact regard sSet */sSet^{\ast/} and sSet fin */sSet^{\ast/}_{fin} as sSet */sSet^{\ast/}-enriched categories.

The category that supports a model structure for excisive functors is the sSet */sSet^{\ast/}-enriched functor category

[sSet fin */,sSet */]. [sSet^{\ast/}_{fin}, sSet^{\ast/}] \,.

(Lydakis 98, example 3.8, def. 4.4)

Proposition

The adjunction

(ι *ι *):[sSet fin */,sSet */] Lyι *ι *[StdSpheres,sSet */]() seqSeqSpec(sSet) BF (\iota_\ast \dashv \iota^\ast) \;\colon\; [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly} \stackrel{\overset{\iota_\ast}{\longleftarrow}}{\underset{\iota^\ast}{\longrightarrow}} [StdSpheres, sSet^{\ast/}] \underoverset{\simeq}{(-)^{seq}}{\longrightarrow} SeqSpec(sSet)_{BF}

(given by restriction ι *\iota^\ast along the defining inclusion ι\iota of def. and by left Kan extension ι *\iota_\ast along ι\iota, and combined with the equivalence () seq(-)^{seq} of prop. ) is a Quillen adjunction and in fact a Quillen equivalence between the Bousfield-Friedlander model structure on sequential prespectra and Lydakis’ model structure for excisive functors.

(Lydakis 98, theorem 11.3) For more details see at model structure for excisive functors. The analogous statement for spectra in TopTop is in (MMSS 00).

Remark

Prop. shows why plain sequential spectra do not carry a symmetric smash product of spectra:

By this remark at smash product of spectra the graded-commutativity implicit in the braiding of the smash product of n-spheres is not reflected after restricting from (sSet */,)(sSet^{\ast/}, \wedge) to the non-full subcategory StdSpheresStdSpheres.

Smash product

The smash product of spectra realized on sequential spectra never has good proprties before passage to the stable homotopy category or lift to better models (see here), but it may still be defined in various ways:

Definition

For X,YX,Y two sequential spectra, def. , their smash product XYX \wedge Y is the sequential spectrum which in even degrees is given by the smash product fo the pointed component spaces of half that degree

(XY) 2nX nY n (X\wedge Y)_{2n} \coloneqq X_n \wedge Y_n

and in odd degree by

(XY) 2n+1S 1X nY n (X\wedge Y)_{2n+1} \coloneqq S^1 \wedge X_n \wedge Y_n

with structure maps being in even degree the identity

σ 2n XY:S 1(XY) 2n=S 1X nY n=(XY) 2n+1 \sigma^{X \wedge Y}_{2 n} \colon S^1 \wedge (X \wedge Y)_{2n} = S^1 \wedge X_n \wedge Y_n = (X \wedge Y)_{2n+1}

and in odd degree as the composite

σ 2n+1 XY:S 1(XY) 2n+1S 1S 1X nY nS 1X nS 1Y nσ n Xσ n YX n+1Y n+1(XY) 2n+2. \sigma^{X\wedge Y}_{2n+1} \colon S^1 \wedge (X \wedge Y)_{2n+1} \simeq S^1 \wedge S^1 \wedge X_n \wedge Y_n \simeq S^1 \wedge X_n \wedge S^1 \wedge Y_n \stackrel{\sigma_n^X \wedge \sigma^Y_n}{\longrightarrow} X_{n+1} \wedge Y_{n+1} \simeq (X\wedge Y)_{2n+2} \,.

(Lydakis 98, def. 10.20, Lydakis 98b, def. 5.9, MMSS 00, def. 11.6)

Proposition

Under the Quillen equivalence of prop. the symmetric monoidal Day convolution product on pre-excisive functors as well as the symmetric monoidal smash product of orthogonal spectra is identified with the smash product of spectra realized on sequential spectra via def. .

(Lydakis 98, theorem 12.5, MMSS 00, prop. 11.9)

References

Symmetric spectra in more general model categories (using the Bousfield-Friedlander theorem) are discussed in

  • Stefan Schwede, section 3 of Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 104 (pdf)

  • Mark Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra Volume 165, Issue 1, 23 November 2001, Pages 63–127 (arXiv:math/0004051)

Last revised on September 6, 2017 at 03:38:29. See the history of this page for a list of all contributions to it.