# nLab sequential spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

In stable homotopy theory, a sequential (pre-)spectrum $E$ (also Boardman spectrum, after (Boardman 65)) is a sequence of pointed homotopy types (pointed topological spaces, pointed simplicial sets) $E_n$, for $n \in \mathbb{N}$, together with maps $\Sigma E_n \to E_{n+1}$ 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 $sSet^{\ast/}$ the category $\ast/sSet$ 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 $X_\bullet$ equipped with morphisms $\sigma_n \colon S^1 \wedge X_n \to X_{n+1}$ for all $n \in \mathbb{N}$, where $S^1 \coloneqq \Delta/\partial\Delta$ is the minimal simplicial circle, and where $\wedge$ is the smash product of pointed objects.

A homomorphism $f \colon X \to Y$ of sequential prespectra is a collection $f_\bullet \colon X_\bullet \to Y_\bullet$ of homomorphisms of pointed simplicial sets, such that all diagrams of the form

$\array{ S^1 \wedge X_n &\stackrel{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma_n^X}} && \downarrow^{\mathrlap{\sigma_n^Y}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} }$

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

###### Example

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

$(X \wedge K_+)_n \coloneqq (X_n \wedge K_+)$

and whose structure maps are given by

$S^1 \wedge (X_n \wedge K_+) \simeq (S^1 \wedge X_n) \wedge K_+ \stackrel{\sigma_n \wedge K_+}{\longrightarrow} X_{n+1}\wedge K_+ \,.$
###### Proposition

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

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

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

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

of the structure maps ${\vert \sigma_n\vert}$ are weak homotopy equivalences.

### As diagram spectra

###### Definition

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

Write

$\iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}$

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

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

• hom-objects are

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

###### Proposition

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

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

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

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

and with structure maps

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

given by

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

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

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

Given

$X_\bullet \;\colon\; I \longrightarrow SeqSpec(Top_{cg})$

a diagram of sequential spectra, then:

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

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

$(\underset{\longleftarrow}{\lim}_i X(i))_n \simeq \underset{\longleftarrow}{\lim}_i X(i)_n \,;$

moreover:

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

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

where the first isomorphism exhibits that $S^1 \wedge(-)$ 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

$\underset{\longleftarrow}{\lim}_i X(i)_n \overset{\underset{\longleftarrow}{\lim}_i \tilde \sigma_n^{X(i)}}{\longrightarrow} \underset{\longleftarrow}{\lim}_i Maps(S^1, X(i)_n)_\ast \simeq Maps(S^1, \underset{\longleftarrow}{\lim}_i X(i)_n)_\ast$

where the last isomorphism exhibits that $Maps(S^1,-)_\ast$ 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 $X, Y \in SeqSpec(Top_{cg})$, called the wedge sum of spectra

$X \vee Y \coloneqq X \sqcup Y$

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

$(X \vee Y)_n = X_n \vee Y_n$

with structure maps

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

### Tensoring and powering over pointed spaces

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

###### Definition

Let $X$ be a sequential spectrum and $K$ a pointed topological space/pointed simplicial set. Then

1. $X \wedge K$ is the sequential spectrum with

• $(X \wedge K)_n \coloneqq X_n \wedge K$ (smash product)

• $\sigma_n^{X\wedge K} \coloneqq \sigma_n^{X} \wedge id_{K}$.

2. $X^K$ is the sequential spectrum with

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

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

### 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. :

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

### 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 $X$ a sequential spectrum and $k \in \mathbb{Z}$, the $k$-fold shifted spectrum of $X$ is the sequential spectrum denoted $X[k]$ given by

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

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

###### Definition

For $X$ a sequential spectrum, then

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

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

###### Definition

For $X$ a sequential spectrum, then

1. the fake suspension of $X$ is the sequential spectrum $\Sigma X$ with

1. $(\Sigma X)_n \coloneqq S^1 \wedge X_n$

2. $\sigma_n^{\Sigma X} \coloneqq S^1 \wedge (\sigma_n)$.

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

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

2. $\tilde \sigma_n^{\Omega X} \coloneqq (\sigma_n)^{S^1}$.

Here $\tilde \Sigma_n$ denotes the $(\Sigma\dashv \Omega)$-adjunct of $\sigma_n$.

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 $\Sigma (X)$ etc. we may omit the parenthesis.

###### Definition

The canonical morphism

1. $\Sigma X \longrightarrow X$ is given in degree $n$ by $\sigma_n^X$.

2. $X[-1] \longrightarrow \Omega X$ is given in degree $n$ by $\tilde \sigma^X_{n-1}$.

###### Proposition

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

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

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

3. $\Sigma \dashv \Omega$.

###### Proof

The first is immediate from the definition.

The second is just degreewise the adjunction smash product$\dashv$pointed 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$\dashv$pointed mapping space-adjunction isomorphism twice, like so:

Morphisms $f\colon \Sigma X \to Y$ are in components given by commuting diagrams of this form:

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

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

$\array{ S^1 \wedge X_n &\overset{f_n}{\longrightarrow}& Y_n \\ {}^{\mathllap{\sigma^X_n}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma^Y_n}} \\ X_{n+1} &\underset{\tilde f_{n+1}}{\longrightarrow}& (Y_{n+1})^{S^1} } \,.$

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

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

This finally equivalently exhibits morphisms of the form

$X \longrightarrow \Omega Y \,.$
###### Example

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

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

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

1. the structure maps constitute a homomorphism

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

and this is a stable equivalence.

2. the adjunct structure maps constitute a homomorphism

$X \longrightarrow \Omega X \,.$

If $X$ 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

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

and in degree $n \geq 1$ these:

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

But in all these cases commutativity it trivially satisfied.

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

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

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

If $X$ 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 $Y$ then by the adjunctions in prop.

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

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

$X\wedge S^1 \longrightarrow \Sigma X \longrightarrow X$

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

If each $X_n$ is a Kan complex, then there are stable equivalences

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

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

### 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

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

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

Write

$\wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}$

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

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

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

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

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 $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ as $sSet^{\ast/}$-enriched categories.

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

$[sSet^{\ast/}_{fin}, sSet^{\ast/}] \,.$
###### Proposition

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

(given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. and by left Kan extension $\iota_\ast$ along $\iota$, and combined with the equivalence $(-)^{seq}$ 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 $Top$ 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 $(sSet^{\ast/}, \wedge)$ to the non-full subcategory $StdSpheres$.

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

$(X\wedge Y)_{2n} \coloneqq X_n \wedge Y_n$

and in odd degree by

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

with structure maps being in even degree the identity

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

and in odd degree as the composite

$\sigma^{X\wedge Y}_{2n+1} \colon S^1 \wedge (X \wedge Y)_{2n+1} \simeq S^1 \wedge S^1 \wedge X_n \wedge Y_n \simeq S^1 \wedge X_n \wedge S^1 \wedge Y_n \stackrel{\sigma_n^X \wedge \sigma^Y_n}{\longrightarrow} X_{n+1} \wedge Y_{n+1} \simeq (X\wedge Y)_{2n+2} \,.$
###### 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. .

(mostly on a model structure on excisive functors on simplicial sets)

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)