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).
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.
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 $S^1 \coloneqq \Delta[1]/\partial\Delta[1]$ 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
This gives a category $SeqSpec(sSet)$ of sequential prespectra.
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
and whose structure maps are given by
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
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
of the structure maps ${\vert \sigma_n\vert}$ are weak homotopy equivalences.
Write $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/}$ for the standard minimal pointed simplicial 1-sphere.
Write
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
(Lydakis 98, def. 4.2), see also (MMSS 00) this example.
There is an $sSet^{\ast/}$-enriched functor
(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
and with structure maps
given by
This is an $sSet^{\ast/}$ enriched equivalence of categories.
(Lydakis 98, prop. 4.3), see also (MMSS 00)
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.
The category $SeqSpec(Top_{cq})$ of sequential spectra (def. ) has all limits and colimits, and they are computed objectwise:
Given
a diagram of sequential spectra, then:
its colimiting spectrum has component spaces the colimit of the component spaces formed in $Top_{cg}$ (via this prop. and this corollary):
its limiting spectrum has component spaces the limit of the component spaces formed in $Top_{cg}$ (via this prop. and this corollary):
moreover:
the colimiting spectrum has structure maps in the sense of def. given by
where the first isomorphism exhibits that $S^1 \wedge(-)$ preserves all colimits, since it is a left adjoint by prop. ;
the limiting spectrum has adjunct structure maps in the sense of def. given by
where the last isomorphism exhibits that $Maps(S^1,-)_\ast$ preserves all limits, since it is a right adjoint by prop. .
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.
The coproduct of spectra $X, Y \in SeqSpec(Top_{cg})$, called the wedge sum of spectra
is componentwise the wedge sum of pointed topological spaces (exmpl.)
with structure maps
The following defines tensoring and powering of sequential spectra over pointed topological spaces/pointed simplicial sets.
Let $X$ be a sequential spectrum and $K$ a pointed topological space/pointed simplicial set. Then
$X \wedge K$ is the sequential spectrum with
$(X \wedge K)_n \coloneqq X_n \wedge K$ (smash product)
$\sigma_n^{X\wedge K} \coloneqq \sigma_n^{X} \wedge id_{K}$.
$X^K$ is the sequential spectrum with
$(X^K)_n \coloneqq (X_n)^K$ (pointed mapping space)
$\sigma_n^{(X^k)} \colon S^1 \wedge X_n^K \to (S^1 \wedge X_n)^K \overset{(\sigma_n)^K}{\longrightarrow} (X_{n+1})^K$.
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. :
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.
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
$(X[k])_n \coloneqq \left\{ \array{X_{n+k} & for \; n+k \geq 0 \\ \ast & otherwise } \right.$;
$\sigma_n^{X[k]} \coloneqq \left\{ \array{ \sigma^X_{n+k} & for \; n+k \geq 0 \\ 0 & otherwise} \right.$.
For $X$ a sequential spectrum, then
For $X$ a sequential spectrum, then
the fake suspension of $X$ is the sequential spectrum $\Sigma X$ with
$(\Sigma X)_n \coloneqq S^1 \wedge X_n$
$\sigma_n^{\Sigma X} \coloneqq S^1 \wedge (\sigma_n)$.
the fake looping of $X$ is the sequential spectrum $\Omega X$ with
$(\Omega X)_n \coloneqq (X_n)^{S^1}$;
$\tilde \sigma_n^{\Omega X} \coloneqq (\sigma_n)^{S^1}$.
Here $\tilde \Sigma_n$ denotes the $(\Sigma\dashv \Omega)$-adjunct of $\sigma_n$.
e.g. (Jardine 15, section 10.4).
The looping and suspension operations in def. and def. commute with shifting, def. . Therefore in expressions like $\Sigma (X[1])$ etc. we may omit the parenthesis.
The canonical morphism
$\Sigma X \longrightarrow X[1]$ is given in degree $n$ by $\sigma_n^X$.
$X[-1] \longrightarrow \Omega X$ is given in degree $n$ by $\tilde \sigma^X_{n-1}$.
The constructions from def. , def. and def. form pairs of adjoint functors $SeqSpec \to SeqSpec$ like so:
$(-)[1] \;\dashv\; (-)[-1] \;\dashv\; (-)[1] \;\dashv\; \cdots$;
$(-)\wedge S^1 \dashv (-)^{S^1}$;
$\Sigma \dashv \Omega$.
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:
Applying the adjunction isomorphism diagonally gives a bijection to diagrams of this form:
Then applying the same isomorphism diagonally once more gives a further bijection to commuting diagrams of this form:
This finally equivalently exhibits morphisms of the form
For $X$ a sequential spectrum, then $X[-1][1] = X$ while $X[1][-1]$ is $X$ with its 0-th component space set to the point. The adjunction unit $X \to X[1][-1]$ has components
For $X$ a sequential spectrum, then (using remark to suppress parenthesis)
the structure maps constitute a homomorphism
and this is a stable equivalence.
the adjunct structure maps constitute a homomorphism
If $X$ is an Omega-spectrum (def. ) then this is a weak equivalence in the strict model structure, hence in particular a stable equivalence.
The diagrams that need to commute for the structure maps to give a homomorphism as claimed are in degree 0 this one
and in degree $n \geq 1$ these:
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
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.
For $X$ a sequential spectrum in simplicial sets. Then there are stable equivalences
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
between the real looping (def. ), the fake looping (def. ) and the shift by -1 (def. ) of $X$.
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.
Write
sSet for the category of simplicial sets;
$sSet^{\ast/}$ for the category of pointed simplicial sets;
$sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the full subcategory of pointed simplicial finite sets.
Write
for the free-forgetful adjunction, where the left adjoint functor $(-)_+$ freely adjoins a base point.
Write
for the smash product of pointed simplicial sets, similarly for its restriction to $sSet_{fin}^{\ast}$:
This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a closed monoidal category and we write
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
(Lydakis 98, example 3.8, def. 4.4)
The adjunction
(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).
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 $StdSpheres$.
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:
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
and in odd degree by
with structure maps being in even degree the identity
and in odd degree as the composite
(Lydakis 98, def. 10.20, Lydakis 98b, def. 5.9, MMSS 00, def. 11.6)
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)
Daniel Kan, Semisimplicial spectra, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. (euclid.ijm/1255644953)
Michael Boardman, Stable homotopy theory, mimeographed notes, University of Warwick, 1965 onward
Frank Adams, Part III, section 2 Stable homotopy and generalised homology, 1974
Robert Switzer, chapter 8 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Aldridge Bousfield, Eric Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)
Manos Lydakis, Simplicial functors and stable homotopy theory Preprint, 1998 (Hopf archive pdf, pdf)
(mostly on a model structure on excisive functors on simplicial sets)
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, section 11 of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)
Stefan Schwede, Symmetric spectra, 2012 (pdf)
(mostly on symmetric spectra)
John F. Jardine, section 10 of: Local homotopy theory, 2016
(with regards to sheaves of spectra)
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)
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