model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
The Bousfield-Friedlander model structure (Bousfield-Friedlander 78, section 2) is a model structure for spectra, specifically it is a standard model structure on sequential spectra in simplicial sets. An immediate variant works for sequential spectra in topological spaces, see at model structure on topological sequential spectra.
As such, the Bousfield-Friedlander model structure presents the stable (infinity,1)-category of spectra of stable homotopy theory, hence, in particular, its homotopy category is the classical stable homotopy category.
Write $S^1 \coloneqq \Delta[1]/\partial\Delta[1]$ for the minimal simplicial circle. Write
for the smash product of pointed simplicial sets.
A sequential prespectrum in simplicial sets, or just sequential spectrum for short (or even just spectrum), 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 $n \in \mathbb{N}$.
A homomorphism $f \colon X \to Y$ of spectra is a sequence $f_\bullet \colon X_\bullet \to Y_\bullet$ of homomorphisms of pointed simplicial sets, such that all diagrams of the form
Write $SeqSpec(sSet)$ for this category of sequential spectra.
For $X \in SeqSpec(sSet)$ and $K \in$ sSet, hence $K_+ \in sSet^{\ast/}$ then $X \wedge K_+$ is the sequential spectrum degreewise given by the smash product of pointed objects
and with structure maps 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
The stable homotopy groups of a sequential spectrum $X$, def. , is the $\mathbb{Z}$-graded abelian groups given by the colimit of homotopy groups of geometric realizations of the component spaces
This constitutes a functor
A morphism $f \colon X \longrightarrow Y$ 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
A Omega-spectrum is a sequential spectrum $X$, def. , such that after geometric realization/Kan fibrant replacement the (smash product $\dahsv$ pointed mapping space)-adjuncts
of the structure maps ${\vert \sigma_n\vert}$ are weak homotopy equivalences.
If a sequential spectrum $X$ is an Omega-spectrum, def. , then its colimiting stable homotopy groups, def. , are attained as the actual homotopy groups of its components:
The canonical $\Omega$-spectrification $Q X$ of a sequential spectrum $X$ of simplicial sets, def. , is the operation of forming degreewise the colimit of higher loop space objects $\Omega(-)\coloneqq (-)^{S^1}$
where $Sing$ denotes the singular simplicial complex functor.
This constitutes an endofunctor
Write
for the natural transformation given in degree $n$ by the $({\vert-\vert}\dashv Sing)$-adjunction unit followed the 0-th component map of the colimiting cocone:
The spectrification of def. satisfies
$Q X$ is an Omega-spectrum, def. ;
$\eta_X \colon X \longrightarrow Q X$ is a stable weak homotopy equivalence, def. ;
if for a homomorphims of sequential spectra $f \colon X \longrightarrow Y$ each $f_n$ is a weak homotopy equivalence, then also each $(Q X)_n$ is a weak homotopy equivalence;
$(Q\eta_X)$ is degreewise a weak homotopy equivalence.
A homomorphism of sequential spectra, def. , is a stable weak homotopy equivalence, def. , precisely if its spectrification $Q f$ , def. , is degreewise a weak homotopy equivalence.
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. It’s main point is that the stable model structure of interest arises fromit via left Bousfield localization.
Say that a homomorphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. is
a strict weak equivalence if each component $f_n \colon X_n \to Y_n$ is a weak equivalence in the classical model structure on simplicial sets (hence a weak homotopy equivalence of geometric realizations);
a strict weak equivalence if each component $f_n \colon X_n \to Y_n$ is a fibration in the classical model structure on simplicial sets (hence a Kan fibration);
a strict cofibration if the simplicial maps $f_0\colon X_0 \to Y_0$ as well as all pushout products of $f_n$ with the structure maps of $X$
are cofibrations of simplicial sets in the classical model structure on simplicial sets (i.e.: monomorphisms of simplicial sets);
The classes of morphisms in def. give the structure of a model category $SeqSpec(sSet)_{strict}$, called the strict model structure on sequential spectra.
Moreover, this is
a simplicial model category with respect to the simplicial enrichment of prop. .
(Bousfield-Friedlander 78, prop. 2.2).
The representation of sequential spectra as diagram spectra says that the category of sequential spectra is equivalently an enriched functor category
(this proposition). Accordingly, this carries the projective model structure on enriched functors, and unwinding the definitions, this 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
By definition, this is equivalently a $\mathbb{N}$-collection of commuting diagrams of simplicial sets of the form
such that all structure maps are respected.
Hence a lifting in the original diagram is a lifting in each degree $n$, such that the lifting in degree $n+1$ 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. Then assume a lifting $l_n$ in degree $n$ has been found
the lifting $l_{n+1}$ in the next degree has to also make the following diagram commute
This is a cocone under the the commuting square for the structure maps, and therefore the outer diagram is equivalently a morphism out of the domain of the pushout product $f_n \Box \sigma_n^X$, while the compatible lift $l_{n+1}$ is equivalently a lift against this pushout product:
Say that a homomorphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. is
a stable weak equivalence if it is a stable weak homotopy equivalence, def. ;
a stable cofibration if the simplicial maps $f_0\colon X_0 \to Y_0$ as well as all pushout products of $f_n$ with the structure maps of $X$
are cofibrations of simplicial sets in the classical model structure on simplicial sets (i.e.: monomorphisms of simplicial sets);
a stable fibration if it is degreewise a fibration of simplicial sets, hence degreewise a Kan fibration, and if in addition the naturality squares of the spectrification, def. ,
are homotopy pullback squares (with respect to the classical model structure on simplicial sets).
The classes of morphisms in def. give the structure of a model category $SeqSpec(sSet)_{stable}$, called the stable model structure on sequential spectra.
Moreover, this is
a simplicial model category with respect to the simplicial enrichment of prop. .
(Bousfield-Friedlander 78, theorem 2.3).
By corollary , the stable model structure $SeqSpectra(sSet)_{stable}$ is, if indeed it exists, the left Bousfield localization of the strict model structure of prop. at the morphisms that become weak equivalences under the spectrification functor $Q \colon SeqSpectra(sSet) \longrightarrow SeqSpectra(sSet)$, def. . By prop. $Q$ satisfies the conditions of the Bousfield-Friedlander theorem, and this implies the claim.
A spectrum $X \in SeqSpec(sSet)_{stable}$ is
fibrant precisely if it is an Omega-spectrum, def. , and each $X_n$ is a Kan complex;
cofibrant precisely if all the structure maps $S^1 \wedge X_n \to X_{n+1}$ are cofibrations of simplicial sets, i.e. monomorphisms.
A sequential spectrum $X\in SeqSpec(sSet)_{stable}$ is cofibrant precisely if all its structure morphisms $S^1 \wedge X_n \to X_{n+1}$ are monomorphisms.
A morphism $\ast \to X$ is a cofibration according to def. (in either the strict or stable model structure, they have the same cofibrations) if
$X_0$ is cofibrant; this is no condition in sSet;
is a cofibration. But in this case the pushout reduces to just its second summand, and so this is now equivalent to
being cofibrations; hence inclusions.
There is a zig-zag of Quillen equivalences relating the Bousfield-Friedlander model structure $SeqSpec(sSet)_{stable}$, def. , prop. with standard model structures on sequential spectra in topological spaces (the model structure on topological sequential spectra) and with Kan’s combinatorial spectra.
(Bousfield-Friedlander 78, section 2.5).
There is a Quillen equivalence to the model structure on symmetric spectra (Hovey-Shipley-Smith 00, section 4.3, Mandell-May-Schwede-Shipley 01, theorem 0.1).
There is a Quillen equivalence between the Bousfield-Friedlander model structure and a model structure for excisive functors (Lydakis 98).
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)
In order to compare this to to sequential spectra consider also the following variant.
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
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 spectra in sSet, def. ) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential spectrum $X^{seq}$ with components
and with structure maps
given by
This is an $sSet^{\ast/}$ enriched equivalence of categories.
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 spectra and Lydakis’ model structure for excisive functors.
(Lydakis 98, theorem 11.3) For more details see at model structure for excisive functors.
The original construction is due to
Generalization of this model structure from sequential pre-spectra in sSet$^{\ast/}$ to sequential spectra in more general proper pointed simplicial model categories is in
Discussion of the Quillen equivalence to the model structure on excisive functors (which does have a symmetric smash product of spectra) is in
Discussion of the Quillen equivalence to the model structure on symmetric spectra is in
Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, theorem 0.1 of Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)