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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Bousfield-Friedlander model structure} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory} [[!include stable homotopy theory - contents]] \hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory} [[!include model category theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{background_on_sequential_spectra}{Background on sequential spectra}\dotfill \pageref*{background_on_sequential_spectra} \linebreak \noindent\hyperlink{sequential_prespectra}{Sequential pre-spectra}\dotfill \pageref*{sequential_prespectra} \linebreak \noindent\hyperlink{omegaspectra}{Omega-spectra}\dotfill \pageref*{omegaspectra} \linebreak \noindent\hyperlink{TheStrictModelStructure}{The strict model structure on sequential spectra}\dotfill \pageref*{TheStrictModelStructure} \linebreak \noindent\hyperlink{TheStableModelStructure}{The stable model structure on sequential spectra}\dotfill \pageref*{TheStableModelStructure} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{FibrantAndCofibrantObjects}{Fibrations and cofibrations}\dotfill \pageref*{FibrantAndCofibrantObjects} \linebreak \noindent\hyperlink{relation_to_sequential_spectra_in__and_to_combinatorial_spectra}{Relation to sequential spectra in $Top$ and to combinatorial spectra}\dotfill \pageref*{relation_to_sequential_spectra_in__and_to_combinatorial_spectra} \linebreak \noindent\hyperlink{relation_to_symmetric_spectra}{Relation to symmetric spectra}\dotfill \pageref*{relation_to_symmetric_spectra} \linebreak \noindent\hyperlink{RelationToExcisiveFunctors}{Relation to excisive functors}\dotfill \pageref*{RelationToExcisiveFunctors} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{Bousfield-Friedlander model structure} (\hyperlink{BousfieldFriedlander78}{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 \emph{[[model structure on topological sequential spectra]]}. As such, the Bousfield-Friedlander model structure [[presentable (infinity,1)-category|presents]] the [[stable (infinity,1)-category of spectra]] of [[stable homotopy theory]], hence, in particular, its [[homotopy category]] is the classical [[stable homotopy category]]. \hypertarget{background_on_sequential_spectra}{}\subsection*{{Background on sequential spectra}}\label{background_on_sequential_spectra} \hypertarget{sequential_prespectra}{}\subsubsection*{{Sequential pre-spectra}}\label{sequential_prespectra} Write $S^1 \coloneqq \Delta[1]/\partial\Delta[1]$ for the minimal [[simplicial set|simplicial]] [[circle]]. Write \begin{displaymath} \wedge \;\colon\; sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/} \end{displaymath} for the [[smash product]] of [[pointed simplicial sets]]. \begin{defn} \label{SequentialSpectra}\hypertarget{SequentialSpectra}{} A \textbf{[[sequential spectrum|sequential]] [[prespectrum]] in [[simplicial sets]]}, or just \textbf{[[sequential spectrum]]} for short (or even just \textbf{[[spectrum]]), is} \begin{itemize}% \item an $\mathbb{N}$-[[graded object|graded]] [[pointed simplicial set]] $X_\bullet$ \item equipped with morphisms $\sigma_n \colon S^1 \wedge X_n \to X_{n+1}$ for all $n \in \mathbb{N}$. \end{itemize} 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 \begin{displaymath} \itexarray{ 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} } \end{displaymath} [[commuting diagram|commute]]. Write $SeqSpec(sSet)$ for this [[category]] of sequential spectra. \end{defn} \begin{example} \label{SmashProductOfSpectrumWithSimplicialSet}\hypertarget{SmashProductOfSpectrumWithSimplicialSet}{} 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 \begin{displaymath} (X \wedge K_+)_n \coloneqq (X_n \wedge K_+) \end{displaymath} and with structure maps given by \begin{displaymath} 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_+ \,. \end{displaymath} \end{example} \begin{prop} \label{SimplicialEnrichment}\hypertarget{SimplicialEnrichment}{} The category $SeqSpec$ of def. \ref{SequentialSpectra} becomes a [[simplicially enriched category]] (in fact an $sSet^{\ast/}$-[[enriched category]]) with [[hom objects]] $[X,Y]\in sSet$ given by \begin{displaymath} [X,Y]_n \coloneqq Hom_{SeqSpec(sSet)}(X\wedge \Delta[n]_+,Y) \,. \end{displaymath} \end{prop} \begin{defn} \label{StableHomotopyGroups}\hypertarget{StableHomotopyGroups}{} The [[stable homotopy groups]] of a [[sequential spectrum]] $X$, def. \ref{SequentialSpectra}, is the $\mathbb{Z}$-[[graded abelian groups]] given by the [[colimit]] of [[homotopy groups]] of [[geometric realizations]] of the component spaces \begin{displaymath} \pi_\bullet(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{\bullet+k}({\vert X_n \vert}) \,. \end{displaymath} This constitutes a [[functor]] \begin{displaymath} \pi_\bullet \;\colon\; SeqSpec(sSet) \longrightarrow Ab^{\mathbb{Z}} \,. \end{displaymath} \end{defn} \begin{defn} \label{StableWeakEquivalenceOfSequentialsSetSpectra}\hypertarget{StableWeakEquivalenceOfSequentialsSetSpectra}{} A morphism $f \colon X \longrightarrow Y$ of [[sequential spectra]], def. \ref{SequentialSpectra}, is called a [[stable weak homotopy equivalence]], if its image under the [[stable homotopy group]]-functor of def. \ref{StableHomotopyGroups} is an [[isomorphism]] \begin{displaymath} \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y) \,. \end{displaymath} \end{defn} \hypertarget{omegaspectra}{}\subsubsection*{{Omega-spectra}}\label{omegaspectra} \begin{defn} \label{OmegaSpectrum}\hypertarget{OmegaSpectrum}{} A \emph{[[Omega-spectrum]]} is a sequential spectrum $X$, def. \ref{SequentialSpectra}, such that after [[geometric realization]]/[[Kan fibrant replacement]] the ([[smash product]] $\dahsv$ [[pointed mapping space]])-[[adjuncts]] \begin{displaymath} {\vert X_n\vert} \stackrel{}{\longrightarrow} {\vert X^{n+1}\vert}^{{\vert S^1\vert}} \end{displaymath} of the structure maps ${\vert \sigma_n\vert}$ are [[weak homotopy equivalences]]. \end{defn} \begin{remark} \label{StableHomotopyGroupsOfOmegaSpectrum}\hypertarget{StableHomotopyGroupsOfOmegaSpectrum}{} If a [[sequential spectrum]] $X$ is an [[Omega-spectrum]], def. \ref{OmegaSpectrum}, then its colimiting [[stable homotopy groups]], def. \ref{StableHomotopyGroups}, are attained as the actual homotopy groups of its components: \begin{displaymath} \pi_k(X) \simeq \simeq \left\{ \itexarray{ \pi_k {\vert X_0 \vert} & if\; k \geq 0 \\ \pi_0 {\vert X_k \vert} & if \; k \lt 0 } \right. \,. \end{displaymath} \end{remark} \begin{defn} \label{Spectrification}\hypertarget{Spectrification}{} The canonical \emph{$\Omega$-[[spectrification]]} $Q X$ of a [[sequential spectrum]] $X$ of [[simplicial sets]], def. \ref{SequentialSpectra}, is the operation of forming degreewise the [[colimit]] of higher [[loop space objects]] $\Omega(-)\coloneqq (-)^{S^1}$ \begin{displaymath} (Q X)_n \coloneqq \underset{\longrightarrow}{\lim}_{k } Sing \Omega^k {\vert X_{n+k}\vert } \,, \end{displaymath} where $Sing$ denotes the [[singular simplicial complex]] functor. This constitutes an [[endofunctor]] \begin{displaymath} Q \;\colon\; SeqSpec(sSet) \longrightarrow SeqSpec(sSet) \,. \end{displaymath} Write \begin{displaymath} \eta \;\colon\; id \longrightarrow Q \end{displaymath} 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]]: \begin{displaymath} (\eta_X)_n \;\colon\; X_n \longrightarrow Sing{\vert X_n\vert} \stackrel{\iota_0}{\longrightarrow} \underset{\longrightarrow}{\lim}_{k } Sing \Omega^k {\vert X_{n+k}\vert } \,. \end{displaymath} \end{defn} \begin{prop} \label{PropertiesOfStandardSpectrification}\hypertarget{PropertiesOfStandardSpectrification}{} The [[spectrification]] of def. \ref{Spectrification} satisfies \begin{enumerate}% \item $Q X$ is an [[Omega-spectrum]], def. \ref{OmegaSpectrum}; \item $\eta_X \colon X \longrightarrow Q X$ is a [[stable weak homotopy equivalence]], def. \ref{StableWeakEquivalenceOfSequentialsSetSpectra}; \item 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; \item $(Q\eta_X)$ is degreewise a weak homotopy equivalence. \end{enumerate} \end{prop} \begin{cor} \label{StableWeakHomotopyEquivalencesofSeqsSetSpectraIsDegreewsieWeakHomotopyEquivalencesOfSpectrification}\hypertarget{StableWeakHomotopyEquivalencesofSeqsSetSpectraIsDegreewsieWeakHomotopyEquivalencesOfSpectrification}{} A homomorphism of [[sequential spectra]], def. \ref{SequentialSpectra}, is a [[stable weak homotopy equivalence]], def. \ref{StableWeakEquivalenceOfSequentialsSetSpectra}, precisely if its [[spectrification]] $Q f$ , def. \ref{PropertiesOfStandardSpectrification}, is degreewise a [[weak homotopy equivalence]]. \end{cor} \hypertarget{TheStrictModelStructure}{}\subsection*{{The strict model structure on sequential spectra}}\label{TheStrictModelStructure} The [[model category]] structure on [[sequential spectra]] which [[presentable (infinity,1)-category|presents]] [[stable homotopy theory]] is the ``stable model structure'' discussed \hyperlink{TheStableModelStructure}{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 [[Bousfield localization of model categories|left Bousfield localization]]. \begin{defn} \label{ClassesOfMorphismsOfTheStrictModelStructureOnSequentialSpectra}\hypertarget{ClassesOfMorphismsOfTheStrictModelStructureOnSequentialSpectra}{} Say that a homomorphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. \ref{SequentialSpectra} is \begin{itemize}% \item a \textbf{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]]); \item a \textbf{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]]); \item a \textbf{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 $X$ \begin{displaymath} X_{n+1}\underset{S^1 \wedge X_n}{\coprod} S^1 \wedge Y_n \longrightarrow Y_{n+1} \end{displaymath} are cofibrations of simplicial sets in the [[classical model structure on simplicial sets]] (i.e.: [[monomorphisms]] of simplicial sets); \end{itemize} \end{defn} \begin{prop} \label{StrictModelStructureOnSequentialPrespectraIsModelCategory}\hypertarget{StrictModelStructureOnSequentialPrespectraIsModelCategory}{} The classes of morphisms in def. \ref{ClassesOfMorphismsOfTheStrictModelStructureOnSequentialSpectra} give the structure of a [[model category]] $SeqSpec(sSet)_{strict}$, called the \textbf{strict model structure} on sequential spectra. Moreover, this is \begin{itemize}% \item a [[proper model category]]; \item a [[simplicial model category]] with respect to the simplicial enrichment of prop. \ref{SimplicialEnrichment}. \end{itemize} \end{prop} (\hyperlink{BousfieldFriedlander78}{Bousfield-Friedlander 78, prop. 2.2}). \begin{proof} The representation of \href{sequential%20spectrum#AsDiagramSpectra}{sequential spectra as diagram spectra} says that the category of sequential spectra is [[equivalence of categories|equivalently]] an [[enriched functor category]] \begin{displaymath} SeqSpec(sSet) \simeq [StdSpheres, sSet^{\ast/}] \end{displaymath} (\href{sequential+spectrum#SequentialSpectraAsDiagramSpectra}{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 \begin{displaymath} \itexarray{ X_\bullet &\stackrel{h_\bullet}{\longrightarrow}& A_\bullet \\ \downarrow^{\mathrlap{f_\bullet}} && \downarrow \\ Y_\bullet &\longrightarrow& B_\bullet } \,. \end{displaymath} By definition, this is equivalently a $\mathbb{N}$-collection of commuting diagrams of simplicial sets of the form \begin{displaymath} \itexarray{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} && \downarrow \\ Y_n &\longrightarrow& B_n } \end{displaymath} such that all structure maps are respected. \begin{displaymath} \itexarray{ X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}} \\ Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1} \\ & \searrow && \searrow \\ && B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1} } \;\;\; \Rightarrow \;\;\; \itexarray{ X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ & \searrow^{\mathrlap{h_n}} && \searrow^{\mathrlap{h_{n+1}}} \\ && A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1} \\ && \downarrow && \downarrow \\ && B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1} } \,. \end{displaymath} 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 \begin{displaymath} \itexarray{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} &\nearrow_{\mathrlap{l_n}}& \downarrow \\ Y_n &\longrightarrow& B_n } \end{displaymath} the lifting $l_{n+1}$ in the next degree has to also make the following diagram commute \begin{displaymath} \itexarray{ X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{h_{n+1}}} & \searrow \\ Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1} && \\ & \searrow^{\mathrlap{l_n}} && \searrow^{\mathrlap{l_{n+1}}} & \downarrow \\ && A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1} } \,. \end{displaymath} 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: \begin{displaymath} \itexarray{ Y_n \underset{X_n}{\sqcup} X_{n+1} &\stackrel{(\sigma_n^A l_n,h_{n+1})}{\longrightarrow}& A_{n+1} \\ \downarrow &{}^{\mathllap{l_{n+1}}}\nearrow& \downarrow \\ Y_{n+1} &\stackrel{}{\longrightarrow}& B_{n+1} } \,. \end{displaymath} \end{proof} \hypertarget{TheStableModelStructure}{}\subsection*{{The stable model structure on sequential spectra}}\label{TheStableModelStructure} \begin{defn} \label{ClassesOfMorphismsOfTheStableModelStructureOnSequentialSpectra}\hypertarget{ClassesOfMorphismsOfTheStableModelStructureOnSequentialSpectra}{} Say that a homomorphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. \ref{SequentialSpectra} is \begin{itemize}% \item a \textbf{stable weak equivalence} if it is a [[stable weak homotopy equivalence]], def. \ref{StableWeakEquivalenceOfSequentialsSetSpectra}; \item a \textbf{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 $X$ \begin{displaymath} X_{n+1}\underset{S^1 \wedge X_n}{\coprod} S^1 \wedge Y_n \longrightarrow Y_{n+1} \end{displaymath} are cofibrations of simplicial sets in the [[classical model structure on simplicial sets]] (i.e.: [[monomorphisms]] of simplicial sets); \item a \textbf{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. \ref{Spectrification}, \begin{displaymath} \itexarray{ X_n &\stackrel{}{\longrightarrow}& (Q X)_n \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{Q f_n}} \\ Y_n &\stackrel{}{\longrightarrow}& (Q Y)_n } \end{displaymath} are [[homotopy pullback]] squares (with respect to the [[classical model structure on simplicial sets]]). \end{itemize} \end{defn} \begin{prop} \label{StableModelStructureOnSequentialSpectraIsModelCategory}\hypertarget{StableModelStructureOnSequentialSpectraIsModelCategory}{} The classes of morphisms in def. \ref{ClassesOfMorphismsOfTheStableModelStructureOnSequentialSpectra} give the structure of a [[model category]] $SeqSpec(sSet)_{stable}$, called the \textbf{stable model structure} on sequential spectra. Moreover, this is \begin{itemize}% \item a [[proper model category]]; \item a [[simplicial model category]] with respect to the simplicial enrichment of prop. \ref{SimplicialEnrichment}. \end{itemize} \end{prop} (\hyperlink{BousfieldFriedlander78}{Bousfield-Friedlander 78, theorem 2.3}). \begin{proof} By corollary \ref{StableWeakHomotopyEquivalencesofSeqsSetSpectraIsDegreewsieWeakHomotopyEquivalencesOfSpectrification}, the stable model structure $SeqSpectra(sSet)_{stable}$ is, if indeed it exists, the [[left Bousfield localization]] of the strict model structure of prop. \ref{StrictModelStructureOnSequentialPrespectraIsModelCategory} at the morphisms that become weak equivalences under the [[spectrification]] functor $Q \colon SeqSpectra(sSet) \longrightarrow SeqSpectra(sSet)$, def. \ref{Spectrification}. By prop. \ref{PropertiesOfStandardSpectrification} $Q$ satisfies the conditions of the [[Bousfield-Friedlander theorem]], and this implies the claim. \end{proof} \begin{remark} \label{}\hypertarget{}{} A spectrum $X \in SeqSpec(sSet)_{stable}$ is \begin{itemize}% \item fibrant precisely if it is an [[Omega-spectrum]], def. \ref{OmegaSpectrum}, and each $X_n$ is a [[Kan complex]]; \item cofibrant precisely if all the structure maps $S^1 \wedge X_n \to X_{n+1}$ are cofibrations of simplicial sets, i.e. [[monomorphisms]]. \end{itemize} \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{FibrantAndCofibrantObjects}{}\subsubsection*{{Fibrations and cofibrations}}\label{FibrantAndCofibrantObjects} \begin{prop} \label{}\hypertarget{}{} 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]]. \end{prop} \begin{proof} A morphism $\ast \to X$ is a cofibration according to def. \ref{ClassesOfMorphismsOfTheStrictModelStructureOnSequentialSpectra} (in either the strict or stable model structure, they have the same cofibrations) if \begin{enumerate}% \item $X_0$ is cofibrant; this is no condition in [[sSet]]; \item \begin{displaymath} \ast_{n+1}\underset{S^1 \wedge \ast_n}{\coprod} S^1 \wedge X_n \longrightarrow X_{n+1} \end{displaymath} is a cofibration. But in this case the [[pushout]] reduces to just its second summand, and so this is now equivalent to \begin{displaymath} S^1 \wedge X_n \longrightarrow X_{n+1} \end{displaymath} being cofibrations; hence inclusions. \end{enumerate} \end{proof} \hypertarget{relation_to_sequential_spectra_in__and_to_combinatorial_spectra}{}\subsubsection*{{Relation to sequential spectra in $Top$ and to combinatorial spectra}}\label{relation_to_sequential_spectra_in__and_to_combinatorial_spectra} \begin{prop} \label{}\hypertarget{}{} There is a [[zig-zag]] of [[Quillen equivalences]] relating the Bousfield-Friedlander model structure $SeqSpec(sSet)_{stable}$, def. \ref{ClassesOfMorphismsOfTheStableModelStructureOnSequentialSpectra}, prop. \ref{StableModelStructureOnSequentialSpectraIsModelCategory} with standard model structures on [[sequential spectra]] in [[topological spaces]] (the [[model structure on topological sequential spectra]]) and with Kan's [[combinatorial spectra]]. \end{prop} (\hyperlink{BousfieldFriedlander78}{Bousfield-Friedlander 78, section 2.5}). \hypertarget{relation_to_symmetric_spectra}{}\subsubsection*{{Relation to symmetric spectra}}\label{relation_to_symmetric_spectra} There is a [[Quillen equivalence]] to the [[model structure on symmetric spectra]] (\hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, section 4.3}, \hyperlink{MandellMaySchwedeShipley01}{Mandell-May-Schwede-Shipley 01, theorem 0.1}). \hypertarget{RelationToExcisiveFunctors}{}\subsubsection*{{Relation to excisive functors}}\label{RelationToExcisiveFunctors} There is a [[Quillen equivalence]] between the Bousfield-Friedlander model structure and a [[model structure for excisive functors]] (\hyperlink{Lydakis98}{Lydakis 98}). \begin{defn} \label{SimplicialSetsPointedAndFinite}\hypertarget{SimplicialSetsPointedAndFinite}{} Write \begin{itemize}% \item [[sSet]] for the [[category]] of [[simplicial sets]]; \item $sSet^{\ast/}$ for the category of [[pointed object|pointed]] simplicial sets; \item $sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the [[full subcategory]] of [[pointed object|pointed]] [[simplicial object|simplicial]] [[finite sets]]. \end{itemize} Write \begin{displaymath} sSet^{\ast/} \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}} sSet \end{displaymath} for the [[free-forgetful adjunction]], where the [[left adjoint]] functor $(-)_+$ freely adjoins a base point. Write \begin{displaymath} \wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/} \end{displaymath} for the [[smash product]] of [[pointed object|pointed]] [[simplicial sets]], similarly for its restriction to $sSet_{fin}^{\ast}$: \begin{displaymath} X \wedge Y \coloneqq cofib\left( \; \left(\, (u(X),\ast) \sqcup (\ast, u(Y)) \,\right) \longrightarrow u(X) \times u(Y) \; \right) \,. \end{displaymath} This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a [[closed monoidal category]] and we write \begin{displaymath} [-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/} \end{displaymath} for the corresponding [[internal hom]], the pointed [[function complex]] functor. \end{defn} We regard all the categories in def. \ref{SimplicialSetsPointedAndFinite} 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]] \begin{displaymath} [sSet^{\ast/}_{fin}, sSet^{\ast/}] \,. \end{displaymath} (\hyperlink{Lydakis98}{Lydakis 98, example 3.8, def. 4.4}) In order to compare this to to [[sequential spectra]] consider also the following variant. \begin{defn} \label{CategoriesOfStandardSpheres}\hypertarget{CategoriesOfStandardSpheres}{} Write $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/}$ for the standard minimal pointed simplicial [[1-sphere]]. Write \begin{displaymath} \iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin} \end{displaymath} for the non-full $sSet^{\ast/}$-[[enriched category|enriched]] [[subcategory]] of pointed [[simplicial object|simplicial]] [[finite sets]], def. \ref{SimplicialSetsPointedAndFinite} whose \begin{itemize}% \item [[objects]] are the [[smash product]] powers $S^n_{std} \coloneqq (S^1_{std})^{\wedge^n}$ (the standard minimal simplicial [[n-spheres]]); \item [[hom-objects]] are \begin{displaymath} [S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres} \coloneqq \left\{ \itexarray{ \ast & for & k \lt 0 \\ im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}}) & otherwise } \right. \end{displaymath} \end{itemize} \end{defn} (\hyperlink{Lydakis98}{Lydakis 98, def. 4.2}) \begin{prop} \label{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}\hypertarget{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}{} There is an $sSet^{\ast/}$-[[enriched functor]] \begin{displaymath} (-)^seq \;\colon\; [StdSpheres,sSet^{\ast/}] \longrightarrow SeqSpec(sSet) \end{displaymath} (from the category of $sSet^{\ast/}$-[[enriched presheaves|enriched copresheaves]] on the categories of standard simplicial spheres of def. \ref{CategoriesOfStandardSpheres} to the category of sequential spectra in [[sSet]], def. \ref{Spectra}) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential spectrum $X^{seq}$ with components \begin{displaymath} X^{seq}_n \coloneqq X(S^n_{std}) \end{displaymath} and with structure maps \begin{displaymath} \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}]} \end{displaymath} given by \begin{displaymath} 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}] \,. \end{displaymath} This is an $sSet^{\ast/}$ [[enriched category theory|enriched]] [[equivalence of categories]]. \end{prop} (\hyperlink{Lydakis98}{Lydakis 98, prop. 4.3}) \begin{prop} \label{}\hypertarget{}{} The [[adjunction]] \begin{displaymath} (\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)_{stable} \end{displaymath} (given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. \ref{CategoriesOfStandardSpheres} and by left [[Kan extension]] $\iota_\ast$ along $\iota$ and combined with the equivalence $(-)^{seq}$ of prop. \ref{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}) 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]]. \end{prop} (\hyperlink{Lydakis98}{Lydakis 98, theorem 11.3}) For more details see at \emph{[[model structure for excisive functors]]}. \hypertarget{references}{}\subsection*{{References}}\label{references} The original construction is due to \begin{itemize}% \item [[Aldridge Bousfield]], [[Eric Friedlander]], \emph{Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets}, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf}{pdf}) \end{itemize} Generalization of this model structure from sequential pre-spectra in [[sSet]]$^{\ast/}$ to sequential spectra in more general [[proper model category|proper]] [[pointed category|pointed]] [[simplicial model categories]] is in \begin{itemize}% \item [[Stefan Schwede]], \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 77-104 (\href{http://www.math.uni-bonn.de/people/schwede/modelspec.pdf}{pdf}) \end{itemize} Discussion of the [[Quillen equivalence]] to the [[model structure on excisive functors]] (which does have a [[symmetric smash product of spectra]]) is in \begin{itemize}% \item Lydakis, \emph{Simplicial functors and stable homotopy theory} Preprint, available via Hopf archive, 1998 (\href{http://hopf.math.purdue.edu/Lydakis/s_functors.pdf}{pdf}) \end{itemize} Discussion of the [[Quillen equivalence]] to the [[model structure on symmetric spectra]] is in \begin{itemize}% \item [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], \emph{Symmetric spectra}, J. Amer. Math. Soc. 13 (2000), 149-208 (\href{http://arxiv.org/abs/math/9801077}{arXiv:math/9801077}) \item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], theorem 0.1 of \emph{Model categories of diagram spectra}, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (\href{http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf}{pdf}) \end{itemize} [[!redirects Bousfield-Friedlander model category]] [[!redirects Bousfield-Friedlander model structures]] \end{document}