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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*{simplicial model category} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory} [[!include model category theory - contents]] \hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory} [[!include enriched category theory contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{EnrichmentTensoringCotensoring}{Enrichment, tensoring, and cotensoring}\dotfill \pageref*{EnrichmentTensoringCotensoring} \linebreak \noindent\hyperlink{derived_homspaces}{Derived hom-spaces}\dotfill \pageref*{derived_homspaces} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{SimpEquivMods}{Simplicial Quillen equivalent models}\dotfill \pageref*{SimpEquivMods} \linebreak \noindent\hyperlink{combinatorial_simplicial_model_categories}{Combinatorial simplicial model categories}\dotfill \pageref*{combinatorial_simplicial_model_categories} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The term \emph{simplicial model category} is short for \emph{[[model structure on simplicial sets|sSet]]${}_{Quillen}$-[[enriched model category]]}. A simplicial model category is a model or presentation for an [[(∞,1)-category]] that is half way in between a bare [[model category]] and a [[Kan complex]]-[[enriched category]]. Specifically, a simplicial model category is an [[sSet]]-[[enriched category]] $C$ together with the structure of a [[model category]] on its underlying [[category]] $C_0$ such that both structures are compatible in a reasonable way. One important use of simplicial model categories comes from the fact that the full [[sSet]]-subcategory $C^\circ \hookrightarrow C$ on the fibrant-cofibrant objects -- which is not just [[sSet]]-enriched but actually [[Kan complex]]-enriched -- is the [[(∞,1)-category]]-enhancement of the [[homotopy category]] of the [[model category]] $C_0$. For generalizations of this construction with [[sSet]] replaced by another [[monoidal model category]] see [[enriched homotopical category]]. \begin{remark} \label{}\hypertarget{}{} The term \emph{simplicial model category} for the notion described here is entirely standard, but in itself a bit suboptimal. More properly one would speak of [[simplicially enriched category]], which is a proper special case of a [[simplicial object]] in [[Cat]] (that for which the simplicial set of objects is discrete). The other caveat is that there are different model category structures on [[sSet]] and hence even the term $sSet$-[[enriched model category]] is ambiguous. For instance the [[model structure for quasi-categories]] is an $sSet$-[[enriched model category]], but not for the standard Quillen model structure on the enriching category: since $sSet_{Joyal}$ is a [[closed monoidal category|closed]] [[monoidal model category]] it is enriched over itself, hence is a $sSet_{Joyal}$-enriched model category, not an $sSet_{Quillen}$-enriched one. So in the standard terminology, $sSet_{Joyal}$ is \emph{not} a ``simplicial model category''. \end{remark} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} A \textbf{simplicial model category} is an [[enriched model category]] which is enriched over $sSet_{Quillen}$: the category [[sSet]] equipped with its standard [[model structure on simplicial sets]]. \end{defn} Spelled out, this means that a simplicial model category is \begin{itemize}% \item an [[sSet]]-[[enriched category]] \begin{itemize}% \item which is [[power|powered]] and [[tensoring|tensored]] over [[sSet]] \end{itemize} \item with the structure of a [[model category]] on the underlying category $C_0$ \item such that for every cofibration $i : A \to B$ and every fibration $p : X \to Y$ in $C_0$ the [[pullback powering]] of [[simplicial sets]] $C(B,X) \stackrel{i^* \times p_*}{\to} C(A,X) \times_{C(A,Y)} C(B,Y)$ is a [[Kan fibration]]; \begin{itemize}% \item and such that this fibration is an acyclic fibration whenever either $i$ or $p$ are acyclic. \end{itemize} \end{itemize} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{EnrichmentTensoringCotensoring}{}\subsubsection*{{Enrichment, tensoring, and cotensoring}}\label{EnrichmentTensoringCotensoring} Let $\mathcal{C}$ be a category equipped with the structure of a [[model category]] and with that of an [[sSet]]-[[enriched category]] with is [[tensoring|tensored]] and [[cotensoring|cotensored]] over [[sSet]]. \begin{prop} \label{EquivalenceOfConditions}\hypertarget{EquivalenceOfConditions}{} The following conditions -- that each make $\mathcal{C}$ into a \emph{simplicial model category} -- are equivalent: \begin{enumerate}% \item the [[tensoring]] $\otimes : \mathcal{C} \times sSet \to \mathcal{C}$ is a left [[Quillen bifunctor]]; \item for any cofibration $X \to Y$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism \begin{displaymath} \mathcal{C}(Y, A) \to \mathcal{C}(X, A) \times_{\mathcal{C}(X,B)} \mathcal{C}(Y,B) \end{displaymath} is a fibration, and is in addition a weak equivalence if either of the two morphisms is; \item for any cofibration $X \to Y$ in $sSet$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism \begin{displaymath} A^Y \to A^X \times_{B^X} B^Y \end{displaymath} is a fibration, and is in addition a weak equivalence if either of the two morphisms is. \end{enumerate} \end{prop} This follows directly from the defining properties of [[tensoring]] and [[cotensoring]]. We list in the following some implications of these equivalent conditions. Let $\mathcal{C}$ now be a simplicial model category. \begin{cor} \label{}\hypertarget{}{} If $A \in \mathcal{C}$ is fibrant, and $X \hookrightarrow Y$ is a cofibration in [[sSet]], then \begin{displaymath} A^{Y} \to A^{X} \end{displaymath} is a fibration in $\mathcal{C}$. \end{cor} \begin{proof} Apply prop. \ref{EquivalenceOfConditions} to the case of the cofibration $X \to Y$ and the fibration $A \to *$, where ``$*$'' denotes the [[terminal object]]. This yields that \begin{displaymath} A^Y \to A^X \times_{{*}^X} {*}^Y \end{displaymath} is a fibration. But ${*}^Y = {*}^X = {*}$ and hence the claim follows. \end{proof} Similarly we have \begin{cor} \label{}\hypertarget{}{} If $X \in \mathcal{C}$ is cofibrant and $A \in \mathcal{C}$ is fibrant, then $\mathcal{C}(X,A)$ is fibrant in [[sSet]], hence is a [[Kan complex]]. \end{cor} \begin{proof} Apply prop. \ref{EquivalenceOfConditions} to the cofibration $\emptyset \to X$, where ``$\emptyset$'' denotes the [[initial object]], and to the fibration $A \to *$ to find that \begin{displaymath} \mathcal{C}(X, A) \to \mathcal{C}(\emptyset, A) \times_{\mathcal{C}(\emptyset,*)} \mathcal{C}(X,*) \end{displaymath} is a fibration. But since $\emptyset$ is initial and $*$ is terminal, all three simplicial sets in the fiber product on the right are the point, hence this is a fibration \begin{displaymath} \mathcal{C}(X,A) \to * \,. \end{displaymath} \end{proof} \hypertarget{derived_homspaces}{}\subsubsection*{{Derived hom-spaces}}\label{derived_homspaces} \begin{prop} \label{DerivedHomSpaceBySimplicialFunctionComplex}\hypertarget{DerivedHomSpaceBySimplicialFunctionComplex}{} For $X$ and $A$ any two objects and $Q X$ and $P A$ a cofibrant and fibrant replacement, respectively, $\mathcal{C}(Q X, P A)$ is the correct [[derived hom-space]] between $X$ and $A$ (see the discussion there). In particular the full $sSet$-enriched [[subcategory]] on cofibrant fibrant objects is therefore an [[simplicially enriched category|sSet-enriched category]] which is fibrant in the [[model structure on simplicially enriched categories]]. Its [[homotopy coherent nerve]] is a [[quasi-category]]. All this are intrinsic incarnatons of the [[(∞,1)-category]] that is [[presentable (∞,1)-category|presented]] by $C$. \end{prop} Although $\mathcal{C}(X,A)$ need not have the correct homotopy type for the mapping space if $X$ is not cofibrant or $A$ is not fibrant, simplicial notions of homotopy are still \emph{sufficient} to detect the model-categorical ones. \begin{prop} \label{SimplicialHomotopyImpliesHomotopy}\hypertarget{SimplicialHomotopyImpliesHomotopy}{} If $f,g:X\to A$ are simplicially homotopic (i.e. in the same connected component of the simplicial set $\mathcal{C}(X,A)$), then they represent the same map in the [[homotopy category]] of $\mathcal{C}$. Therefore, any [[simplicial homotopy equivalence]] in $\mathcal{C}$ is a weak equivalence. \end{prop} \begin{proof} This is Lemma 9.5.15 and Proposition 9.5.16 of \hyperlink{Hirschhorn}{Hirschhorn}. \end{proof} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item The [[classical model structure on simplicial sets]] $sSet_{Quillen}$ is a [[closed monoidal category|closed]] [[monoidal model category]] and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category. \item The [[classical model structure on topological spaces]] for [[compactly generated topological spaces]] (\href{classical+model+structure+on+topological+spaces#ModelStructureOnCompactlyGeneratedTopologicalSpaces}{here}) is similarly enriched over itself. Under [[geometric realization]] this also makes it a simplicial model category. (See also for instance \hyperlink{GoerssSchemmerhorn06}{Goerss-Schemmerhorn 06, p. 26}) \item For $C$ any small [[sSet]]-[[enriched category]] and $A$ [[simplicial combinatorial model category]], the [[global model structure on functors]] $[C^{op}, A]_{proj}$ and $[C^{op},A]_{inj}$ are themselved simplicial combinatorial model categories. See \emph{[[model structure on simplicial presheaves]]}. \item The [[Bousfield localization of model categories|left Bousfield localization]] of a [[combinatorial simplicial model category]] at any set of morphisms is again a combinatorial simplicial model category. Large classes of examples arise this way. \end{itemize} \hypertarget{SimpEquivMods}{}\subsubsection*{{Simplicial Quillen equivalent models}}\label{SimpEquivMods} While many model categories do not admit an $sSet_{Quillen}$-enrichment, for large classes of model categories one can find a [[Quillen equivalence]] to a model category that does have an $sSet_{Quillen}$-enrichment. These are constructed as [[Bousfield localization of model categories|Bousfield localization]] of [[Reedy model structures]] on the [[category of simplicial objects]] in the given model category. \begin{defn} \label{LocalizationOfStructureOnSimplicalObjects}\hypertarget{LocalizationOfStructureOnSimplicalObjects}{} Let $C$ be a \begin{itemize}% \item [[left proper model category|left proper]] \item [[combinatorial model category]]. \end{itemize} By the discussion at \emph{[[cofibrantly generated model category]]} in the section \emph{\href{http://ncatlab.org/nlab/show/cofibrantly+generated+model+category#PresentationAndGeneration}{Presentation and generation}} there exists a [[small set]] $E \subset Obj(C)$ of objects that detect weak equivalences. For some such choice of $E$, let \begin{displaymath} S := \{ \; e \cdot (\Delta[k] \to \Delta[l]) \; \}_{e \in E, ([k] \to [l]) \in \Delta} \subset Mor([\Delta^{op}, sSet]) \end{displaymath} where $e \cdot \Delta[k] : [n] \mapsto \coprod_{\Delta([n],[k])} e$. Write \begin{displaymath} [\Delta^{op}, C]_{proj, S} \stackrel{\longleftarrow}{\longrightarrow} [\Delta^{op}, C]_{proj} \end{displaymath} for the [[Bousfield localization of model categories|left Bousfield localization]] of the projective [[model structure on functors]] at this set $S$ of morphisms. Similarly, write \begin{displaymath} [\Delta^{op}, C]_{Reedy, S} \stackrel{\leftarrow}{\longrightarrow} [\Delta^{op}, C]_{Reedy} \end{displaymath} for the left Bousfield localization of the [[Reedy model structure]] at $S$. \end{defn} \begin{lemma} \label{HocolimOverHomotopyConstantSimplicialDiagram}\hypertarget{HocolimOverHomotopyConstantSimplicialDiagram}{} Let $C$ be a [[cofibrantly generated model category]]. If $X \in [\Delta^{op}, C]$ is degreewise cofibrant and has all structure maps being weak equivalences, then all $X_i \to hocolim X$ are weak equivalences. Hence $X \to const\,hocolim X$ is a weak equivalence. \end{lemma} This appears as (\hyperlink{Dugger}{Dugger, prop. 5.4 corollary 5.5}). \begin{theorem} \label{LocalizationOfReedyOnSimplicialObjects}\hypertarget{LocalizationOfReedyOnSimplicialObjects}{} The model structures from def. \ref{LocalizationOfStructureOnSimplicalObjects} have the following properties. \begin{enumerate}% \item The weak equivalences in both are precisely those morphisms which become weak equivalences under [[homotopy colimit]] over $\Delta^{op}$. \item The fibrant objects in both are precisely those objects that are fibrant in the corresponding unlocalized structures, and such that all the face and degeneracy maps are weak equivalences in $C$. \item The [[colimit]]/constant [[adjoint functors]] \begin{displaymath} (\lim_{\to} \dashv const) \colon C \stackrel{\overset{\lim_\to}{\longleftarrow}}{\underset{const}{\longrightarrow}} [\Delta^{op}, C]_{proj, S} \end{displaymath} constitute a [[Quillen equivalence]], the identity functors constitute a Quillen equivalence \begin{displaymath} [\Delta^{op}, C]_{Reedy, S} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} [\Delta^{op}, C]_{proj, S} \,, \end{displaymath} and the constant/[[limit]] [[adjoint functors]] (since $\Delta^{op}$ has an [[initial object]] the limit is evaluation in degree 0) constitute a Quillen equivalence \begin{displaymath} (const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\overset{const}{\longleftarrow}}{\underset{ev_0}{\longrightarrow}} C \,; \end{displaymath} \item The canonical [[sSet]]-[[enriched category|enrichment]]/[[tensoring]]/[[powering]] of the [[category of simplicial objects]] $[\Delta^{op}, C]$ makes $[\Delta^{op}, C]_{Reedy,S}$ (but \emph{not} in general $[\Delta^{op}, C]_{proj,S}$) into a simplicial model category. \end{enumerate} \end{theorem} This is (\hyperlink{Dugger}{Dugger, theorem 5.2, theorem 5.7, theorem 6.1}). So in particular every [[proper model category|left proper]] [[combinatorial model category]] is [[Quillen equivalence|Quillen equivalent]] to a simplicial model category. \begin{proof} We first show that the fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the objectwise fibrant objects all whose structure maps are weak equivalences in $C$. The argument for the fibrant objects in $[\Delta^{op}, C]_{Reedy,S}$ is directly analogous. By general properties of [[Bousfield localization of model categories|left Bousfield localization]], the fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the projective fibrant objects $X$ for which all induced morphisms on [[derived hom spaces]] \begin{displaymath} \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}(s \cdot(\Delta[k] \to \Delta[l]), X) \end{displaymath} are weak equivalences. Since $s$ is cofibrant in $C$ by definition, also $s \cdot \Delta[k]$ is cofibrant in $[\Delta^{op}, C]_{proj}$. So for $X \in [\Delta^{op}, C]_{proj}$ fibrant, let $X_\bullet \in [\Delta^{op}, [\Delta^{op}, C]]$ be a \href{derived+hom+space#Framings}{simplicial framing} for it. Notice that this means that for all $n \in \mathbb{N}$ also $X_\bullet([n])$ is a simplicial framing for $X([n])$. This is because \begin{enumerate}% \item $const X \to X_\bullet$ being a weak equivalence means that for all $n$ the morphism $X \to X_n$ is a weak equivalence, which means that for all $k$ the morphism $X([k]) \to X_n([k])$ is a weak equivalence. \item $X_\bullet$ being fibrant in $[\Delta^{op}, [\Delta^{op}, C]_{proj}]_{Reedy}$ means that for all $n\in \mathbb{N}$ the morphism $X_{\Delta[n]} \to X_{\partial \Delta[n]}$ is a fibration in $[\Delta^{op}, C]_{proj}$, hence that for all $k \in \mathbb{N}$ the morphism $X_{\Delta[n]}([k]) \to X_{\partial \Delta[n]}([k])$ is a fibration in $C$, hence that $X([k])$ is Reedy fibrant. \end{enumerate} Then we find \begin{displaymath} \begin{aligned} \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}( s \cdot(\Delta[k] \to \Delta[l]), X) & \simeq Hom_{[\Delta^{op}, C]}(s \cdot(\Delta[k] \to \Delta[l]), X_\bullet) \\ & \simeq Hom_{C}(s , X_\bullet([l]) \to X_\bullet([k])) \\ & \simeq \mathbb{R}Hom_C(s, X([l]) \to X([k])) \,. \end{aligned} \end{displaymath} By assumption on the set $S$, this implies the claim. $\,$ Now we show that the weak equivalences in $[\Delta^{op}, C]_{proj,S}$ are precisely those morphisms that become weak equivalences under the [[homotopy colimit]]. By functorial [[cofibrant resolution]] and [[two-out-of-three]], it is sufficient to show that this holds for morphisms between cofibrant objects. By lemma \ref{HocolimOverHomotopyConstantSimplicialDiagram}, we have weak equivalences \begin{displaymath} \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X) \stackrel{\simeq}{\to} \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z) \stackrel{\simeq}{\to} \mathbb{R}Hom_{C}(\lim_\to A , Z) \end{displaymath} seen by computing the derived homs by \href{derived+hom+space#Framings}{simplicial framings}. Now, by properties of left [[nLab:Bousfield localization of model categories|Bousfield localization]], $A \to B$ is a weak equivalence if for all $S$-[[local objects]] $X$ the morphism $\mathbb{R}Hom(A \to B, X)$ is a weak equivalence. Looking at the diagram \begin{displaymath} \itexarray{ \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{C}(\lim_\to A , Z) \\ \uparrow && \uparrow && \uparrow \\ \mathbb{R}Hom_{[\Delta^{op}, C]}(B,X) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{[\Delta^{op}, C]}(B, const Z) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{C}(\lim_\to B , Z) } \end{displaymath} we see that this is the case precisely if the vertical morphism on the right is a weak equivalence for all fibrant $Z \in C$, which is the case if $\lim_\to A \to \lim_\to B$ is a weak equivalence. Since $A$ and $B$ here are cofibrant in $[\Delta^{op}, C]_{proj}$, the colimits here are indeed [[homotopy colimits]] (as discussed there). $\,$ Now we discuss that $(\lim_\to \dashv const): C \to [\Delta^{op}, C]_{proj,S}$ is a [[Quillen equivalence]]. First observe that on the global model structure $const : C \to [\Delta^{op}, C]_{proj}$ is clearly a right Quillen functor, hence we have a Quillen adjunction on the unlocalized structure. Moreover, by definition and by the above discussion, the [[derived functor]] of the [[left adjoint]] $\lim_\to$, namely the [[homotopy colimit]], takes the localizing set $S$ to weak equivalences in $C$. Therefore the assumptions of the discussion at \emph{\href{http://ncatlab.org/nlab/show/Quillen+adjunction#BehaviourUnderLocalization}{Quillen equivalence - Behaviour under localization}} are met, and hence it follows that $(\lim_\to \dashv const)$ descends as a Quillen adjunction also to the localization. To see that this is a Quillen equivalence, it is sufficient to show that for $A \in [\Delta^{op}, C]_{proj}$ cofibrant and $Z \in C$ fibrant, a morphism $\lim_\to A \to Z$ is a weak equivalence in $C$ precisely if the [[adjunct]] $A \to const Z$ becomes a weak equivalence under the homotopy colimit. For this notice that we have a commuting diagram \begin{displaymath} \itexarray{ hocolim A &\to& hocolim(const Z) \\ \downarrow && \downarrow \\ colim A &\to& colim (const Z) & \simeq Z } \end{displaymath} and so our statement follows (by 2-out-of-3) once we know that the vertical morphisms here are weak equivalences. The left one is because $A$ is cofibrant, by assumption, as before. To see that the right one is, too, consider the factorization \begin{displaymath} Z \to (const Z)_i \to hocolim (const Z) \to Z \end{displaymath} of the identity on $Z$, for any $i \in \mathbb{N}$. By lemma \ref{HocolimOverHomotopyConstantSimplicialDiagram} the first morphism is a weak equivalence, and hence so is the morphism in question. $\,$ Now we show that the weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$ are the hocolim-equivalences. By a general result on \href{Bousfield+localization+of+model+categories#FunctorialityOfLocalization}{functoriality of localization}, we have that the $(id \dashv id ) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{proj,S}$ is at least a [[Quillen adjunction]]. Let then $A \to B$ be a morphism in $[\Delta^{op}, C]$ and consider two fibrant replacements \begin{displaymath} \itexarray{ A &\to& \bar A &\to & \hat A \\ \downarrow && \downarrow && \downarrow \\ B &\to& \bar B &\to& \hat B } \,, \end{displaymath} where the first one ($\bar A \to \bar B$) is taken in $[\Delta^{op}, C]_{proj,S}$ and the second ($\hat A \to \hat B$) in $[\Delta^{op}, C]_{Reedy}$. Assume first that $A \to B$ is a hocolim-equivalence. Then so is $\hat A \to \hat B$, because the horizontal morphisms are all objectwise weak equivalences. But $\hat A$ and $\hat B$ are fibrant in $[\Delta^{op}, C]_{Reedy}$, hence in $[\Delta^{op}, C]_{proj}$ by construction and at the same time all their structure maps are weak equivalences (use 2-out-of-3), so that they are in fact fibrant in $[\Delta^{op}, C]_{proj,S}$. By general properties of left Bousfield localization, weak equivalences between local fibrant objects are already weak equivalences in the unlocalized structure -- so $\hat A \to \hat B$ is indeed even an objectwise weak equivalence. It follows then that so is $\bar A \to \bar B$, which is therefore in partiular a weak equivalence in $[\Delta^{op}, C]_{Reedy, S}$. Finally the left horizontal morphisms are also weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$, by the above Quillen adjunction. So finally by 2-out-of-3 in $[\Delta^{op}, C]_{Reedy,S}$ it follows that also $A \to B$ is a weak equivalence there. By an analogous diagram chase, one shows the converse implication holds, that $A \to B$ being a weak equivalence in $[\Delta^{op}, C]_{Reedy,S}$ implies that it is a hocolim-equivalence. With this now it is clear that the identity adjunction above is in fact a Quillen equivalence. $\,$ Finally we show that $(const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} C$ is a Quillen equivalence. First, it is immediate to check that $const : C \to [\Delta^{op}, C]_{Reedy}$ is left Quillen, and since $id : [\Delta^{op}, C]_{Reedy} \to [\Delta^{op}, C]_{Reedy,S}$ is left Quillen by definition of Bousfield localization, the above is at least a Quillen adjunction. To see that it is a Quillen equivalence, let $A \in C$ be cofibrant and $X \in [\Delta^{op}, C]_{Reedy,S}$ be fibrant -- which by the above means that it is a simplicial resolution -- and consider a morphism $const A \to X$. We need to show that this is a weak equivalence, hence, by the above, that its hocolim is a weak equivalence, precisely if $A \to X_0$ is a weak equivalence in $C$. To that end, find a cofibrant resolution $const \tilde A \to \tilde X$ of $const A \to X$ in $[\Delta^{op}, C]_{proj}$ and consider the diagram \begin{displaymath} \itexarray{ A &\stackrel{\simeq}{\leftarrow}& \tilde A &\stackrel{\simeq}{\to}& colim(const \tilde A) \\ \downarrow && \downarrow && \downarrow \\ X_0 &\stackrel{\simeq}{\leftarrow}& \tilde X_0 &\stackrel{\simeq}{\to}& colim \tilde X } \,. \end{displaymath} The colimits on the right compute the homotopy colimit. By 2-out-of-3 it follows that the right vertical morphism is a weak equivalence precisely if the left vertical morphisms is. $\,,$ Finally it remains to show that $[\Delta^{op}, C]_{Reedy,S}$ is a simplicially enriched model category. (\ldots{}) \end{proof} \begin{theorem} \label{AUniquenessTheorem}\hypertarget{AUniquenessTheorem}{} \textbf{(uniqueness)} Let $C$ be a [[model category]]. Then there is a unique model category structure on $s C = [\Delta^{op}, C]$ such that \begin{itemize}% \item every morphism that is degreewise a weak equivalence in $C$ is a weak equivalence; \item the cofibrations are those of the [[Reedy model structure]]; \item the fibrant objects are the Reedy-fibrant objects whose face and degeneracy maps are weak equivalences in $C$. \end{itemize} \end{theorem} This is (\hyperlink{RezkSchwedeShipley}{Rezk-Schwede-Shipley, theorem 3.1}). \begin{proof} By theorem \ref{LocalizationOfReedyOnSimplicialObjects} at least one such model structure exists. By the discussion at \emph{\href{model%20category#RedundancyInTheAxioms}{model category -- Redundancy of the axioms}}, the classes of cofibrations and fibrant objects already determine a model category structure. \end{proof} \begin{cor} \label{DerivedHomBySimplicialReplacement}\hypertarget{DerivedHomBySimplicialReplacement}{} For $C$ any [[proper model category|left proper]] [[combinatorial model category]], the [[derived hom-space]] between two objects $X, A$ may be computed by \begin{itemize}% \item choosing a cofibrant replacement $\hat X$ of $X$ in $C$; \item choosing a Reedy fibrant replacement $\hat A$ of $const A$ in $[\Delta^{op}, C]$ such that all face and degeneracy maps are weak equivalences, \end{itemize} setting \begin{displaymath} Maps(X,A) : [n] \mapsto Hom_C(\hat X, \hat A_n) \,. \end{displaymath} \end{cor} \begin{proof} By theorem \ref{LocalizationOfReedyOnSimplicialObjects} we may compute the [[derived hom space]] in $[\Delta^{op}, C]_{Reedy,S}$ after the inclusion $const : C \to [\Delta^{op}, C]$. Since by that theorem $[\Delta^{op}, C]_{Reedy,S}$ is a simplicial model category, by prop. \ref{DerivedHomSpaceBySimplicialFunctionComplex} the derived hom space is given by the simplicial function complex between a cofibrant replacement of $const X$ and a fibrant replacement of $const A$. If $\hat X$ is cofibrant, then $const \hat X$ is already Reedy cofibrant, and by the theorem $\hat A$ as stated is a a fibrant resolution of $const A$. Finally, the theorem says that the simplicial function complex is given by \begin{displaymath} \begin{aligned} [\Delta^{op}, C](const \hat X, \hat A)_n & = Hom_{[\Delta^{op}, C]}((const \hat X) \cdot \Delta[n], \hat A) \\ & \simeq Hom_{[\Delta^{op}, C]}((const \hat X) , \hat A^{\Delta[n]}) \\ & \simeq Hom_C(\hat X, \hat A_n) \end{aligned} \,. \end{displaymath} \end{proof} There is also a version for stable model categories: \begin{theorem} \label{}\hypertarget{}{} Every [[proper model category|proper]] [[cofibrantly generated model category|cofibrantly generated]] [[stable model category]] is [[Quillen equivalence|Quillen equivalent]] to a simplicial model category \end{theorem} This is (\hyperlink{RezkSchwedeShipley}{Rezk-Schwede-Shipley, prop 1.3}). \hypertarget{combinatorial_simplicial_model_categories}{}\subsubsection*{{Combinatorial simplicial model categories}}\label{combinatorial_simplicial_model_categories} A particularly important type of simplicial model categories are those that are also [[combinatorial model category|combinatorial model categories]]. A [[combinatorial simplicial model category]] is precisely a \emph{presentation} for a [[locally presentable (∞,1)-category]]. See there for more details. \hypertarget{references}{}\subsection*{{References}}\label{references} The definition appears in \begin{itemize}% \item [[Dan Quillen]], chapter II, section 2 of \emph{Homotopical algebra}, Lecture Notes in Mathematics \textbf{43}, Springer-Verlag 1967, iv+156 pp. \end{itemize} Textbook references include \begin{itemize}% \item [[Philip Hirschhorn]], section 9.1.5 of \emph{Model categories and their localizations}, volume 99 of Mathematical Surveys and Monographs , American Mathematical Society, 2009. \item [[Jacob Lurie]], section A.3 in \emph{[[Higher Topos Theory]]} \end{itemize} Further review includes \begin{itemize}% \item [[Paul Goerss]], [[Kristen Schemmerhorn]], \emph{Model categories and simplicial methods} (\href{http://arxiv.org/abs/math/0609537}{arXiv:math/0609537}) \end{itemize} Further developments include \begin{itemize}% \item [[Charles Rezk]], [[Stefan Schwede]], [[Brooke Shipley]], \emph{Simplicial structures on model categories and functors}, American Journal of Mathematics, Vol. 123, No. 3 (Jun., 2001), pp. 551-575 (\href{http://arxiv.org/abs/math/0101162}{arXiv:0101162}, \href{http://www.jstor.org/stable/pdfplus/25099071.pdf}{jstor}) \item [[Dan Dugger]], \emph{Replacing model categories with simplicial ones}, Trans. Amer. Math. Soc. vol. 353, number 12 (2001), 5003-5027. (\href{http://hopf.math.purdue.edu/Dugger/smod.pdf}{pdf}) \end{itemize} [[!redirects simplicial model categories]] [[!redirects sSet-model category]] [[!redirects sSet-model categories]] [[!redirects simplicial model structure]] [[!redirects simplicial model structures]] \end{document}