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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*{Reedy model structure} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \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{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{plain_version}{Plain version}\dotfill \pageref*{plain_version} \linebreak \noindent\hyperlink{enriched_version}{Enriched version}\dotfill \pageref*{enriched_version} \linebreak \noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{EnrichedModelStructure}{Enriched model structure}\dotfill \pageref*{EnrichedModelStructure} \linebreak \noindent\hyperlink{relation_to_other_model_structures}{Relation to other model structures}\dotfill \pageref*{relation_to_other_model_structures} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{over_the_arrow_category}{Over the arrow category}\dotfill \pageref*{over_the_arrow_category} \linebreak \noindent\hyperlink{OverTheTowerCategory}{Over the tower category}\dotfill \pageref*{OverTheTowerCategory} \linebreak \noindent\hyperlink{SimplexCategory}{Over the simplex category}\dotfill \pageref*{SimplexCategory} \linebreak \noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak \noindent\hyperlink{fibrant_and_cofibrant_objects}{Fibrant and cofibrant objects}\dotfill \pageref*{fibrant_and_cofibrant_objects} \linebreak \noindent\hyperlink{OverDeltaWithValuesInSimplicialSets}{With values in simplicial sets}\dotfill \pageref*{OverDeltaWithValuesInSimplicialSets} \linebreak \noindent\hyperlink{with_values_in_an_arbitrary_model_category}{With values in an arbitrary model category}\dotfill \pageref*{with_values_in_an_arbitrary_model_category} \linebreak \noindent\hyperlink{EnrichmentOverTheSimplexCategory}{Enrichment}\dotfill \pageref*{EnrichmentOverTheSimplexCategory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{Reedy model structure} is a [[global model structure on functors]]: given a [[Reedy category]] $R$ and a [[model category]] $C$ the \textbf{Reedy model structure} is a [[model category]] structure on the [[functor category]] $[R,C] = Func(R,C)$. As opposed to the [[global model structure on functors|projective and injective model structure]] on functors this does not require any further structure on $C$, but instead makes a strong assumption on $R$. If all three exist, then, in a precise sense, the Reedy model structure sits \emph{in between} the injective and the projective model structure. As such, it has the advantage that \emph{both} the cofibrations as well as the fibrations can be fairly explicitly described and detected in terms of cofibrations and fibrations in $C$. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{plain_version}{}\subsubsection*{{Plain version}}\label{plain_version} \begin{theorem} \label{model}\hypertarget{model}{} If $R$ is a [[Reedy category]] and $C$ is a [[model category]], then there is a canonical induced [[model category|model structure]] on the [[functor category]] $C^R$ in which the weak equivalences are the objectwise weak equivalences in $C$. \end{theorem} The basic idea is as follows. Given a diagram $X:R\to C$ and an object $r\in R$, define its \textbf{latching object} to be \begin{displaymath} L_r X = \colim_{s \overset{+}{\to} r} X_s \end{displaymath} where the colimit is over the full subcategory of $R_+/r$ containing all objects except the identity $1_r$. Dually, define its \textbf{matching object} to be \begin{displaymath} M_r X = \lim_{r \overset{-}{\to} s} X_s \end{displaymath} where the limit is over the full subcategory of $r/R_-$ containing all objects except $1_r$. There are evident canonical, and natural, morphisms \begin{displaymath} L_r X\to X_r \to M_r X. \end{displaymath} Note that $L_0 X = 0$ is the initial object and $M_0 X$ is the terminal object, since there are no objects of degree $\lt 0$. In the case $R=\Delta^{op}$, the latching object $L_n X$ can be thought of as the object of \emph{degenerate} $n$-simplices sitting inside the object $X_n$ of all $n$-simplices. When $R=\alpha$ is an ordinal, then $L_{n+1} X = X_n$ and $M_n X = 1$, and dually for $R=\alpha^{op}$. We now define a morphism $f:X\to Y$ in $M^R$ to be a cofibration or trivial cofibration if for all $r$, the map \begin{displaymath} L_r Y \amalg_{L_r X} X_r \to Y_r \end{displaymath} is a cofibration or trivial cofibration in $M$, respectively, and to be a fibration or trivial fibration if for all $r$, the map \begin{displaymath} X_r \to M_r X \times_{M_r Y} Y_r \end{displaymath} is a fibration or trivial fibration in $M$, respectively. Define $f$ to be a weak equivalence if each $f_r$ is a weak equivalence in $M$. One then verifies that this defines a model structure; the details can be found in (for instance) Hovey and Hirschhorn's books. In particular, to factor a morphism $f:X\to Y$ in either of the two necessary ways, we construct the factorization $f_r = g_r h_r$ inductively on $r$, by factoring the induced morphism \begin{displaymath} L_r Z \amalg_{L_r X} X_r \to M_r Z \times_{M_r Y} Y_r \end{displaymath} in the appropriate way in $M$. \hypertarget{enriched_version}{}\subsubsection*{{Enriched version}}\label{enriched_version} For $V$ a [[cosmos|suitable enriching category]], there is a refinement of the notion of Reedy category to a notion of \emph{$V$-[[enriched Reedy category]]} such that if $C$ is a $V$-[[enriched model category]] -- in particular when it is a [[simplicial model category]] for $V =$ [[SSet]] -- the [[enriched functor category]] $[R,C]$ is itself a $V$-[[enriched model category]] (see \hyperlink{Angeltveit}{Angeltveit}). In the case that we do have extra assumptions on the codomain in that \begin{itemize}% \item $C$ is a [[combinatorial simplicial model category]] \item with $C^\circ$ the [[(∞,1)-category]] [[presentable (∞,1)-category|presented]] by $C$ \item and with the [[Reedy category]] $R$ an ordinary category regarded as a [[SSet]]-[[enriched category]], \end{itemize} the Reedy model structure, having the same weak equivalences as the [[global model structure on functors]], presents similarly the [[(∞,1)-category of (∞,1)-functors]] $Func_\infty(R,C^\circ)$, from $C$ into the [[(∞,1)-category]] [[presentable (∞,1)-category|presented by]] $C$. \hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks} \begin{itemize}% \item Any Reedy cofibration or fibration is, in particular, an objectwise one, but the converse does not generally hold. \item An object $X$ is Reedy cofibrant if and only if each map $L_r X \to X_r$ is a cofibration in $M$. In particular, this implies that each $X_r$ is cofibrant in $M$. \item For some $M$, $M^R$ also admits a [[projective model structure|projective]] or [[injective model structure|injective]] model structures. For instance for $M =$ [[SSet]] this is the [[global model structure on simplicial presheaves]]. In general the Reedy structure will not be the same as either, but will be a kind of mixture of both. If $R = R_+$ then the Reedy model structure coincides with the [[projective model structure]], if $R = R_-$ it coincides with the [[injective model structure]]. For a detailed comparison of Reedy and global projective/injective model structures see around example A.2.9.22 in [[Higher Topos Theory|HTT]]. \item In addition to its existing for all $C$, another advantage of the Reedy structure is the explicit nature of its cofibrations, fibrations, and factorizations. \item If $R$ admits more than one structure of Reedy category, then $C^R$ will have more than one Reedy model structure. For instance, if $R = (\cdot\to\cdot)$ is the [[walking arrow]], then we can regard it as either the ordinal $2$ or its opposite $2^{op}$, resulting in two different Reedy model structures on $C^2$. \item For a general Reedy category $R$, the diagonal functor $C\to C^R$ need not be either a right or a left [[Quillen adjunction|Quillen functor]] (although, of course, it has left and right adjoints given by colimits and limits over $R$). One can, however, characterize those Reedy categories for which one or the other is the case, and in this case one can construct [[homotopy limit|homotopy limits]] and colimits using the derived functors of these Quillen adjunctions. \end{itemize} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{EnrichedModelStructure}{}\subsubsection*{{Enriched model structure}}\label{EnrichedModelStructure} \begin{prop} \label{}\hypertarget{}{} For $C$ a [[Reedy category]] and $A$ a [[symmetric monoidal category|symmetric]] [[monoidal model category]], the Reedy model structure on $[C,A]_{Reedy}$ is naturally an $A$-[[enriched model category]]. If in addition $A$ is a $V$-[[enriched model category]] for some symmetric monoidal model category $V$, then so is $[C,A]_{Reedy}$ \end{prop} This appears as (\hyperlink{Barwick}{Barwick, lemma 4.2, corollary 4.3}). (\ldots{}check assumptions\ldots{}) \hypertarget{relation_to_other_model_structures}{}\subsubsection*{{Relation to other model structures}}\label{relation_to_other_model_structures} \begin{prop} \label{}\hypertarget{}{} Let $C$ be a [[combinatorial model category]] and $R$ a [[Reedy category]]. The identity functors provide left [[Quillen equivalences]] \begin{displaymath} [R,C]_{proj} \stackrel{\simeq_{Quillen}}{\to} [R,C]_{Reedy} \stackrel{\simeq_{Quillen}}{\to} [R,C]_{inj} \end{displaymath} from the projective [[model structure on functors]] to the injective one. \end{prop} See also [[Higher Topos Theory|HTT, remark A.2.9.23]] \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{over_the_arrow_category}{}\subsubsection*{{Over the arrow category}}\label{over_the_arrow_category} The simplest nontrivial example is obtained for \begin{displaymath} R = I = \{1 \to 0\} \end{displaymath} the [[interval category]]. In this case the [[functor category]] $[I,C]$ is the [[arrow category]] $C$. We take the degree on the objects to be as indicated. Then $R_- = R$ and $R_+$ contains only the identity morphisms. For $F : I \to C$ a functor, i.e. a morphism $F(1) \to F(0)$ in $C$, we find \begin{itemize}% \item the latching object $latch_0 F = colim_{(s \stackrel{+}{\to} 0)} F(s) = \emptyset$; \item the latching object $latch_1 F = colim_{(s\stackrel{+}{\to}1)} F(s) = \emptyset$; \item the matching object $match_0 F = lim_{(0 \stackrel{-}{\to}s)} F(s) = {*}$; \item the matching object $match_1 F = lim_{(1 \stackrel{-}{\to}s)} F(s) = F(0)$ \end{itemize} where $\emptyset$ denotes the [[initial object]] and ${*}$ the [[terminal object]] (being the [[colimit]] and [[limit]] over the empty [[diagram]], respectively). From this we find that for a [[natural transformation]] $\eta : F \to G$ \begin{displaymath} \itexarray{ F(1) &\stackrel{\eta_1}{\to}& G(1) \\ \downarrow && \downarrow \\ F(0) &\stackrel{\eta_0}{\to}& G(0) } \end{displaymath} that \begin{itemize}% \item it is a Reedy cofibration in $[I,C]$ if \begin{itemize}% \item $\eta_0 : F(0) \coprod_{\emptyset} \emptyset = F(0) \to G(0)$ is a cofibration \end{itemize} and \begin{itemize}% \item $\eta_1 : F(1) \coprod_{\emptyset} \emptyset = F(1) \to G(1)$ is a cofibration \end{itemize} \item it is a Reedy fibration in $[I,C]$ if \begin{itemize}% \item $\eta_0 : F(0) \to G(0) \times_{*} {*} = G(0)$ is a fibration \item the universal morphism $F(1) \to G(1) \times_{G(0)} F(0)$ \begin{displaymath} \itexarray{ F(1) \\ & \searrow \\ && F(0) \times_{G(0)} G(1) &\to& G(1) \\ && \downarrow && \downarrow \\ && F(0) &\stackrel{\eta_0}{\to}& G(0) } \end{displaymath} is a fibration. \end{itemize} Notice that since fibrations are preserved by pullbacks and under composition with themselves, it follows that also $\eta_1 : F(1) \to G(1)$ is a fibration. \item The cofibrant objects in $[I,C]$ are those arrows $F(1) \to F(0)$ in $C$ for which $F(1)$ and $F(0)$ are cofibrant; \item The fibrant objects in $[I,C]$ are those arrows $F(1) \to F(0)$ in $C$ that are fibrations between fibrant objects in $C$. \end{itemize} So in accord with the proposition above one finds that this Reedy model structure on $[I,C]$ coincides with the \emph{injective} [[global model structure on functors]] on $I$. \hypertarget{OverTheTowerCategory}{}\subsubsection*{{Over the tower category}}\label{OverTheTowerCategory} Let $R = \mathbb{N}^{op} = \{\cdots \to 2 \to 1 \to 0\}$ be the natural numbers regarded as a [[poset]] using the greater-than relation. With the degree as indicated, this is a Reedy category with $R_- = R$ and $R_+$ containing only identity morphisms. Now the [[functor category]] $[R,C]$ is the category of \emph{towers} of morphisms in $C$. The analysis of the Reedy model structure on this involves just a repetition of the steps involved in the analysis of the arrow category in the above example. One finds: \begin{itemize}% \item a natural transformation $\eta : F \to G$ is a fibration precisely if \begin{itemize}% \item the component $\eta_0 : F(0) \to G(0)$ is a fibration \item all universal morphisms $F(n) \to F(n-1) \times_{G(n-1)} G(n)$ are fibrations. \end{itemize} \item the fibrant objects are the towers of fibrations on fibrant objects in $C$. \end{itemize} A detailed discussion of the model structure on towers is for instance in (\hyperlink{GoerssJardine}{GoerssJardine, chapter 6}) By duality it follows that analogously there is a model structure on co-towers \begin{displaymath} X_0 \to X_1 \to X_2 \to \cdots \end{displaymath} in a model category $C$, whose fibrations and weak equivalences are the degreewise ones, and whose cofibrations are those transformations that are a cofibration in degree 0 and where the canonical pushout-morphisms in each square are cofibrations. \hypertarget{SimplexCategory}{}\subsubsection*{{Over the simplex category}}\label{SimplexCategory} The motivating and central example of [[Reedy categories]] is the [[simplex category]] $\Delta$. Recall that \begin{itemize}% \item a map $[k] \to [n]$ is in $\Delta_+$ precisely if it is an injection; \item a map $[n] \to [k]$ is in $\Delta_-$ precisely if it is a surjection. \end{itemize} Dually, for the [[opposite category]] $\Delta^{op}$ \begin{itemize}% \item a morphism $[n] \leftarrow [k]$ is in $(\Delta^{op})_-$ precisely if the map underlying it is an injection; \item a morphism $[k] \leftarrow [n]$ is in $(\Delta^{op})_+$ precisely if the map underlying it is a surjection. \end{itemize} Let $C = sSet_{Quillen}$ be the category [[sSet]] equipped with the standard [[model structure on simplicial sets]] and consider the Reedy model structures on $[\Delta^{op}, sSet_{Quillen}]$ $[\Delta,sSet_{Quillen}]$. \hypertarget{properties_2}{}\paragraph*{{Properties}}\label{properties_2} We record some useful facts. \hypertarget{fibrant_and_cofibrant_objects}{}\paragraph*{{Fibrant and cofibrant objects}}\label{fibrant_and_cofibrant_objects} Let $\mathcal{C}$ be a [[model category]] \begin{prop} \label{}\hypertarget{}{} For $X \in [\Delta^{op},\mathcal{C}]$ a [[simplicial object]] and $n \in \mathbb{N}$, \begin{itemize}% \item the [[latching object]] $L_n X$ is the [[union]] of all degenerate $n$-cells; \item the [[matching object]] is $M_n X \simeq X^{\partial \Delta[n]}$, the [[powering]] of the [[boundary of a simplex|boundary of the n-simplex]] into $X$, hence the $n$-cells of the $(n-1)$-[[skeleton]] of $X$. \end{itemize} \end{prop} More details on this are currently at \emph{[[generalized Reedy model structure]]}. \begin{example} \label{EveryBisimplicialSetIsReedyCofibrant}\hypertarget{EveryBisimplicialSetIsReedyCofibrant}{} If $\mathcal{C} = sSet$ is the [[classical model structure on simplicial sets]], then every object in $[\Delta^{op}, \mathcal{C}]_{Reedy}$ ([[bisimplicial sets]]) is cofibrant (\hyperlink{Hirschhorn02}{Hirschhorn 02, corollary 15.8.8}). \end{example} \hypertarget{OverDeltaWithValuesInSimplicialSets}{}\paragraph*{{With values in simplicial sets}}\label{OverDeltaWithValuesInSimplicialSets} \begin{prop} \label{}\hypertarget{}{} Every simplicial set $X$ regarded as a simplicial diagram \begin{displaymath} X : \Delta^{op} \to Set \hookrightarrow sSet \end{displaymath} is Reedy cofibrant in $[\Delta^{op}, sSet]$. \end{prop} \begin{proof} The latching object of $X$ at $n$ is \begin{displaymath} L_n(X) = \lim_{\to} \left( ([n]\to [k]\;surj.\;in\;\Delta) \mapsto X_k \right) \,. \end{displaymath} The canonical map \begin{displaymath} L_n(X) \to X_n \end{displaymath} identifies $X_k$ along $X([n] \to [k]) : X_k \to X_n$ in $X_n$ as a bunch of degenrate $n$-cells. In total, $L_n(X)$ is identified as the set of all degenerate $n$-cells of $X$. Therefore $L_n(X) \to X_n$ is clearly an injection of sets, hence a monomorphism of (constant) simplicial sets. Monomorphisms are the cofibrations in $sSet_{Quillen}$. \end{proof} \begin{prop} \label{}\hypertarget{}{} The canonical cosimplicial simplicial set \begin{displaymath} \Delta[-] : \Delta \to sSet \end{displaymath} is Reedy cofibrant in $[\Delta,sSet_{Quillen}]$. \end{prop} \begin{proof} The latching object at $n$ is \begin{displaymath} L_n(\Delta[-]) = \lim_\to \left( ([k] \to [n]\; inj.\in\;\Delta) \mapsto \Delta[k] \right) \,. \end{displaymath} This is $\partial \Delta[n]$. The inclusion $\partial \Delta[n] \to \Delta[n]$ is a monomorphism, hence a cofibration in $sSet_{Quillen}$ (in fact these are the generating cofibrations). \end{proof} \begin{prop} \label{}\hypertarget{}{} Every [[simplicial set]] is the [[homotopy colimit]] over its diagram of simplices (with values in the constant simplicial set on the sets of simplicies $X_n$): \begin{displaymath} X \simeq hocolim ( [n] \mapsto const X_n) \,. \end{displaymath} \end{prop} \begin{proof} Because $X_{(-)} : \Delta^{op} \to sSet$ is Reedy cofibrant by the above, by the discussion at [[homotopy colimit]] we can compute the hocolim by the [[coend]] \begin{displaymath} \int^{[n]} Q(*)_n \cdot X_n \,, \end{displaymath} where $Q(*) : \Delta \to sSet$ is a cofibrant [[resolution]] of the point in $[\Delta, sSet_{Quillen}]_{Reedy}$. Using the above observation, we can take this to be $\Delta[-]$ since this is cofibrant by the above observation and clearly $\Delta[-] \to *$ is objectwise a weak equivalence in $sSet_{Quillen}$. Therefore the hocolim is (up to equivalence) represented by the simplicial set \begin{displaymath} \int^{[n]} \Delta[n] \cdot X_n \,. \end{displaymath} But by the [[co-Yoneda lemma]] this is in fact [[isomorphism|isomorphic]] to $X$, hence in particular weakly equivalent to $X$. \end{proof} \begin{remark} \label{}\hypertarget{}{} This kind of argument has many immediate generalizations. For instance for $C = [K^{op}, sSet_{Quillen}]_{inj}$ the injective [[model structure on simplicial presheaves]] over any small category $K$, or any of its left [[Bousfield localization of model categories|Bousfield localizations]], we have that the cofibrations are objectwise those of simplicial sets, hence objectwise monomorphisms, hence it follows that every simplicial presheaf $X$ is the hocolim over its simplicial diagram of component presheaves. \end{remark} For the following write $\mathbf{\Delta} : \Delta \to sSet$ for the [[fat simplex]]. \begin{prop} \label{}\hypertarget{}{} The fat simplex is Reedy cofibrant. \end{prop} \begin{proof} By the discussion at [[homotopy colimit]], the fat simplex is cofibrant in the projective [[model structure on functors]] $[\Delta, sSet_{Quillen}]_{proj}$. By the \hyperlink{Properties}{general properties} of Reedy model structures, the identity functor $[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy}$ is a [[Quillen adjunction|left Quillen functor]], hence preserves cofibrant objects. \end{proof} \hypertarget{with_values_in_an_arbitrary_model_category}{}\paragraph*{{With values in an arbitrary model category}}\label{with_values_in_an_arbitrary_model_category} Let $C$ be a [[model category]]. \begin{prop} \label{}\hypertarget{}{} For $X \in [\Delta^{op}, C]$ a Reedy cofibrant object, the [[Bousfield-Kan map]] \begin{displaymath} \int^{[n]} \mathbf{\Delta}[n] \cdot X_n \to \int^{[n]} \Delta[n] \cdot X_n \end{displaymath} is a weak equivalence in $C$. \end{prop} \begin{proof} The [[coend]] over the [[copower|tensor]] is a left [[Quillen bifunctor]] \begin{displaymath} \int (-)\cdot (-) : [\Delta,sSet_{Quillen}]_{Reedy} \times [\Delta^{op}, C]_{Reedy} \end{displaymath} (as discussed there). Therefore with its second argument fixed and cofibrant it is a [[Quillen adjunction|left Quillen functor]] in the remaining argument. As such, it preserves weak equivalences between cofibrant objects (by the [[factorization lemma]]). By the \hyperlink{OverDeltaWithValuesInSimplicialSets}{above discussion}, both $\mathbf{\Delta}[n]$ and $\Delta[-]$ are indeed cofibrant in $[\Delta,sSet_{Quillen}]_{Reedy}$. Clearly the functor $\mathbf{\Delta}[-] \to \Delta[-]$ is objectwise a weak equivalence in $sSet_{Quillen}$, hence is a weak equivalence. \end{proof} \hypertarget{EnrichmentOverTheSimplexCategory}{}\paragraph*{{Enrichment}}\label{EnrichmentOverTheSimplexCategory} The following proposition should be read as a \textbf{warning} that an obvious idea about simplicial enrichment of Reedy model structures over the simplex category does \emph{not} work. \begin{prop} \label{}\hypertarget{}{} For $C$ a [[model category]], the [[category of simplicial objects]] $[\Delta^{op}, C]$ in $C$ is canonically an [[sSet]]-[[enriched category]]. However, this does \textbf{not} in general harmonize with the Reedy model structure to make $[\Delta^{op}, C]_{Reedy}$ a [[simplicial model category]]. More precisely the following parts of the [[pushout-product axiom]] for the $sSet$-[[tensoring]] hold. Let $f : A \to B$ be a cofibration in $[\Delta^{op}, C]_{Reedy}$ and $s : S \to T$ be a cofibration in $sSet_{Quillen}$. \begin{enumerate}% \item the [[pushout-product]] $f \bar \otimes g$ is a cofibration in $[\Delta^{op}, C]_{Reedy}$ ; \item and it is an acyclic cofibration if $f$ is; \item it is \textbf{not necessarily} acyclic if $s$ is. \end{enumerate} \end{prop} That the first two items do hold is discussed for instance as (\hyperlink{Dugger}{Dugger, prop. 4.4}). A counterexample for the third item is in (\hyperlink{Dugger}{Dugger, remark 4.6}). \begin{remark} \label{}\hypertarget{}{} However, there are [[Bousfield localization of model categories|left Bousfield localizations]] of $[\Delta^{op}, sSet]_{Reedy}$ for which \begin{enumerate}% \item the above $sSet$-enrichment does constitute an $sSet$-[[enriched model category]]; \item the result model structure is Quillen equivalent to $C$ itself. \end{enumerate} This is in fact a useful technique for replacing $C$ by a Quillen equivalent and $sSet$-enriched model structure. More discussion of this point is at \emph{[[simplicial model category]]} in the section \emph{\href{simplicial+model+category#SimpEquivMods}{Simplicial Quillen equivalent models}}. \end{remark} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[generalized Reedy model structure]] \item [[Joyal-Tierney calculus]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The original text is \begin{itemize}% \item [[Chris Reedy]], \emph{Homotopy Theory of Model Categories} (\href{http://www-math.mit.edu/~psh/reedy.pdf}{retyped pdf}) \end{itemize} A quick review is in section A.2.9 of \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]} . \end{itemize} A textbook account is in \begin{itemize}% \item [[Philip Hirschhorn]], chapter 15 of \emph{Model Categories and Their Localizations}, AMS Math. Survey and Monographs Vol 99 (2002) (\href{http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99}{AMS}, \href{http://www.gbv.de/dms/goettingen/360115845.pdf}{pdf toc}, \href{http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf}{pdf}) \end{itemize} Discussion of functoriality of Reedy model structures is in \begin{itemize}% \item [[Clark Barwick]], \emph{On Reedy Model Categories} (\href{http://arxiv.org/abs/0708.2832}{arXiv:0708.2832}) \end{itemize} The discussion of enriched Reedy model structures is in \begin{itemize}% \item Vigleik Angeltveit, \emph{Enriched Reedy categories} (\href{http://arxiv.org/abs/math/0612137}{arXiv}) \end{itemize} The main statement is theorem 4.7 there. The Reedy model structure on towers is discussed for instance in chapter 6 of \begin{itemize}% \item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]} (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{dvi} \href{http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf}{PDF}) \end{itemize} The Reedy model structure on categories of simplicial objects is discussed in more detail for instance in \begin{itemize}% \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 Reedy model category]] [[!redirects Reedy model categories]] [[!redirects latching object]] [[!redirects latching objects]] [[!redirects matching object]] [[!redirects matching objects]] [[!redirects Reedy model structures]] \end{document}