on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A Reedy model structure is a global model structure on functors:
given a Reedy category $R$ and a model category $C$ the Reedy model structure is a model category structure on the functor category $[R,C] = Func(R,C)$.
As opposed to the 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 in between the injective and the projective model structure. As such, it has the advantage that both the cofibrations as well as the fibrations can be fairly explicitly described and detected in terms of cofibrations and fibrations in $C$.
If $R$ is a Reedy category and $C$ is a model category, then there is a canonical induced model structure on the functor category $C^R$ in which the weak equivalences are the objectwise weak equivalences in $C$.
The basic idea is as follows. Given a diagram $X:R\to M$ and an object $r\in R$, define its latching object to be
where the colimit is over the full subcategory of $R_+/r$ containing all objects except the identity $1_r$. Dually, define its matching object to be
where the limit is over the full subcategory of $r/R_-$ containing all objects except $1_r$. There are evident canonical, and natural, morphisms
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 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
is a cofibration or trivial cofibration in $M$, respectively, and to be a fibration or trivial fibration if for all $r$, the map
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
in the appropriate way in $M$.
For $V$ a suitable enriching category, there is a refinement of the notion of Reedy category to a notion of $V$-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 Angeltveit).
In the case that we do have extra assumptions on the codomain in that
with $C^\circ$ the (∞,1)-category presented by $C$
and with the Reedy category $R$ an ordinary category regarded as a SSet-enriched category,
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 presented by $C$.
Any Reedy cofibration or fibration is, in particular, an objectwise one, but the converse does not generally hold.
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 $M$.
For some $M$, $M^R$ also admits a projective or 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 HTT.
In addition to its existing for all $C$, another advantage of the Reedy structure is the explicit nature of its cofibrations, fibrations, and factorizations.
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 $C^2$.
For a general Reedy category $R$, the diagonal functor $C\to C^R$ need not be either a right or a left 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 limits and colimits using the derived functors of these Quillen adjunctions.
For $C$ a Reedy category and $A$ a 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 $[C,A]_{Reedy}$
This appears as (Barwick, lemma 4.2, corollary 4.3).
(…check assumptions…)
Let $C$ be a combinatorial model category and $R$ a Reedy category.
The identity functors provide left Quillen equivalences
from the projective model structure on functors to the injective one.
See also HTT, remark A.2.9.23
The simplest nontrivial example is obtained for
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
the latching object $latch_0 F = colim_{(s \stackrel{+}{\to} 0)} F(s) = \emptyset$;
the latching object $latch_1 F = colim_{(s\stackrel{+}{\to}1)} F(s) = \emptyset$;
the matching object $match_0 F = lim_{(0 \stackrel{-}{\to} 0)} F(s) = {*}$
the matching object $match_1 F = lim_{(1 \stackrel{-}{\to}s)} F(s) = F(0)$
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 $\eta : F \to G$
that
it is a Reedy cofibration in $[I,C]$ if
and
it is a Reedy fibration in $[I,C]$ if
$\eta_0 : F(0) \to G(0) \times_{*} {*} = G(0)$ is a fibration
the universal morphism $F(1) \to G(1) \times_{G(0)} F(0)$
is a fibration.
Notice that since fibrations are preserved by pullbacks and under composition with themselves, it follows that also $\eta_1 : F(1) \to F(0)$ is a fibration.
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;
The fibrant objects in $[I,C]$ are those arrows $F(1) \to F(0)$ in $C$ that are fibrations between fibrant objects in $C$.
So in accord with the proposition above one finds that this Reedy model structure on $[I,C]$ coincides with the injective global model structure on functors on $I$.
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 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:
a natural transformation $\eta : F \to G$ is a fibration precisely if
the component $\eta_0 : F(0) \to G(0)$ is a fibration
all universal morphisms $F(n) \to F(n-1) \times_{G(n-1)} G(n)$ are fibrations.
the fibrant objects are the towers of fibrations on fibrant objects in $C$.
A detailed discussion of the model structure on towers is for instance in (GoerssJardine, chapter 6)
By duality it follows that analogously there is a model structure on co-towers
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.
The motivating and central example of Reedy categories is the simplex category $\Delta$.
Recall that
a map $[k] \to [n]$ is in $\Delta_+$ precisely if it is an injection;
a map $[n] \to [k]$ is in $\Delta_-$ precisely if it is a surjection.
Dually, for the opposite category $\Delta^{op}$
a morphism $[n] \leftarrow [k]$ is in $(\Delta^{op})_-$ precisely if the map underlying it is an injection;
a morphism $[k] \leftarrow [n]$ is in $(\Delta^{op})_+$ precisely if the map underlying it is a surjection.
Let $C = sSet_{Quillen}$ be the category sSet equipped with the standard model structure on simplicial sets and consider the Reedy model structures on $[\Delta^{op}, sSet_{Quillen}]$ $[\Delta,sSet_{Quillen}]$.
We record some useful facts.
Let $\mathcal{C}$ be a model category
For $X \in [\Delta^{op},\mathcal{C}]$ a simplicial object and $n \in \mathbb{N}$,
the latching object $L_n X$ is the union of all degenerate $n$-cells;
the matching object is $M_n X \simeq X^{\partial \Delta[n]}$, the powering of the boundary of the n-simplex into $X$, hence the $n$-cells of the $(n-1)$-skeleton of $X$.
More details on this are currently at generalized Reedy model structure.
If for instance in $\mathcal{C}$ all monomorphisms are cofibrations, then every object in $[\Delta^{op}, \mathcal{C}]_{Reedy}$ is cofibrant.
Every simplicial set $X$ regarded as a simplicial diagram
is Reedy cofibrant in $[\Delta^{op}, sSet]$.
The latching object of $X$ at $n$ is
The canonical map
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 $sSet_{Quillen}$.
The canonical cosimplicial simplicial set
is Reedy cofibrant in $[\Delta,sSet_{Quillen}]$.
The latching object at $n$ is
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).
Every simplicial set is the homotopy colimit over its diagram of simplices (with values in the constant simplicial set on the sets of simplicies $X_n$):
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
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
But by the co-Yoneda lemma this is in fact isomorphic to $X$, hence in particular weakly equivalent to $X$.
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 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.
For the following write $\mathbf{\Delta} : \Delta \to sSet$ for the fat simplex.
The fat simplex is Reedy cofibrant.
By the discussion at homotopy colimit, the fat simplex is cofibrant in the projective model structure on functors $[\Delta, sSet_{Quillen}]_{proj}$. By the general properties of Reedy model structures, the identity functor $[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy}$ is a left Quillen functor, hence preserves cofibrant objects.
Let $C$ be a model category.
For $X \in [\Delta^{op}, C]$ a Reedy cofibrant object, the Bousfield-Kan map
is a weak equivalence in $C$.
The coend over the tensor is a left Quillen bifunctor
(as discussed there). Therefore with its second argument fixed and cofibrant it is a left Quillen functor in the remaining argument. As such, it preserves weak equivalences between cofibrant objects (by the factorization lemma). By the 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.
The following proposition should be read as a warning that an obvious idea about simplicial enrichment of Reedy model structures over the simplex category does not work.
For $C$ a model category, the category of simplicial objects $[\Delta^{op}, C]$ in $C$ is canonically an sSet-enriched category.
However, this does 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 $sSet_{Quillen}$.
the pushout-product $f \bar \otimes g$ is a cofibration in $[\Delta^{op}, C]_{Reedy}$ ;
and it is an acyclic cofibration if $f$ is;
it is not necessarily acyclic if $s$ is.
That the first two items do hold is discussed for instance as (Dugger, prop. 4.4). A counterexample for the third item is in (Dugger, remark 4.6).
However, there are left Bousfield localizations of $[\Delta^{op}, sSet]_{Reedy}$ for which
the above $sSet$-enrichment does constitute an $sSet$-enriched model category;
the result model structure is Quillen equivalent to $C$ itself.
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 simplicial model category in the section Simplicial Quillen equivalent models.
The original text is
A review of Reedy model structures is in section A.2.9 of
Discussion of functoriality of Reedy model structures is in
The discussion of enriched Reedy model structures is in
The main statement is theorem 4.7 there.
The Reedy model structure on towers is discussed for instance in section 6 of
The Reedy model structure on categories of simplicial objects is discussed in more detail for instance in