nLab Reedy model structure

Idea

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 $C$.

Definition

Plain version

For

• $\mathcal{R}$ a Reedy category

• $\mathcal{C}$ a model category.

consider

• the functor category $\mathcal{C}^\mathcal{R}$

  whose objects $X \colon \mathcal{R} \to \mathcal{C}$ we refer to as ($\mathcal{R}$-shaped) diagrams in $\mathcal{C}$.

Definition

(latching and matching objects)
Given

1. a diagram $X \colon \mathcal{R} \to \mathcal{C}$,

2. an object $r \in \mathcal{R}$

then:

The latching object of $X$ at stage $r$ is the colimit in $\mathcal{C}$

(1)$$L_r X \;\coloneqq\; \underset{ s \underset{+}{\to} r }{colim} X_s$$

over the full subcategory of the slice category $R_+/r$ containing all objects except the identity $id_r$.

Dually:

The matching object of $X$ at stage $r$ is the limit in $\mathcal{C}$

(2)$$M_r X \;\coloneqq\; \underset{ r \underset{-}{\to} s } {\lim} X_s \,,$$

over the full subcategory of the coslice category $r/R_-$ containing all objects except $id_r$.

By the universal property of (co)limits there are evident natural transformations:

(3)$$L_r X \longrightarrow X_r \longrightarrow M_r X \,.$$

(e.g. Hovey (1999), Def. 5.2.2; Hirschhorn (2002), Def. 15.2.5)

Example

(latching and matching of simplicial objects)
Consider the case that $\mathcal{R} = \Delta^{op}$ is the opposite simplex category with its canonical Reedy structure (here), where

• $(\Delta^{op})_+ \,=\, (\Delta_-)^{op}$ is the opposite of the surjections $[r] \to [r-k]$,
• $(\Delta^{op})_- \,=\, (\Delta_+)^{op}$ is the opposite of the injections $[r-k] \to [r]$.

For $X \colon \Delta^{op} \to \mathcal{C}$ a simplicial object, its value on a surjection is the corresponding degeneracy map, whereas its value on an injection is the corresponding face map.

This means that the cocone morphism $L_r X \to X_r$ (3) out of the the latching object (1) has as components in degree $r$ all inclusions of degenerate simplices

$$X_{r-k} \overset{\phantom{--}}{\hookrightarrow} X_r$$

which $X_r$ receives. Hence one may often think of the latching object $L_r X$ as the object of degenerate $r$-simplices sitting inside the object $X_r$ of all $r$-simplices. Cf. Prop. and Prop. below.

Dually, the cone morphism $X_r \to M_r X$ (3) to the matching object has as components in degree $r$ all face maps

$$X_r \to X_{r-k}$$

out of $X_r$. Hence one may often think of the matching object $M_r X$ as the object of hollow shells of $r$-simplices that are candidates to have a filler in $X_r$, with the map $X_r \to M_r X$ assigning to each $r$-simplex its boundary.

This intuition carries over to presheaves over other Reedy categories, such as cubical and globular sets, with the caveat that $R_+$ and $R_-$ consider all morphisms raising/lowering degree, including e.g. diagonals of cubes.

Remark

Let $\mathcal{R}_{\lt{}i}$ be the full subcategory of $\mathcal{R}$ on objects of degree strictly less than $i$, and write $I_i^j : \mathcal{R}_{\lt{}i} \to \mathcal{R}_{\lt{}j}$ for the inclusion functor.

Then the functor $(I_i^{i+1})^* : [\mathcal{R}_{\lt{}i+1}, \mathcal{C}] \to [\mathcal{R}_{\lt{}i}, \mathcal{C}]$ forgets the presheaf cells of degree $i$. Assuming the latching and matching objects exist, $(I_i^{i+1})^*$ is part of an adjoint triple, having

• a left adjoint $(I_i^{i+1})_!$ such that

  • $((I_i^{i+1})_! X)_r = X_r$ for $d(r) \lt{} i$ and
  • $((I_i^{i+1})_! X)_r = L_r X$ for $d(r) = i$.

  In other words, $(I_i^{i+1})_!$ endows a presheaf with cells of degree $i$ freely, by taking only the degenerate cells.

• a right adjoint $(I_i^{i+1})_{*}$ such that

  • $((I_i^{i+1})_* X)_r = X_r$ for $d(r) \lt i$ and
  • $((I_i^{i+1})_* X)_r = M_r X$ for $d(r) = i$.

  In other words, $(I_i^{i+1})_*$ endows a presheaf with cells of degree $i$ cofreely, by assigning every candidate boundary a unique filler.

Notice in particular that $L_0 X = 0$ is the initial object and $M_0 X = 1$ is the terminal object, since there are no objects of degree $\lt 0$.

Example

When $\mathcal{R} = \alpha$ is an ordinal (regarded as a poset and thus as a category), then $L_{n+1} X = X_n$ and $M_n X = 1$; and dually for $\mathcal{R}=\alpha^{op}$.

Definition

A morphism $f \colon X \to Y$ in the functor category $\mathcal{C}^{\mathcal{R}}$ (hence a natural transformation between diagrams) is called:

• a Reedy equivalence if each component morphism $f_r$ is a weak equivalence in $\mathcal{C}$;

• a Reedy-cofibration or acyclic Reedy-cofibration if for all $r \in \mathcal{R}$, the map

  (4)$$L_r Y \overset {L_r X} {\amalg} X_r \longrightarrow Y_r$$

  is a cofibration or acyclic cofibration in $\mathcal{C}$, respectively;

• a Reedy-fibration or acyclic Reedy-fibration if for all $r \in \mathcal{R}$, the map

  (5)$$X_r \longrightarrow M_r X \underset {M_r Y} {\times} Y_r$$

  is a fibration or acyclic fibration in $\mathcal{C}$, respectively.

(e.g. Hirschhorn (2002), Def. 15.3.3)

Remark

In particular this means that:

1. An object $X$ is Reedy-cofibrant iff the comparison maps $L_r \to X_r$ (3) from its latching object are cofibrations.

2. An object $X$ is Reedy-fibrant iff the comparison maps $X_r \to M_r X$ (3) to its matching object are fibrations.

Proposition

(Reedy model structure)
The classes of morphisms in Def. constitute a model category-structure on the functor category $\mathcal{C}^{\mathcal{R}}$, called the Reedy model structure.

(e.g. Hovey (1999), Thm. 5.2.5; Hirschhorn (2002), Thm 15.3.4)

Remark

To see the factorization of a morphism $f \colon X\to Y$ in either of the two necessary ways, construct the factorization $f_r = g_r h_r$ inductively on $r$, by factoring the induced morphism

$$L_r Z \amalg_{L_r X} X_r \to M_r Z \times_{M_r Y} Y_r$$

in the appropriate way in $\mathcal{C}$.

Enriched version

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

• $C$ is a combinatorial simplicial model category

• 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 $R$ into the (∞,1)-category presented by $C$.

Remarks

• 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 $\Delta : C\to C^R$ 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 $R$). $\Delta$ is a left Quillen functor if and only if for all objects $r$, the latching category $\partial(r / R_-)$ is either connected or empty. Dually, $\Delta$ is a right Quillen functor if and only if for all objects $r$, the matching category $\partial(R_+ / r)$ is empty or connected. (See Hirschhorn 15.10.2 and 15.10.8, where this property is called having fibrant (resp. cofibrant) constants.) In these cases, one can construct homotopy limits and colimits using the derived functors of the Quillen adjunctions $\mathrm{colim} \dashv \Delta \dashv \mathrm{lim}$.

Properties

Properness

Proposition

For $\mathcal{R}$ a Reedy category and $\mathcal{C}$ a model category which is left or right proper, then also the Reedy model structure on $Func(\mathcal{R}, \mathcal{C})$ (Prop. ) is left or right proper, respectively.

This appears as Hirschorn (2002), Thm. 15.3.4 (2), there attributed to Daniel Kan.

Combinatorial structure

Proposition

(cofibrant generation)
If $\mathcal{C}$ is a cofibrantly generated model category (and $\mathcal{R}$ is a Reedy category) then also the Reedy model structure (Def. ) on the functor category $\mathcal{C}^{\mathcal{R}}$ is cofibrantly generated.

This is due to Hirschhorn (2002), Theorem 15.6.27, assuming here that the domains of generating (acyclic) cofibrations are relatively small objects. In the generality of generalized Reedy categories cf. Berger & Moerdijk (2011), Thm. 7.5

The dual statement concerning fibrant generation is in BHKKRS15, Thm. 5.9.

Corollary

(combinatoriality)
If $\mathcal{C}$ is a combinatorial model category (and $\mathcal{R}$ is a small Reedy category) then also the Reedy model structure (Def. ) on the functor category $\mathcal{C}^{\mathcal{R}}$ is combinatorial.

Proof

Recall that “combinatorial” means “locally presentable and cofibrantly generated”. Prop. gives the cofibrant generation, and local presentability follows from general results on locally presentable functor categories (see there).

Proposition

(left properness)
If $\mathcal{C}$ is

• combinatorial and left proper

or

• cellular and left proper

then also the Reedy model structure on $\mathcal{C}^{\mathcal{R}}$ is (combinatorial, by Cor. , or cellular, respectively, and) left proper.

This is mentioned in Dugger 2001, item 3. on p. 5008 (6 of 25).

Enriched model structure

Proposition

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…)

Relation to other model structures

Proposition

Let $C$ be a combinatorial model category and $R$ a Reedy category.

The identity functors provide left Quillen equivalences

$$[R,C]_{proj} \stackrel{\simeq_{Quillen}}{\to} [R,C]_{Reedy} \stackrel{\simeq_{Quillen}}{\to} [R,C]_{inj}$$

from the projective model structure on functors to the injective one.

See also HTT, remark A.2.9.23

Examples

Over the arrow category

The simplest nontrivial example is obtained for

$$R = I = \{1 \to 0\}$$

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 $match_0 F = lim_{(0 \stackrel{-}{\to}s)} F(s) = {*}$;

• 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 $\eta : F \to G$

$$\array{ F(1) &\stackrel{\eta_1}{\to}& G(1) \\ \downarrow && \downarrow \\ F(0) &\stackrel{\eta_0}{\to}& G(0) }$$

that

• it is a Reedy cofibration in $[I,C]$ if

  • $\eta_0 : F(0) \coprod_{\emptyset} \emptyset = F(0) \to G(0)$ is a cofibration

  and

  • $\eta_1 : F(1) \coprod_{\emptyset} \emptyset = F(1) \to G(1)$ is a cofibration

• 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 $F(1) \to G(1) \times_{G(0)} F(0)$

    $$\array{ F(1) \\ & \searrow \\ && F(0) \times_{G(0)} G(1) &\to& G(1) \\ && \downarrow && \downarrow \\ && F(0) &\stackrel{\eta_0}{\to}& G(0) }$$

    is a fibration.

  Notice that since fibrations are preserved by pullbacks and under composition with themselves, it follows that also $\eta_1 : F(1) \to G(1)$ 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 $I$.

Over the tower category

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

$$X_0 \to X_1 \to X_2 \to \cdots$$

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.

Over the simplex category

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 $[\Delta^{op}, sSet_{Quillen}]$ $[\Delta,sSet_{Quillen}]$.

With values in simplicial sets

Proposition

For $X \in [\Delta^{op},Set]$ 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.

Proposition

If $\mathcal{C} = sSet$ is the classical model structure on simplicial sets, then every object in $[\Delta^{op}, \mathcal{C}]_{Reedy}$ (i.e. every bisimplicial sets) is Reedy-cofibrant.

(Hirschhorn (2002), corollary 15.8.8)

The idea is to observe that the latching objects are the sub-objects of degeenrate cells, so that the comparison morphism (3) is a monomorphism and hence a cofibration in the classical model structure on simplicial sets.

Similarly:

Proposition

The canonical cosimplicial simplicial set

$$\Delta[-] : \Delta \to sSet$$

is Reedy cofibrant in $[\Delta,sSet_{Quillen}]$.

Proof

The latching object at $n$ is

$$L_n(\Delta[-]) = \lim_\to \left( ([k] \to [n]\; inj.\in\;\Delta) \mapsto \Delta[k] \right) \,.$$

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).

Corollary

Every simplicial set is the homotopy colimit over its diagram of simplices (with values in the constant simplicial set on the sets of simplicies $X_n$):

$$X \simeq hocolim ( [n] \mapsto const X_n) \,.$$

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

$$\int^{[n]} Q(*)_n \cdot X_n \,,$$

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

$$\int^{[n]} \Delta[n] \cdot X_n \,.$$

But by the co-Yoneda lemma this is in fact isomorphic to $X$, hence in particular weakly equivalent to $X$.

Remark

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.

Proposition

The fat simplex is Reedy cofibrant.

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 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.

With values in an addive model category

In this section, let $\mathcal{A}$ be an additive model category in which all retractions exhibit direct sums (such as in an abelian category, such as in a model structure on chain complexes).

In this situaiton the Dold-Kan correspondence applies (see there) and can be used to get further information about the Reedy model structure on the category $s\mathcal{A} \coloneqq \mathcal{A}^{(\Delta^{op})}$ of simplicial objects in $\mathcal{A}$.

Proposition

Under the above assumptions, for any simplicial object $X_\bullet \in s\mathcal{A}$ and all $n \in \mathbb{N}$, the comparison morphism (3) from the $n$th latching object (Def. ) of $X_\bullet$ is a (split) monomorphism:

$$L_n X \xhookrightarrow{\phantom{---}} X_n \,.$$

This statement and the following proof spells out a suggestion by Charles Rezk in MO:a/300657.

Proof

Under the given assumption on $\mathcal{A}$ the Dold-Kan correspondence applies (see there) to simplicial objects in $\mathcal{A}$ and says that, up to isomorphism, $X_\bullet$ is of the form $X_\bullet \simeq \Gamma(V_\bullet)$ for some connective chain complex $V_\bullet$ (in fact $V_\bullet \simeq N X_\bullet$ is the normalized chain complex of $X_\bullet$), in that each $X_r$ is the direct sum of one copy of $V_s$ for each surjection $[r] \twoheadrightarrow [s]$ in $\Delta$ (by this Prop.):

$$X_r \;\simeq\; \underset {[r] \twoheadrightarrow [s]} {\bigoplus} V_s \;\simeq\; \underset{ (D X)_r }{ \underbrace{ \left( \underset { {[r] \overset{p}{\twoheadrightarrow} [s]} \atop p \neq id_{[r]} } {\bigoplus} V_s \right) } } \,\oplus\, \underset{ (N X)_r }{ \underbrace{ V_r } } \,.$$

(On the right we have split off the summand corresponding to the identity morphism, just for emphasis, since this is what matters in a moment).

Moreover (still by this Prop.), under this identification the degeneracy maps $X_{[r'] \overset{p}{\twoheadrightarrow} [r]}$ of $X_\bullet$ are simply given on these direct summands by precomposition of their labels with $p$:

$$\array{ X_{r} & \overset{X_{[r'] \overset{p}{\twoheadrightarrow} [r]} }{\longrightarrow} & X_{r'} \\ \big( [r] \overset{q}{\twoheadrightarrow} [s] ,\, v \in V_s \big) &\mapsto& \big( [r'] \overset{q \circ p}{\twoheadrightarrow} [s] ,\, v \in V_s \big) }$$

We claim (argument below) that this implies that the colimit defining $L_r X$ consists of equivalence classes of triples of the form $\Big( [r] \overset{p}{\twoheadrightarrow} [r'] \overset{q}{\twoheadrightarrow} [s] ,\, v \in V_s \Big)$ for non-identity $p$, under the equivalence relation which identifies $(p,q,v)$ with $(p'q',v')$ iff $p \circ q \,=\, p' \circ q'$ and $v = v'$. This will mean that the equivalence classes are simply labeled by pairs $\big([r] {\twoheadrightarrow} [s] ,\, v \in V_s\big)$ where the surjection is non-identity, hence that the colimit is

$$L_r X \;\coloneqq\; \underset{ \underset { { p \colon [r] \twoheadrightarrow [s] } \atop { p \neq id } } {\longrightarrow} }{\lim} X_s \;\simeq\; \underset{ { { p \colon [r] \twoheadrightarrow [s] } \atop { p \neq id } } }{\bigoplus} V_s$$

and that the comparison map $L_r X \to X_r$ (3) is the canonical inclusion of all the non-identity direct summands:

$$L_r X \xhookrightarrow{\phantom{--}} L_r X \,\oplus\, V_r \;\simeq\; X_\bullet \,,$$

which proves the proposition.

Hence it remains to show the claim about the colimit. Since the degeneracy maps on the triples $\big( [r] \overset{p}{\twoheadrightarrow} [r'] \overset{q}{\twoheadrightarrow} [s], \, v \in V_2 \big)$ all act trivially on the $v$-component, we may regard this as a colimit in Set over the $(p,q)$-pairs. This may be computed as the set of all such pairs quotiented by the equivalence relation which is generated from the relation imposed by the morphisms in the diagram, which, unwinding the definitions, is:

$$(p,q) \simeq (p',q') \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\; \exists \;\;\; \array{ && [r] \\ & \mathllap{^p}\swarrow && \searrow\mathrlap{^{p'}} \\ [s] && \longrightarrow && [s'] \\ & \mathllap{_q} \searrow && \swarrow\mathrlap{_{q'}} \\ && [t] \mathrlap{\,.} }$$

This already shows the claim for the case that the horizontal lift seen in the diagram on the right exist.

In general, given a commuting diagram in the $\Delta$ of the form

(without necessarily a horizontal lift), then we may equivalently factor it as a zig-zag of diagrams

where the left half of this zig-zag is

and exhibits that, in the colimit, $(p,q,v)$ is identified with $(q\circ p, id, v) \,=\, (q' \circ p', id, v)$; while, analogously, the right half exhibits that this in turn is identified with $(p',q',v)$.

Corollary

Let $\mathcal{A}$ be a model category whose underlying category is as above, then a morphism $f_\bullet \,\colon\, X_\bullet \to Y_\bullet$ in $s\mathcal{A}$ is a Reedy cofibration (Def. ) if and only if the Dold-Kan-corresponding chain map $N f_\bullet$ of normalized chain complexes is degreewise a cofibration in $\mathcal{A}$.

Proof

By Prop. the maps (4) are of this form

$$\array{ L_r Y \overset{L_r X}{\amalg} X_r &\longrightarrow& Y_r \\ (N X)_r \oplus (D Y)_r &\overset{(N f)_r \oplus id}{\longrightarrow}& (N Y)_r \oplus (D Y)_r }$$

But $(N f)_r \oplus id$ is clearly a cofibration iff $(N f)_r$ is.

Remark

One direction of the implication in Cor. holds generally without any assumptions on $\mathcal{A}$ or $\mathcal{R}$:

If a morphisms $f_\bullet \colon X_\bullet \to Y_\bullet$ of functors $\mathcal{R} \to \mathcal{C}$ is Reedy cofibrant (Def. ) then for each $r \in \mathcal{R}$ the induced morphism

$$X_r/L_r X \overset {\;\; f_r/L_r f \;\;} {\longrightarrow} Y_r/L_r Y$$

is a cofibration in $\mathcal{C}$.

Proof

Consider the following commuting diagram in $\mathcal{C}$:

$$\array{ L_r X &\longrightarrow& X_r \\ \big\downarrow && \big\downarrow & \searrow\mathrlap{{}^{f_r}} \\ L_r Y & \longrightarrow & L_r Y \overset{L_r X}{\sqcup} X_r & \underset{\in Cof}{\longrightarrow} & Y_r \\ \big\downarrow && \big\downarrow && \big\downarrow \\ \ast &\longrightarrow& X_r / L_r X &\underset{f_r / L_r f}{\longrightarrow}& Y_r/L_r Y }$$

Here the top square, the left rectangle and the bottom rectagle are pushouts by definition of the latching object (Def. ) and by definition of the quotients $X_r/L_r X \,\coloneqq\, cofib(L_r X \to X_r)$. Therefore the pasting law implies first that the left bottom square and then that the right bottom square is a pushout. This exhibits $f_r/L_r f$ as the pushout of a cofibration, and hence as a cofibration itself.

With values in an arbitrary model category

Let $C$ be a model category.

Corollary

For $X \in [\Delta^{op}, C]$ a Reedy cofibrant object, the Bousfield-Kan map

$$\int^{[n]} \mathbf{\Delta}[n] \cdot X_n \to \int^{[n]} \Delta[n] \cdot X_n$$

is a weak equivalence in $C$.

Proof

The coend over the tensor is a left Quillen bifunctor

$$\int (-)\cdot (-) : [\Delta,sSet_{Quillen}]_{Reedy} \times [\Delta^{op}, C]_{Reedy}$$

(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.

Enrichment

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.

Proposition

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 $sSet_{Quillen}$.

1. the pushout-product $f \bar \otimes g$ is a cofibration in $[\Delta^{op}, C]_{Reedy}$ ;

2. and it is an acyclic cofibration if $f$ is;

3. it is not necessarily acyclic if $s$ is.

That the first two items do hold is discussed for instance as Dugger 2001, prop. 4.4. A counterexample for the third item is in Dugger 2001, remark 4.6.

Remark

However, there are left Bousfield localizations of $[\Delta^{op}, sSet]_{Reedy}$ for which

1. the above $sSet$-enrichment does constitute an $sSet$-enriched model category;

2. 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.

Related concepts

• generalized Reedy model structure

• Joyal-Tierney calculus

References

The original text is

• Chris Reedy, Homotopy Theory of Model Categories (retyped pdf)

Textbook accounts:

• Mark Hovey, Section 5.2 of: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]

• Philip Hirschhorn, Chapter 15 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf]

• Jacob Lurie, Section A.2.9 in Higher Topos Theory (2009)

• Emily Riehl, Chapter 14 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

An overview stressing the role of weighted colimits is in

• Emily Riehl, Dominic Verity, The theory and practice of Reedy categories, Theory and Applications of Categories 29 9 (2014) 256-301 [tac:29-09,arXiv:1304.6871]

Discussion of functoriality of Reedy model structures:

• Clark Barwick, On Reedy Model Categories (arXiv:0708.2832)

On enriched

• Vigleik Angeltveit, Enriched Reedy categories, Proceedings of the American Mathematical Society 136 7 (2008) [arXiv:math/0612137, AMS:S0002-9939-08-09185-5, jstor:20535419]

and monoidal Reedy model structures:

• Moncef Ghazel, Fethi Kadhi, Reedy diagrams in V-model categories, Applied Categorical Structures 27 (2019) 549–566 [arXiv:1610.06535, doi:10.1007/s10485-019-09566-w]

Discussion of fibrant generation of Reedy model structure:

• Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kedziorek, Emily Riehl, Brooke Shipley, Thm. 5.9 in: Left-induced model structures and diagram categories, in: Women in Topology: Collaborations in Homotopy Theory, Contemporary Mathematics 641 American Mathematical Society (2015) [arXiv:1401.3651, ISBN:978-1-4704-2495-4]

The main statement is theorem 4.7 there.

The Reedy model structure on towers is discussed for instance in chapter 6 of

• Paul Goerss, Rick Jardine, Simplicial homotopy theory (dvi PDF)

The Reedy model structure on categories of simplicial objects is discussed in more detail for instance in

• Dan Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353 12 (2001) 5003-5027 [ams:S0002-9947-01-02661-7, pdf]