nLab Reedy model structure



Model category theory

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A Reedy model structure is a global model structure on functors:

given a Reedy category RR and a model category CC the Reedy model structure is a model category structure on the functor category [R,C]=Func(R,C)[R,C] = Func(R,C).

As opposed to the projective and injective model structure on functors this does not require any further structure on CC, but instead makes a strong assumption on RR.

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 CC.


Plain version



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

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


(latching and matching objects)

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

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


The latching object of XX at stage rr is the colimit in 𝒞\mathcal{C}

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

over the full subcategory of the slice category R +/rR_+/r containing all objects except the identity id rid_r.


The matching object of XX at stage rr is the limit in 𝒞\mathcal{C}

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

over the full subcategory of the coslice category r/R r/R_- containing all objects except id rid_r.

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

(3)L rXX rM rX. L_r X \longrightarrow X_r \longrightarrow M_r X \,.

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


(latching of simplicial objects)
Consider the case that =Δ op\mathcal{R} = \Delta^{op} is the opposite simplex category with its canonical Reedy structure (here), where (Δ op) +=(Δ ) op(\Delta^{op})_+ \,=\, (\Delta_-)^{op} is the opposite of the surjections [r][rk][r] \to [r-k].

For X:Δ op𝒞X \colon \Delta^{op} \to \mathcal{C} a simplicial object, its value on such a morphism is the corresponding degeneracy map.

This means that the cocone morphism (3) out of the the latching object (1) has as components in degree rr all inclusions of degenerate simplices

X rkX r X_{r-k} \overset{\phantom{--}}{\hookrightarrow} X_r

which X rX_r receives.

Hence one may often think of the latching object L rXL_r X of a simplicial object XX as the object of degenerate r r -simplices sitting inside the object X rX_r of all rr-simplices. Cf. Prop. and Prop. below.


Notice that L 0X=0L_0 X = 0 is the initial object and M 0XM_0 X is the terminal object, since there are no objects of degree <0\lt 0.


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


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

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

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

    (4)L rY⨿L rXX rY r L_r Y \overset {L_r X} {\amalg} X_r \longrightarrow Y_r

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

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

    (5)X rM rX×M rYY r X_r \longrightarrow M_r X \underset {M_r Y} {\times} Y_r

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

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


In particular this means that:

  1. An object XX is Reedy-cofibrant iff the comparison maps L rX rL_r \to X_r (3) from its latching object are cofibrations.

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


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

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


To see the factorization of a morphism f:XYf \colon X\to Y in either of the two necessary ways, construct the factorization f r=g rh rf_r = g_r h_r inductively on rr, by factoring the induced morphism

L rZ⨿ L rXX rM rZ× M rYY r L_r Z \amalg_{L_r X} X_r \to M_r Z \times_{M_r Y} Y_r

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

Enriched version

For VV a suitable enriching category, there is a refinement of the notion of Reedy category to a notion of VV-enriched Reedy category such that if CC is a VV-enriched model category – in particular when it is a simplicial model category for V=V = SSet – the enriched functor category [R,C][R,C] is itself a VV-enriched model category (see Angeltveit).

In the case that we do have extra assumptions on the codomain in that

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 (R,C )Func_\infty(R,C^\circ), from RR into the (∞,1)-category presented by CC.


  • Any Reedy cofibration or fibration is, in particular, an objectwise one, but the converse does not generally hold.

  • An object XX is Reedy cofibrant if and only if each map L rXX rL_r X \to X_r is a cofibration in MM. In particular, this implies that each X rX_r is cofibrant in MM.

  • For some MM, M RM^R also admits a projective or injective model structures. For instance for M=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 +R = R_+ then the Reedy model structure coincides with the projective model structure, if R=R 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 CC, another advantage of the Reedy structure is the explicit nature of its cofibrations, fibrations, and factorizations.

  • If RR admits more than one structure of Reedy category, then C RC^R will have more than one Reedy model structure. For instance, if R=()R = (\cdot\to\cdot) is the walking arrow, then we can regard it as either the ordinal 22 or its opposite 2 op2^{op}, resulting in two different Reedy model structures on C 2C^2.

  • For a general Reedy category RR, the diagonal functor Δ:CC R\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 RR). Δ\Delta is a left Quillen functor if and only if for all objects rr, the latching category (r/R )\partial(r / R_-) is either connected or empty. Dually, Δ\Delta is a right Quillen functor if and only if for all objects rr, the matching category (R +/r)\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 colimΔlim\mathrm{colim} \dashv \Delta \dashv \mathrm{lim}.




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


(cofibrant generation)
If 𝒞\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 𝒞 \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.


If 𝒞\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 𝒞 \mathcal{C}^{\mathcal{R}} is combinatorial.


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).


(left properness)
If 𝒞\mathcal{C} is


then also the Reedy model structure on 𝒞 \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


For CC a Reedy category and AA a symmetric monoidal model category, the Reedy model structure on [C,A] Reedy[C,A]_{Reedy} is naturally an AA-enriched model category.

If in addition AA is a VV-enriched model category for some symmetric monoidal model category VV, then so is [C,A] Reedy[C,A]_{Reedy}

This appears as (Barwick, lemma 4.2, corollary 4.3).

(…check assumptions…)

Relation to other model structures


Let CC be a combinatorial model category and RR a Reedy category.

The identity functors provide left Quillen equivalences

[R,C] proj Quillen[R,C] Reedy Quillen[R,C] inj [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


Over the arrow category

The simplest nontrivial example is obtained for

R=I={10} R = I = \{1 \to 0\}

the interval category.

In this case the functor category [I,C][I,C] is the arrow category CC.

We take the degree on the objects to be as indicated. Then R =RR_- = R and R +R_+ contains only the identity morphisms.

For F:ICF : I \to C a functor, i.e. a morphism F(1)F(0)F(1) \to F(0) in CC, we find

  • the latching object latch 0F=colim (s+0)F(s)=latch_0 F = colim_{(s \stackrel{+}{\to} 0)} F(s) = \emptyset;

  • the latching object latch 1F=colim (s+1)F(s)=latch_1 F = colim_{(s\stackrel{+}{\to}1)} F(s) = \emptyset;

  • the matching object match 0F=lim (0s)F(s)=*match_0 F = lim_{(0 \stackrel{-}{\to}s)} F(s) = {*};

  • the matching object match 1F=lim (1s)F(s)=F(0)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 η:FG\eta : F \to G

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


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

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


    • η 1:F(1) =F(1)G(1)\eta_1 : F(1) \coprod_{\emptyset} \emptyset = F(1) \to G(1) is a cofibration
  • it is a Reedy fibration in [I,C][I,C] if

    • η 0:F(0)G(0)× **=G(0)\eta_0 : F(0) \to G(0) \times_{*} {*} = G(0) is a fibration

    • the universal morphism F(1)G(1)× G(0)F(0)F(1) \to G(1) \times_{G(0)} F(0)

      F(1) F(0)× G(0)G(1) G(1) F(0) η 0 G(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 η 1:F(1)G(1)\eta_1 : F(1) \to G(1) is a fibration.

  • The cofibrant objects in [I,C][I,C] are those arrows F(1)F(0)F(1) \to F(0) in CC for which F(1)F(1) and F(0)F(0) are cofibrant;

  • The fibrant objects in [I,C][I,C] are those arrows F(1)F(0)F(1) \to F(0) in CC that are fibrations between fibrant objects in CC.

So in accord with the proposition above one finds that this Reedy model structure on [I,C][I,C] coincides with the injective global model structure on functors on II.

Over the tower category

Let R= op={210}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 =RR_- = R and R +R_+ containing only identity morphisms.

Now the functor category [R,C][R,C] is the category of towers of morphisms in CC.

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 η:FG\eta : F \to G is a fibration precisely if

    • the component η 0:F(0)G(0)\eta_0 : F(0) \to G(0) is a fibration

    • all universal morphisms F(n)F(n1)× G(n1)G(n)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 CC.

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 0X 1X 2 X_0 \to X_1 \to X_2 \to \cdots

in a model category CC, 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 Δ\Delta.

Recall that

  • a map [k][n][k] \to [n] is in Δ +\Delta_+ precisely if it is an injection;

  • a map [n][k][n] \to [k] is in Δ \Delta_- precisely if it is a surjection.

Dually, for the opposite category Δ op\Delta^{op}

  • a morphism [n][k][n] \leftarrow [k] is in (Δ op) (\Delta^{op})_- precisely if the map underlying it is an injection;

  • a morphism [k][n][k] \leftarrow [n] is in (Δ op) +(\Delta^{op})_+ precisely if the map underlying it is a surjection.

Let C=sSet QuillenC = sSet_{Quillen} be the category sSet equipped with the standard model structure on simplicial sets and consider the Reedy model structures on [Δ op,sSet Quillen][\Delta^{op}, sSet_{Quillen}] [Δ,sSet Quillen][\Delta,sSet_{Quillen}].

With values in simplicial sets


For X[Δ op,Set]X \in [\Delta^{op},Set] a simplicial object and nn \in \mathbb{N},

More details on this are currently at generalized Reedy model structure.


If 𝒞=sSet\mathcal{C} = sSet is the classical model structure on simplicial sets, then every object in [Δ op,𝒞] Reedy[\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.



The canonical cosimplicial simplicial set

Δ[]:ΔsSet \Delta[-] : \Delta \to sSet

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


The latching object at nn is

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

This is Δ[n]\partial \Delta[n]. The inclusion Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n] is a monomorphism, hence a cofibration in sSet QuillensSet_{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 nX_n):

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

Because X ():Δ opsSetX_{(-)} : \Delta^{op} \to sSet is Reedy cofibrant by the above, by the discussion at homotopy colimit we can compute the hocolim by the coend

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

where Q(*):ΔsSetQ(*) : \Delta \to sSet is a cofibrant resolution of the point in [Δ,sSet Quillen] Reedy[\Delta, sSet_{Quillen}]_{Reedy}. Using the above observation, we can take this to be Δ[]\Delta[-] since this is cofibrant by the above observation and clearly Δ[]*\Delta[-] \to * is objectwise a weak equivalence in sSet QuillensSet_{Quillen}.

Therefore the hocolim is (up to equivalence) represented by the simplicial set

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

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


This kind of argument has many immediate generalizations. For instance for C=[K op,sSet Quillen] injC = [K^{op}, sSet_{Quillen}]_{inj} the injective model structure on simplicial presheaves over any small category KK, 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 XX is the hocolim over its simplicial diagram of component presheaves.

For the following write Δ:ΔsSet\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 [Δ,sSet Quillen] proj[\Delta, sSet_{Quillen}]_{proj}. By the general properties of Reedy model structures, the identity functor [Δ,sSet Quillen] proj[Δ,sSet Quillen] Reedy[\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 𝒜\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 s𝒜𝒜 (Δ op)s\mathcal{A} \coloneqq \mathcal{A}^{(\Delta^{op})} of simplicial objects in 𝒜\mathcal{A}.


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

L nXX n. L_n X \xhookrightarrow{\phantom{---}} X_n \,.

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

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

X r[r][s]V s([r]p[s]pid [r]V s)(DX) rV r(NX) r. 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]p[r]X_{[r'] \overset{p}{\twoheadrightarrow} [r]} of X X_\bullet are simply given on these direct summands by precomposition of their labels with pp:

X r X [r]p[r] X r ([r]q[s],vV s) ([r]qp[s],vV s) \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 rXL_r X consists of equivalence classes of triples of the form ([r]p[r]q[s],vV s) \Big( [r] \overset{p}{\twoheadrightarrow} [r'] \overset{q}{\twoheadrightarrow} [s] ,\, v \in V_s \Big) for non-identity pp, under the equivalence relation which identifies (p,q,v)(p,q,v) with (pq,v)(p'q',v') iff pq=pqp \circ q \,=\, p' \circ q' and v=vv = v'. This will mean that the equivalence classes are simply labeled by pairs ([r][s],vV s)\big([r] {\twoheadrightarrow} [s] ,\, v \in V_s\big) where the surjection is non-identity, hence that the colimit is

L rXlimp:[r][s]pidX sp:[r][s]pidV s 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 rXX rL_r X \to X_r (3) is the canonical inclusion of all the non-identity direct summands:

L rXL rXV rX , 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 ([r]p[r]q[s],vV 2)\big( [r] \overset{p}{\twoheadrightarrow} [r'] \overset{q}{\twoheadrightarrow} [s], \, v \in V_2 \big) all act trivially on the vv-component, we may regard this as a colimit in Set over the (p,q)(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)(p,q) [r] p p [s] [s] q q [t]. (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 Δ \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)(p,q,v) is identified with (qp,id,v)=(qp,id,v)(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)(p',q',v).


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


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

L rY⨿L rXX r Y r (NX) r(DY) r (Nf) rid (NY) r(DY) r \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 (Nf) rid(N f)_r \oplus id is clearly a cofibration iff (Nf) r(N f)_r is.


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

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

X r/L rXf r/L rfY r/L rY X_r/L_r X \overset {\;\; f_r/L_r f \;\;} {\longrightarrow} Y_r/L_r Y

is a cofibration in 𝒞\mathcal{C}.


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

L rX X r f r L rY L rYL rXX r Cof Y r * X r/L rX f r/L rf Y r/L rY \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 rXcofib(L rXX r)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 rff_r/L_r f as the pushout of a cofibration, and hence as a cofibration itself.

With values in an arbitrary model category

Let CC be a model category.


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

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

is a weak equivalence in CC.


The coend over the tensor is a left Quillen bifunctor

()():[Δ,sSet Quillen] Reedy×[Δ op,C] Reedy \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 Δ[n]\mathbf{\Delta}[n] and Δ[]\Delta[-] are indeed cofibrant in [Δ,sSet Quillen] Reedy[\Delta,sSet_{Quillen}]_{Reedy}. Clearly the functor Δ[]Δ[]\mathbf{\Delta}[-] \to \Delta[-] is objectwise a weak equivalence in sSet QuillensSet_{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 CC a model category, the category of simplicial objects [Δ op,C][\Delta^{op}, C] in CC is canonically an sSet-enriched category.

However, this does not in general harmonize with the Reedy model structure to make [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy} a simplicial model category.

More precisely the following parts of the pushout-product axiom for the sSetsSet-tensoring hold. Let f:ABf : A \to B be a cofibration in [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy} and s:STs : S \to T be a cofibration in sSet QuillensSet_{Quillen}.

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

  2. and it is an acyclic cofibration if ff is;

  3. it is not necessarily acyclic if ss 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.


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

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

  2. the result model structure is Quillen equivalent to CC itself.

This is in fact a useful technique for replacing CC by a Quillen equivalent and sSetsSet-enriched model structure. More discussion of this point is at simplicial model category in the section Simplicial Quillen equivalent models.


The original text is

Textbook accounts:

An overview stressing the role of weighted colimits is in

Discussion of functoriality of Reedy model structures:

On enriched

and monoidal Reedy model structures:

Discussion of fibrant generation of Reedy model structure:

The main statement is theorem 4.7 there.

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

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

Last revised on May 18, 2024 at 05:21:46. See the history of this page for a list of all contributions to it.