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Reedy model structure

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Model category theory

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Contents

Idea

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.

Definition

Plain version

Theorem

If RR is a Reedy category and CC is a model category, then there is a canonical induced model structure on the functor category C RC^R in which the weak equivalences are the objectwise weak equivalences in CC.

The basic idea is as follows. Given a diagram X:RMX:R\to M and an object rRr\in R, define its latching object to be

L rX=colim s+rX s L_r X = \colim_{s \overset{+}{\to} r} X_s

where the colimit is over the full subcategory of R +/rR_+/r containing all objects except the identity 1 r1_r. Dually, define its matching object to be

M rX=lim rsX s M_r X = \lim_{r \overset{-}{\to} s} X_s

where the limit is over the full subcategory of r/R r/R_- containing all objects except 1 r1_r. There are evident canonical, and natural, morphisms

L rXX rM rX.L_r X\to X_r \to M_r X.

Note 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.

In the case R=Δ opR=\Delta^{op}, the latching object L nXL_n X can be thought of as the object of degenerate nn-simplices sitting inside the object X nX_n of all nn-simplices. When R=αR=\alpha is an ordinal, then L n+1X=X nL_{n+1} X = X_n and M nX=1M_n X = 1, and dually for R=α opR=\alpha^{op}.

We now define a morphism f:XYf:X\to Y in M RM^R to be a cofibration or trivial cofibration if for all rr, the map

L rY⨿ L rXX rY rL_r Y \amalg_{L_r X} X_r \to Y_r

is a cofibration or trivial cofibration in MM, respectively, and to be a fibration or trivial fibration if for all rr, the map

X rM rX× M rYY r X_r \to M_r X \times_{M_r Y} Y_r

is a fibration or trivial fibration in MM, respectively. Define ff to be a weak equivalence if each f rf_r is a weak equivalence in MM.

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:XYf:X\to Y in either of the two necessary ways, we 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 MM.

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 CC into the (∞,1)-category presented by CC.

Remarks

  • 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 CC RC\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). 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.

Properties

Enriched model structure

Observation

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

Observation

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

Examples

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

that

  • 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

    and

    • 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) η 1 G(1) F(0) η 0 G(0) \array{ F(1) \\ & \searrow \\ && F(0) \times_{G(0)} G(1) &\stackrel{\eta_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)F(0)\eta_1 : F(1) \to F(0) 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}].

Properties

We record some useful facts.

Fibrant and cofibrant objects

Let 𝒞\mathcal{C} be a model category

Proposition

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

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

Example

If for instance in 𝒞\mathcal{C} all monomorphisms are cofibrations, then every object in [Δ op,𝒞] Reedy[\Delta^{op}, \mathcal{C}]_{Reedy} is cofibrant.

With values in simplicial sets
Observation

Every simplicial set XX regarded as a simplicial diagram

X:Δ opSetsSet X : \Delta^{op} \to Set \hookrightarrow sSet

is Reedy cofibrant in [Δ op,sSet][\Delta^{op}, sSet].

Proof

The latching object of XX at nn is

L n(X)=lim (([n][k]surj.inΔ)X k). L_n(X) = \lim_{\to} \left( ([n]\to [k]\;surj.\;in\;\Delta) \mapsto X_k \right) \,.

The canonical map

L n(X)X n L_n(X) \to X_n

identifies X kX_k along X([n][k]):X kX nX([n] \to [k]) : X_k \to X_n in X nX_n as a bunch of degenrate nn-cells. In total, L n(X)L_n(X) is identified as the set of all degenerate nn-cells of XX.

Therefore L n(X)X nL_n(X) \to X_n is clearly an injection of sets, hence a monomorphism of (constant) simplicial sets. Monomorphisms are the cofibrations in sSet QuillensSet_{Quillen}.

Observation

The canonical cosimplicial simplicial set

Δ[]:ΔsSet \Delta[-] : \Delta \to sSet

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

Proof

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).

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 nX_n):

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

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.

Remark

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.

Observation

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 [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 arbitrary model category

Let CC be a model category.

Corollary

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.

Proof

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.

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 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, prop. 4.4). A counterexample for the third item is in (Dugger, remark 4.6).

Remark

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.

References

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

  • Vigleik Angeltveit, Enriched Reedy categories (arXiv)

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

  • Dan Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. vol. 353, number 12 (2001), 5003-5027. (pdf)

Revised on May 18, 2012 20:05:17 by Urs Schreiber (89.204.137.144)