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 $[R,C]=\mathrm{Func}(R,C)$.

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 $R$.

There is a refinement 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. (Reference Ang below.)

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

• $C$ 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 ${\mathrm{Func}}_{\mathrm{\infty }}(R,C^{\circ })$, from $C$ into the (∞,1)-category presented by $C$.

Definition

Theorem

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

$$L_{r}X={colim}_{s\stackrel{+}{\to }r}{X}_s$$L_r X = \colim_{s \overset{+}{\to} r} X_s

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

$$M_{r}X=\underset{r\stackrel{-}{\to }s}{\lim }{X}_s$$M_r X = \lim_{r \overset{-}{\to} s} X_s

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

$$L_{r}X\to X_r\to M_{r}X.$$L_r X\to X_r \to M_r X.

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 $<0$.

In the case $R={\Delta }^{\mathrm{op}}$, 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 $R={\alpha }^{\mathrm{op}}$.

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

$$L_{r}Y{⨿}_{L_{r}X}{X}_r\to Y_r$$L_r Y \amalg_{L_r X} X_r \to Y_r

is a cofibration or trivial cofibration in $M$, respectively, and to be a fibration or trivial fibration if for all $r$, the map

$$X_r\to M_{r}X{\times }_{M_{r}Y}{Y}_r$$X_r \to M_r X \times_{M_r Y} Y_r

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

$$L_{r}Z{⨿}_{L_{r}X}{X}_r\to M_{r}Z{\times }_{M_{r}Y}{Y}_r$$L_r Z \amalg_{L_r X} X_r \to M_r Z \times_{M_r Y} Y_r

in the appropriate way in $M$.

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 $2^{\mathrm{op}}$, 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.

Properties

Observation

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

The identity functors provide left Quillen equivalences

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

See also HTT, remark A.2.9.23

Examples

Arrow category

The simplest nontrivial example is obtained for

$$R=I=\{1\to 0\}$$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 ${\mathrm{latch}}_{0}F={\mathrm{colim}}_{(s\stackrel{+}{\to }0)}F(s)=\varnothing$;

• the latching object ${\mathrm{latch}}_{1}F={\mathrm{colim}}_{(s\stackrel{+}{\to }1)}F(s)=\varnothing$;

• the matching object ${\mathrm{match}}_{0}F={\lim }_{(0\stackrel{-}{\to }0)}F(s)=*$

• the matching object ${\mathrm{match}}_{1}F={\lim }_{(1\stackrel{-}{\to }s)}F(s)=F(0)$

where $\varnothing$ 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 $\eta :F\to G$

$$\begin{array}{ccc}F(1)& \stackrel{{\eta }_1}{\to }& G(1)\\ \downarrow & & \downarrow \\ F(0)& \stackrel{{\eta }_0}{\to }& G(0)\end{array}$$\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 $[I,C]$ of

  • ${\eta }_0:F(0){\coprod }_{\varnothing }\varnothing =F(0)\to G(0)$ is a cofibration

  and

  • ${\eta }_1:F(1){\coprod }_{\varnothing }\varnothing =F(1)\to G(1)$ 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 $F(1)\to G(1){\times }_{G(0)}F(0)$

    $$\begin{array}{c}F(0)\\ & \searrow \\ & & F(0){\times }_{G(0)}G(1)& \stackrel{{\eta }_1}{\to }& G(0)\\ & & \downarrow & & \downarrow \\ & & F(0)& \stackrel{{\eta }_0}{\to }& G(0)\end{array}$$\array{ F(0) \\ & \searrow \\ && F(0) \times_{G(0)} G(1) &\stackrel{\eta_1}{\to}& G(0) \\ && \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 ${\eta }_1:F(1)\to F(0)$ 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 $I$.

Tower category

Let $R={\mathbb{N}}^{\mathrm{op}}=\{\cdots \to 2\to 1\to 0\}$ be the natural numbers regarded as a poset using the great-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 invoilved 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

  • all components ${\eta }_n:F(n)\to G(n)$ are fibrations

  • 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 $C$.

References

A review of Reedy model structures is in section A.2.9 of

• Jacob Lurie, Higher Topos Theory .

The discussion of enriched Reedy model structures is in

• Vigleik Angeltveit, Enriched Reedy categories (arXiv)

The main statement is theorem 4.7 there.