model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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 $C$.
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 $\mathcal{C}$.
(latching and matching objects)
Given
then:
The latching object of $X$ at stage $r$ is the colimit in $\mathcal{C}$
over the full subcategory of the slice category $R_+/r$ containing all objects except the identity $id_r$.
The matching object of $X$ at stage $r$ is the limit in $\mathcal{C}$
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:
(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
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
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
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.
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
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
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 $\lt 0$.
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 $\mathcal{R}=\alpha^{op}$.
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
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
is a fibration or acyclic fibration in $\mathcal{C}$, respectively.
In particular this means that:
(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.
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
in the appropriate way in $\mathcal{C}$.
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 $R$ 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 $\Delta : C\to C^R$ 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 $\mathrm{colim} \dashv \Delta \dashv \mathrm{lim}$.
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.
(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.
The dual statement concerning fibrant generation is in BHKKRS15, Thm. 5.9.
(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.
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 $\mathcal{C}$ is
or
then also the Reedy model structure on $\mathcal{C}^{\mathcal{R}}$ is (combinatorial, by Cor. , or cellular, respectively, and) left proper.
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 $match_0 F = lim_{(0 \stackrel{-}{\to}s)} F(s) = {*}$;
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 $\eta : F \to G$
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 $\eta_1 : F(1) \to G(1)$ 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 $I$.
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 $[\Delta^{op}, sSet_{Quillen}]$ $[\Delta,sSet_{Quillen}]$.
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.
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:
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.
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 $\mathcal{A}$.
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:
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.):
(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 $p$:
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
and that the comparison map $L_r X \to X_r$ (3) is the canonical inclusion of all the non-identity direct summands:
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:
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 $(p',q',v)$.
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 $\mathcal{A}$.
By Prop. the maps (4) are of this form
But $(N f)_r \oplus id$ is clearly a cofibration iff $(N f)_r$ is.
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
is a cofibration in $\mathcal{C}$.
Consider the following commuting diagram in $\mathcal{C}$:
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.
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 2001, prop. 4.4. A counterexample for the third item is in Dugger 2001, 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
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
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 September 9, 2024 at 12:08:21. See the history of this page for a list of all contributions to it.